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_posts/2020-04-02-rsa3.md
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@ -83,7 +83,7 @@ The two biggest "implementations" of public-key cryptography vary only in the ma
and how the numbers are ["trapdoored"](https://en.wikipedia.org/wiki/Trapdoor_function) to decrypt if you have the correct key.
I will discuss the differences in approach here.
If you want to skip to the next article where I show you how to encrypt your own documents using RSA, see [this link](/2020/04/06/rsa4.html).
If you want to skip to the next article where I show you how to encrypt your own documents using RSA, see [this link](/2020/04/06/rsa4/).
### RSA

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_site/2020/04/02/rsa3/index.html
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@ -116,7 +116,7 @@ If you trust nothing and no one, this is your perfered method of security.</p>
and how the numbers are <a href="https://en.wikipedia.org/wiki/Trapdoor_function">“trapdoored”</a> to decrypt if you have the correct key.</p>
<p>I will discuss the differences in approach here.
If you want to skip to the next article where I show you how to encrypt your own documents using RSA, see <a href="/2020/04/06/rsa4.html">this link</a>.</p>
If you want to skip to the next article where I show you how to encrypt your own documents using RSA, see <a href="/2020/04/06/rsa4/">this link</a>.</p>
<h3 id="rsa">RSA</h3>

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_site/feed.xml
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<?xml version="1.0" encoding="utf-8"?><feed xmlns="http://www.w3.org/2005/Atom" ><generator uri="https://jekyllrb.com/" version="4.2.0">Jekyll</generator><link href="/feed.xml" rel="self" type="application/atom+xml" /><link href="/" rel="alternate" type="text/html" /><updated>2021-09-22T08:19:13-06:00</updated><id>/feed.xml</id><entry><title type="html">How To Produce Semantically Correct MathML From XaTeX/LaTeX (and other accessibility ideas)</title><link href="/2021/09/18/how-to-generate-proper-content-mathml-from-katex-or-latex/" rel="alternate" type="text/html" title="How To Produce Semantically Correct MathML From XaTeX/LaTeX (and other accessibility ideas)" /><published>2021-09-18T00:00:00-06:00</published><updated>2021-09-18T00:00:00-06:00</updated><id>/2021/09/18/how-to-generate-proper-content-mathml-from-katex-or-latex</id><content type="html" xml:base="/2021/09/18/how-to-generate-proper-content-mathml-from-katex-or-latex/">&lt;p&gt;During a recent run-in with the Simon Fraser Fraser University accessibility department,
<?xml version="1.0" encoding="utf-8"?><feed xmlns="http://www.w3.org/2005/Atom" ><generator uri="https://jekyllrb.com/" version="4.2.0">Jekyll</generator><link href="/feed.xml" rel="self" type="application/atom+xml" /><link href="/" rel="alternate" type="text/html" /><updated>2021-09-22T11:33:06-06:00</updated><id>/feed.xml</id><entry><title type="html">How To Produce Semantically Correct MathML From XaTeX/LaTeX (and other accessibility ideas)</title><link href="/2021/09/18/how-to-generate-proper-content-mathml-from-katex-or-latex/" rel="alternate" type="text/html" title="How To Produce Semantically Correct MathML From XaTeX/LaTeX (and other accessibility ideas)" /><published>2021-09-18T00:00:00-06:00</published><updated>2021-09-18T00:00:00-06:00</updated><id>/2021/09/18/how-to-generate-proper-content-mathml-from-katex-or-latex</id><content type="html" xml:base="/2021/09/18/how-to-generate-proper-content-mathml-from-katex-or-latex/">&lt;p&gt;During a recent run-in with the Simon Fraser Fraser University accessibility department,
I learned that they’re writers are so well-trained as to write “image” where a simple diagram is shown,
and “print out picture of output” where a piece of code lies.
I figure the geniuses over there could use some help creating files for the visually impaired.

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_site/melody/cmpt-225/01/01-intro.html
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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<title> | tait.tech</title>
<link rel="stylesheet" href="/assets/css/style.css">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
</head>
<body>
<main>
<div id="wrapper">
<h1 id="cmpt-225---introduction">CMPT 225 - Introduction</h1>
<h2 id="cmpt-225-d100--data-structures--programming">CMPT: 225, D100 — Data Structures &amp; Programming</h2>
<h2 id="graded-work">Graded Work</h2>
<ul>
<li>4 Assignments, 32%
<ul>
<li>Programming &amp; Written Components</li>
<li>Programs:
<ul>
<li>In C++</li>
<li>running on ESIL Linux machines</li>
<li>using command line tools</li>
</ul>
</li>
</ul>
</li>
<li>3 Tests, 33%
<ul>
<li>In class</li>
</ul>
</li>
<li>1 Final Exam, 35%</li>
</ul>
<h2 id="lectures">Lectures</h2>
<ul>
<li>Slides + audio recording will be posted <em>after</em> lectures.</li>
</ul>
<h2 id="labs">Labs</h2>
<ul>
<li>There are <em>help times</em> (first labs are <em>next week</em>)</li>
</ul>
<h2 id="instructor-office-hours">Instructor Office Hours</h2>
<ul>
<li>Will be primarily online — times TBA.</li>
</ul>
<h2 id="instructor-contact">Instructor Contact</h2>
<ul>
<li>Via Canvas or email (for email: write from your SFU account, use subject “225: …”)</li>
</ul>
<h2 id="some-basic-rules">Some Basic Rules</h2>
<ul>
<li>In class, do not distract others.</li>
<li>In class, no taking photos/videos/recording</li>
<li>Help each other, but submit your own work.</li>
<li>Masks are mandatory in class:
<ul>
<li>may be removed while asking a question</li>
<li>if you cannot wear a mask, please email me</li>
</ul>
</li>
<li>Please stay on “your” side of the table
<ul>
<li>and stay 6’ away if you don’t have a mask on</li>
</ul>
</li>
</ul>
<h2 id="course-content">Course Content</h2>
<ul>
<li>Algorithms: processes that operate on data</li>
<li>Programs: implementations of algorithms</li>
<li>Data in programs: stored in data structures
<ul>
<li>simple D.S.: variables</li>
<li>most algorithms required compound D.S.s: e.g. arrays, lists, …</li>
<li>most non-trivial applications require non-trivial D.S.s</li>
</ul>
</li>
<li>Data Type: collection of values + operations on these values</li>
<li>Multiple data types:
<ul>
<li>Cleaner/simpler slg. design</li>
<li>better code</li>
</ul>
</li>
<li>Abstract Data Type: defined by values + operations without reference to how things are implemented</li>
<li>ADTs provide an abstract view of data:
<ul>
<li>let us reason about algorithms at a high level, ignoring implementation details</li>
</ul>
</li>
<li>Good ADT choices reduce program complexity &amp; increase likelihood of success ( which is good).</li>
<li>a[i] vs. a(i)</li>
</ul>
<h2 id="we-will-look-at">We will look at:</h2>
<ul>
<li>fundamental ADTs</li>
<li>fundamental data structures (to implement them)</li>
<li>sorting algorithms</li>
<li>with attention to:
<ul>
<li>correctness</li>
<li>efficiency/speed</li>
<li>how to implement</li>
<li>how to choose (for an application)</li>
</ul>
</li>
</ul>
<h2 id="pre-regs">Pre-regs:</h2>
<ul>
<li>Basic Programming (e.g. CMPT 125)</li>
<li>Discrete Math (e.g. MACM 101)</li>
</ul>
<p>(students get these in many different ways)</p>
<h2 id="to-do">To Do</h2>
<ul>
<li>Read chapters 1 + 3 of text</li>
<li>Lear about CSIL</li>
</ul>
<h2 id="end">End</h2>
<footer>
</footer>
</div>
</main>
</body>
</html>

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_site/melody/cmpt-225/02/02-stacks-and-queues.html
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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<title> | tait.tech</title>
<link rel="stylesheet" href="/assets/css/style.css">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
</head>
<body>
<main>
<div id="wrapper">
<h1 id="stacks--queues---1">Stacks &amp; Queues - 1</h1>
<p>CMPT 225, Fall 2021, Lecture 2</p>
<h2 id="stack">Stack</h2>
<ul>
<li>ADT that stores a collection of objects/values</li>
<li>Insertions and removals follow last-in-first-out pattern</li>
<li>Fundamental Operations:
<ul>
<li>push: inserts an element on the “top”</li>
<li>pop: removes + returns the top element</li>
</ul>
</li>
</ul>
<p>e.g.</p>
<table>
<tbody>
<tr>
<td>Stack index</td>
<td>Value</td>
</tr>
</tbody>
</table>
<p>push a</p>
<table>
<tbody>
<tr>
<td>Stack index</td>
<td>Value</td>
</tr>
<tr>
<td>0</td>
<td>a</td>
</tr>
</tbody>
</table>
<p>push b</p>
<table>
<tbody>
<tr>
<td>Stack index</td>
<td>Value</td>
</tr>
<tr>
<td>0</td>
<td>b</td>
</tr>
<tr>
<td>1</td>
<td>a</td>
</tr>
</tbody>
</table>
<p>push c</p>
<table>
<tbody>
<tr>
<td>Stack index</td>
<td>Value</td>
</tr>
<tr>
<td>0</td>
<td>c</td>
</tr>
<tr>
<td>1</td>
<td>b</td>
</tr>
<tr>
<td>2</td>
<td>a</td>
</tr>
</tbody>
</table>
<p>pop</p>
<table>
<tbody>
<tr>
<td>Stack index</td>
<td>Value</td>
</tr>
<tr>
<td>0</td>
<td>b</td>
</tr>
<tr>
<td>1</td>
<td>a</td>
</tr>
</tbody>
</table>
<ul>
<li>conventient optional operations
<ul>
<li>size: return # of elements on stack // encapsulation</li>
<li>empty: check for emptiness // better than “size=0?”</li>
<li>top: return top element, but don’t remove it // better than x = pop(); push(x) then use x</li>
</ul>
</li>
</ul>
<h2 id="algorithm-applications">Algorithm Applications</h2>
<ul>
<li>parasing/evaluation/transformation of expressions</li>
<li>speech recognition coding/decoding</li>
<li>shared network access control</li>
<li>programming languages &amp; execution
<ul>
<li>Postscript, Forth, the call stack</li>
</ul>
</li>
<li>things we do daily with computers</li>
</ul>
<h2 id="parenthasies-checking">Parenthasies Checking</h2>
<p>Parenthasies: <code class="language-plaintext highlighter-rouge">(()(((()())()()(()))</code></p>
<p>A diagram showing:
For every left parenthasies, add one to a counter.
For every right parthenthasies, remove one from the counter.
If the counter is not zero by the end of the sequence, there are too many on one side of the parenthasies.</p>
<p>Can be done by counting unmached left parenthasies:</p>
<p>Successful example:</p>
<table>
<tbody>
<tr>
<td>Character</td>
<td>Counter</td>
</tr>
<tr>
<td> </td>
<td>0</td>
</tr>
<tr>
<td>(</td>
<td>1</td>
</tr>
<tr>
<td>(</td>
<td>2</td>
</tr>
<tr>
<td>)</td>
<td>1</td>
</tr>
<tr>
<td>(</td>
<td>2</td>
</tr>
<tr>
<td>(</td>
<td>3</td>
</tr>
<tr>
<td>)</td>
<td>2</td>
</tr>
<tr>
<td>)</td>
<td>1</td>
</tr>
<tr>
<td>)</td>
<td>0</td>
</tr>
<tr>
<td>Yay!</td>
<td>0</td>
</tr>
</tbody>
</table>
<p>Failed example:</p>
<table>
<tbody>
<tr>
<td>Character</td>
<td>Country</td>
</tr>
<tr>
<td> </td>
<td>0</td>
</tr>
<tr>
<td>(</td>
<td>1</td>
</tr>
<tr>
<td>(</td>
<td>2</td>
</tr>
<tr>
<td>)</td>
<td>1</td>
</tr>
<tr>
<td>(</td>
<td>2</td>
</tr>
<tr>
<td>)</td>
<td>1</td>
</tr>
<tr>
<td> </td>
<td>Oh no it’s not zero!</td>
</tr>
</tbody>
</table>
<p>Another fail:</p>
<table>
<tbody>
<tr>
<td>Character</td>
<td>Counter</td>
</tr>
<tr>
<td> </td>
<td>0</td>
</tr>
<tr>
<td>(</td>
<td>1</td>
</tr>
<tr>
<td>)</td>
<td>0</td>
</tr>
<tr>
<td>(</td>
<td>1</td>
</tr>
<tr>
<td>)</td>
<td>0</td>
</tr>
<tr>
<td>)</td>
<td>-1</td>
</tr>
<tr>
<td> </td>
<td>Negative 1 this time!</td>
</tr>
</tbody>
</table>
<p>Do this one yourself:</p>
<table>
<tbody>
<tr>
<td>Character</td>
<td>Counter</td>
</tr>
<tr>
<td>(</td>
<td> </td>
</tr>
<tr>
<td>)</td>
<td> </td>
</tr>
<tr>
<td>)</td>
<td> </td>
</tr>
<tr>
<td>(</td>
<td> </td>
</tr>
</tbody>
</table>
<p>But consider multiple grouping symbols: <code class="language-plaintext highlighter-rouge">(\{\{)</code>, <code class="language-plaintext highlighter-rouge">(\{)\{</code>, <code class="language-plaintext highlighter-rouge">(\{[( ... ]</code>.
Counting is not enough.</p>
<h2 id="parenthasies-checking-1">Parenthasies Checking</h2>
<p>Diagram showing failure on the last character of this sequence: <code class="language-plaintext highlighter-rouge">(()((()()))()(())))</code></p>
<p>Can be done by counting unmatched left parenthasies:</p>
<table>
<tbody>
<tr>
<td>Character</td>
<td>Counter</td>
</tr>
<tr>
<td>(</td>
<td>1</td>
</tr>
<tr>
<td>(</td>
<td>2</td>
</tr>
<tr>
<td>)</td>
<td>1</td>
</tr>
<tr>
<td>(</td>
<td>2</td>
</tr>
<tr>
<td>(</td>
<td>3</td>
</tr>
<tr>
<td>)</td>
<td>2</td>
</tr>
<tr>
<td>)</td>
<td>1</td>
</tr>
<tr>
<td>)</td>
<td>0</td>
</tr>
</tbody>
</table>
<p>Example 2:</p>
<table>
<tbody>
<tr>
<td>Character</td>
<td>Counter</td>
</tr>
<tr>
<td>(</td>
<td>1</td>
</tr>
<tr>
<td>(</td>
<td>2</td>
</tr>
<tr>
<td>)</td>
<td>1</td>
</tr>
<tr>
<td>(</td>
<td>2</td>
</tr>
<tr>
<td>)</td>
<td>1</td>
</tr>
</tbody>
</table>
<p>Example 3:</p>
<table>
<tbody>
<tr>
<td>Character</td>
<td>Counter</td>
</tr>
<tr>
<td>(</td>
<td>1</td>
</tr>
<tr>
<td>)</td>
<td>0</td>
</tr>
<tr>
<td>(</td>
<td>1</td>
</tr>
<tr>
<td>)</td>
<td>0</td>
</tr>
<tr>
<td>)</td>
<td>-1</td>
</tr>
</tbody>
</table>
<p>Example 4:</p>
<table>
<tbody>
<tr>
<td>Character</td>
<td>Counter</td>
</tr>
<tr>
<td>(</td>
<td>1</td>
</tr>
<tr>
<td>)</td>
<td>0</td>
</tr>
<tr>
<td>)</td>
<td>-1</td>
</tr>
<tr>
<td>(</td>
<td>0</td>
</tr>
</tbody>
</table>
<p>But consider multiple grouping symbols:</p>
<ul>
<li>sequence <code class="language-plaintext highlighter-rouge">(\{\})</code> will and should work</li>
<li>sequence <code class="language-plaintext highlighter-rouge">(\{)\}</code> will work and should not</li>
<li>sequence <code class="language-plaintext highlighter-rouge">(\{[( ...</code> ???</li>
</ul>
<p>Counting is not enough.</p>
<h2 id="stack-based-algorithm-for-checking-grouping-symbols">Stack-based Algorithm for Checking Grouping Symbols</h2>
<p>Pseudo-code:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>I is an input stream of symbols.
S is a new empty stack.
while there are signals to read from I {
c is next symbol from I
if c is a left grouping signal {
push c on S
} else if c is a right grouping signal{
if s in empty { report error and stop }
d is pop S
if c and d do not match { report error and stop }
endif // if c is not a grouping symbol, ignore it
end while
if S is not empty { report error and exit }
report ok
</code></pre></div></div>
<p>Diagram showing how each set of parenthasies are matched, innermost first torwards the outermost.</p>
<h2 id="stack-partially-filled-array-implementation">Stack: Partially-Filled Array Implementation</h2>
<p>Variables:</p>
<ul>
<li>A - an array of stack elements</li>
<li>capacity - size of array</li>
<li>top - index in array of top stack element
<ul>
<li>-1 if stack is empty</li>
</ul>
</li>
</ul>
<p>A starts out as:</p>
<table>
<tbody>
<tr>
<td>Index</td>
<td>0</td>
<td>1</td>
<td>2</td>
<td>…</td>
<td>capacity-1</td>
</tr>
<tr>
<td>Value</td>
<td>a</td>
<td>b</td>
<td>c</td>
<td>…</td>
<td> </td>
</tr>
</tbody>
</table>
<ul>
<li>Capacity = &lt;size of A&gt;</li>
<li>top = 2</li>
</ul>
<p>push e</p>
<p>A now equals</p>
<table>
<tbody>
<tr>
<td>Index</td>
<td>0</td>
<td>1</td>
<td>2</td>
<td>3</td>
<td>4</td>
<td>…</td>
<td>capacity-1</td>
</tr>
<tr>
<td>Value</td>
<td>a</td>
<td>b</td>
<td>c</td>
<td>d</td>
<td>e</td>
<td>…</td>
<td> </td>
</tr>
</tbody>
</table>
<ul>
<li>capacity = &lt;size of A&gt;</li>
<li>top = 3</li>
</ul>
<h2 id="the-queue-adt">The Queue ADT</h2>
<p>A container that stores a collection of items with insertion and removal in “first-in-first-out” order.</p>
<p>Stack: a digram showing circled being placed on top of eachother, pushing circles into the tube, then popping them out from the same size (the top). Like a pringles can of elements.</p>
<p>Queue: a diagram showing circles being unqueued in on the left and dequeued on the right. Like waiting in line for food.</p>
<p>Basic operations:</p>
<ul>
<li>enqueue: add an item to the back of a list</li>
<li>dequeue: remove &amp; return the item at the front of the list</li>
</ul>
<h2 id="example-palindrome-checking">Example: Palindrome Checking</h2>
<p>Sequence <code class="language-plaintext highlighter-rouge">aba</code> should succeed.
Sequence <code class="language-plaintext highlighter-rouge">aab</code> should not.</p>
<p>In an array: a diagram showing the following pattern with the variable <code class="language-plaintext highlighter-rouge">i</code> set to the leftmost element and <code class="language-plaintext highlighter-rouge">j</code> set to the rightmost element.</p>
<table>
<tbody>
<tr>
<td>Values</td>
<td>a</td>
<td>b</td>
<td>c</td>
<td>…</td>
<td>s</td>
<td>b</td>
<td>a</td>
</tr>
</tbody>
</table>
<p>With a stack and queue:</p>
<ol>
<li>read symbols, starting each in a stack and a queue</li>
<li>while the stack, queue are not empty:
<ul>
<li>pop and element from the stack + delete an element from the queue</li>
<li>if different -&gt; not a palindrome</li>
</ul>
</li>
<li>palindrome</li>
</ol>
<h2 id="queue-array-implementations">Queue: Array Implementations</h2>
<p>Enqueue: a,b,c,d.
Front is now a, last is now d.</p>
<table>
<tbody>
<tr>
<td>Values</td>
<td>a</td>
<td>b</td>
<td>c</td>
<td>d</td>
<td>…</td>
</tr>
</tbody>
</table>
<p>Dequeue returns and deletes first element.
Front is now b, last is now d.</p>
<table>
<tbody>
<tr>
<td>Values</td>
<td>?</td>
<td>b</td>
<td>c</td>
<td>d</td>
<td>…</td>
</tr>
</tbody>
</table>
<p>Many enquques/dequeues later:</p>
<table>
<tbody>
<tr>
<td>Values</td>
<td>s</td>
<td>…</td>
<td>m</td>
<td>n</td>
<td>o</td>
</tr>
</tbody>
</table>
<p>Front element is now <code class="language-plaintext highlighter-rouge">m</code>, last element is now s.</p>
<table>
<tbody>
<tr>
<td>Values</td>
<td>?</td>
<td>t</td>
<td>u</td>
<td>v</td>
<td>w</td>
<td>…</td>
</tr>
</tbody>
</table>
<p>A similar slide is here, but I can’t see the differnece.</p>
<p>Variables:</p>
<ul>
<li>array A is array of size capacity</li>
<li>front is 0</li>
<li>back is 0</li>
<li>size is 0</li>
<li>capacity = &lt;size of array&gt;</li>
</ul>
<p>Pseudo-code:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>array A = array of size capacity // holds queue contents
front = 0 // index in A of front element, if queue not empty
back = 0 // index where next unqueued element will go
size = 0 // number of elements in queue
capacity = &lt;size of array&gt; // ???(can't read) queue size supported
enqueue(x) { // adds x to the back of the queue
// requires size&lt;capacity&gt;
A[back] = x
size = size + 1
back = (back + 1) % capacity
}
dequeue() { // removes front element of queue and returns it
// required size &gt; 0
temp = A[front]
front = (front + 1) % capacity
size = size - 1
return temp
}
</code></pre></div></div>
<p>Is <code class="language-plaintext highlighter-rouge">size == front - back</code>?</p>
<p>Diagram showing only true if front and back haven’t been switched around due to wrapping around back to the beginning o the array.</p>
<h2 id="circular-array-queue-example">“Circular Array” Queue Example</h2>
<ul>
<li>size = 0</li>
<li>front = 0</li>
<li>back = 0</li>
</ul>
<table>
<tbody>
<tr>
<td>Values</td>
<td>empty</td>
<td>empty</td>
<td>empty</td>
<td>empty</td>
<td>empty</td>
<td>empty</td>
</tr>
</tbody>
</table>
<p>enqueue: a,b,c,d</p>
<ul>
<li>front = 0</li>
<li>back = 4</li>
</ul>
<table>
<tbody>
<tr>
<td>Values</td>
<td>a</td>
<td>b</td>
<td>c</td>
<td>d</td>
<td>empty</td>
<td>empty</td>
</tr>
</tbody>
</table>
<p>dequque 3 times</p>
<ul>
<li>front = 3</li>
<li>back = 4</li>
</ul>
<table>
<tbody>
<tr>
<td>Values</td>
<td>empty</td>
<td>empty</td>
<td>empty</td>
<td>d</td>
<td>empty</td>
<td>empty</td>
</tr>
</tbody>
</table>
<p>enqueue e,f,g,h</p>
<ul>
<li>front = 3</li>
<li>back = 2</li>
</ul>
<table>
<tbody>
<tr>
<td>Values</td>
<td>g</td>
<td>h</td>
<td>empty</td>
<td>d</td>
<td>e</td>
<td>f</td>
</tr>
</tbody>
</table>
<p>unqueue j</p>
<ul>
<li>size = 6</li>
<li>font = 3</li>
<li>back = 3</li>
</ul>
<table>
<tbody>
<tr>
<td>Values</td>
<td>g</td>
<td>h</td>
<td>j</td>
<td>d</td>
<td>e</td>
<td>f</td>
</tr>
</tbody>
</table>
<p>dequeue 3 times</p>
<ul>
<li>back = 3</li>
<li>front = 0</li>
</ul>
<table>
<tbody>
<tr>
<td>Values</td>
<td>g</td>
<td>h</td>
<td>j</td>
<td>empty</td>
<td>empty</td>
<td>empty</td>
</tr>
</tbody>
</table>
<p>dequeue 3 times</p>
<ul>
<li>back = 3</li>
<li>front = 3</li>
</ul>
<table>
<tbody>
<tr>
<td>Values</td>
<td>empty</td>
<td>empty</td>
<td>empty</td>
<td>empty</td>
<td>empty</td>
<td>empty</td>
</tr>
</tbody>
</table>
<h2 id="end">End</h2>
<footer>
</footer>
</div>
</main>
</body>
</html>

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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<title> | tait.tech</title>
<link rel="stylesheet" href="/assets/css/style.css">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
</head>
<body>
<main>
<div id="wrapper">
<h1 id="memory--pointers-1">Memory &amp; Pointers 1</h1>
<p>CMPT-225, Fall 2021</p>
<h2 id="computer-memory">Computer Memory</h2>
<ul>
<li>A sequence of locations</li>
<li>Indexed by address: 0,1,2,…</li>
<li>Each location stores a data <em>byte</em></li>
<li>Processor can <em>read</em> or <em>write</em> the byte at each address.</li>
<li>Regions of memory are allocated to processes <em>as needed</em>, according to some scheme.</li>
</ul>
<p>Diagram displaying “Code + Data for running part of OS” at the start of memory, “Free memory” in the middle, and “Code + Data for running user processes” and the end.</p>
<h2 id="variables--memory">Variables &amp; Memory</h2>
<p>A variable is (roughly) and names &amp; tagged collection of bytes:</p>
<p>Diagram showing:</p>
<ul>
<li>“int x; x = 6” taking 4 bytes of memory,</li>
<li>“char c; c = ‘x’;” taking 1 byte of memory</li>
<li>
<p>“bool b; b = ‘true’;” taking 1 byte of memory</p>
</li>
<li>So, <em>at run time</em>, each variable has an <em>address</em> in memory.</li>
<li>In C, C++ we can:
<ul>
<li>access the address of a variable</li>
<li>access a variable or memory location by its address</li>
<li>declare variables by storing addresses (pointers).</li>
</ul>
</li>
</ul>
<h2 id="addresses--pointers---by-example">Addresses &amp; Pointers - By Example</h2>
<ul>
<li>“int i = 5;”
<ul>
<li>allocate space for an <em>int</em>,</li>
<li>name the space “i”,</li>
<li>store 5 there</li>
</ul>
</li>
<li>“int *p;”
<ul>
<li>allocate space for an address,</li>
<li>name it p,</li>
<li>record its type as “pointer to int”</li>
</ul>
</li>
<li>“p = ∧i;”
<ul>
<li>“&amp;i” is the address of “i”</li>
<li>store ∧i in p</li>
<li>so, p becomes a <em>pointer to i</em></li>
</ul>
</li>
<li>“cout « i;”
<ul>
<li>outputs the value stored in i, <em>5</em></li>
</ul>
</li>
<li>“cout « p;”
<ul>
<li>outputs the address of i: 0xbffffbbc</li>
</ul>
</li>
<li>“cout « *p;”
<ul>
<li>”*” <em>dereferences</em> p. That is, *p is the value <em>pointed to by p</em>. In this case <em>5</em></li>
</ul>
</li>
</ul>
<p>Diagram of memory:</p>
<table>
<tbody>
<tr>
<td>Address</td>
<td>Name</td>
<td>Value</td>
</tr>
<tr>
<td>…</td>
<td>…</td>
<td>…</td>
</tr>
<tr>
<td><span id="point">254</span></td>
<td>i</td>
<td>5</td>
</tr>
<tr>
<td>…</td>
<td>…</td>
<td>…</td>
</tr>
<tr>
<td>923</td>
<td>p</td>
<td><a href="#point">254</a></td>
</tr>
</tbody>
</table>
<p>Second diagram of memory:</p>
<table>
<tbody>
<tr>
<td>Address</td>
<td>Name</td>
<td>Value</td>
<td>Size</td>
</tr>
<tr>
<td>0x6fffbe</td>
<td>i</td>
<td>5</td>
<td>4 bytes</td>
</tr>
<tr>
<td>…</td>
<td>…</td>
<td>…</td>
<td>…</td>
</tr>
<tr>
<td>???</td>
<td>p</td>
<td>0x6fffbe</td>
<td>4 bytesa</td>
</tr>
</tbody>
</table>
<p>Slide showing program code: see pointers.c</p>
<p>Slide showing output: see output.txt</p>
<p>Slide showing same program with different highlights.</p>
<p>Slide showing same output with different highlights.</p>
<h2 id="end">End</h2>
<footer>
</footer>
</div>
</main>
</body>
</html>

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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<title> | tait.tech</title>
<link rel="stylesheet" href="/assets/css/style.css">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
</head>
<body>
<main>
<div id="wrapper">
<h1 id="memory--pointers-1">Memory &amp; Pointers 1</h1>
<p>CMPT-225, Fall 2021</p>
<h2 id="computer-memory">Computer Memory</h2>
<ul>
<li>A sequence of locations</li>
<li>Indexed by address: 0,1,2,…</li>
<li>Each location stores a data <em>byte</em></li>
<li>Processor can <em>read</em> or <em>write</em> the byte at each address.</li>
<li>Regions of memory are allocated to processes <em>as needed</em>, according to some scheme.</li>
</ul>
<p>Diagram displaying “Code + Data for running part of OS” at the start of memory, “Free memory” in the middle, and “Code + Data for running user processes” and the end.</p>
<h2 id="variables--memory">Variables &amp; Memory</h2>
<p>A variable is (roughly) and names &amp; tagged collection of bytes:</p>
<p>Diagram showing:</p>
<ul>
<li>“int x; x = 6” taking 4 bytes of memory,</li>
<li>“char c; c = ‘x’;” taking 1 byte of memory</li>
<li>
<p>“bool b; b = ‘true’;” taking 1 byte of memory</p>
</li>
<li>So, <em>at run time</em>, each variable has an <em>address</em> in memory.</li>
<li>In C, C++ we can:
<ul>
<li>access the address of a variable</li>
<li>access a variable or memory location by its address</li>
<li>declare variables by storing addresses (pointers).</li>
</ul>
</li>
</ul>
<h2 id="addresses--pointers---by-example">Addresses &amp; Pointers - By Example</h2>
<ul>
<li>“int i = 5;”
<ul>
<li>allocate space for an <em>int</em>,</li>
<li>name the space “i”,</li>
<li>store 5 there</li>
</ul>
</li>
<li>“int *p;”
<ul>
<li>allocate space for an address,</li>
<li>name it p,</li>
<li>record its type as “pointer to int”</li>
</ul>
</li>
<li>“p = ∧i;”
<ul>
<li>“&amp;i” is the address of “i”</li>
<li>store ∧i in p</li>
<li>so, p becomes a <em>pointer to i</em></li>
</ul>
</li>
<li>“cout « i;”
<ul>
<li>outputs the value stored in i, <em>5</em></li>
</ul>
</li>
<li>“cout « p;”
<ul>
<li>outputs the address of i: 0xbffffbbc</li>
</ul>
</li>
<li>“cout « *p;”
<ul>
<li>”*” <em>dereferences</em> p. That is, *p is the value <em>pointed to by p</em>. In this case <em>5</em></li>
</ul>
</li>
</ul>
<p>Diagram of memory:</p>
<table>
<tbody>
<tr>
<td>Address</td>
<td>Name</td>
<td>Value</td>
</tr>
<tr>
<td>…</td>
<td>…</td>
<td>…</td>
</tr>
<tr>
<td><span id="point">254</span></td>
<td>i</td>
<td>5</td>
</tr>
<tr>
<td>…</td>
<td>…</td>
<td>…</td>
</tr>
<tr>
<td>923</td>
<td>p</td>
<td><a href="#point">254</a></td>
</tr>
</tbody>
</table>
<p>Second diagram of memory:</p>
<table>
<tbody>
<tr>
<td>Address</td>
<td>Name</td>
<td>Value</td>
<td>Size</td>
</tr>
<tr>
<td>0x6fffbe</td>
<td>i</td>
<td>5</td>
<td>4 bytes</td>
</tr>
<tr>
<td>…</td>
<td>…</td>
<td>…</td>
<td>…</td>
</tr>
<tr>
<td>???</td>
<td>p</td>
<td>0x6fffbe</td>
<td>4 bytesa</td>
</tr>
</tbody>
</table>
<p>Slide showing program code: see pointers.c</p>
<p>Slide showing output: see output.txt</p>
<p>Slide showing same program with different highlights.</p>
<p>Slide showing same output with different highlights.</p>
<h2 id="end">End</h2>
<footer>
</footer>
</div>
</main>
</body>
</html>

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main() begin
XXXX = 111111111
XXXX is stored at &XXXX = 0xfffff40750e0
YYYY is a pointer to XXXX: YYYY = 0xfffff40750e0
* dereferences the pointer: *YYYY = 111111111
Array AAAA can be accessed with array notaions:
AAAA[0] = 222222222
AAAA[1] = 333333333
AAAA[2] = 444444444
Array variable AAAA is a pointer to A[0]: AAAA = 0xfffff40750f8
So, dereferencing AAAA should give us A[0]: *AAAA = 222222222
Adding 1 to an int pointer makes it point to the next int
AAAA = 0xfffff40750f8
AAAA+1 = 0xfffff40750fc
*(AAAA+1) = 333333333
We can look at contents of a chunk of memory:
Peeking at the memory in the neighbourhood of &XXXX, we see:
Address Contents in Hex Contents in Decimal
0xfffff40750fc: 13de4355 = 333333333
0xfffff40750f8: d3ed78e = 222222222
0xfffff40750f4: ffff = 65535
0xfffff40750f0: f40750e0 = -200847136
0xfffff40750ec: ffff = 65535
0xfffff40750e8: f40750e8 = -200847128
0xfffff40750e4: 6 = 6
0xfffff40750e0: 69f6bc7 = 111111111
0xfffff40750dc: ffff = 65535
0xfffff40750d8: a478f4e0 = -1535576864
0xfffff40750d4: aaaa = 43690
0xfffff40750d0: c46a13e0 = -999681056
0xfffff40750cc: ffff = 65535
0xfffff40750c8: a478f538 = -1535576776
0xfffff40750c4: ffff = 65535
main() ends

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/* a small program to demonstrate C++ addresses, pointers and arrays */
#include <iostream> /* Tait: missing name for import */
#include <iomanip> /* Tait: missing name from import */
using namespace std;
int main(){
cout << "main() begin\n";
int XXXX = 111111111;
cout << "XXXX = " << XXXX << endl;
cout << "XXXX is stored at &XXXX = " << &XXXX << endl;
int * YYYY = &XXXX;
cout << "YYYY is a pointer to XXXX: YYYY = " << YYYY << endl;
cout << "* dereferences the pointer: *YYYY = " << *YYYY << endl;
int AAAA[3] = { 222222222, 333333333, 444444444 };
cout << "Array AAAA can be accessed with array notaions: " << endl;
cout << " AAAA[0] = " << AAAA[0] << endl;
cout << " AAAA[1] = " << AAAA[1] << endl;
cout << " AAAA[2] = " << AAAA[2] << endl << endl;
cout << "Array variable AAAA is a pointer to A[0]: AAAA = " << AAAA << endl;
cout << "So, dereferencing AAAA should give us A[0]: *AAAA = " << *AAAA << endl << endl;
cout << "Adding 1 to an int pointer makes it point to the next int" << endl;
cout << "AAAA = " << AAAA << endl;
cout << "AAAA+1 = " << (AAAA+1) << endl;
cout << "*(AAAA+1) = " << *(AAAA+1) << endl << endl;
cout << "We can look at contents of a chunk of memory:" << endl;
cout << "Peeking at the memory in the neighbourhood of &XXXX, we see: " << endl << endl;
cout << "Address Contents in Hex Contents in Decimal " << endl;
int * p = (&XXXX)+7;
for (int i = 0; i < 15; i++) {
cout << p << ": " << setw(8) << hex << *p << " = " << setw(11) << dec << *p << endl;
p -= 1;
}
cout << "main() ends" << endl;
}

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Title: Activity - Code
Video of CERT Secure Coding Initiative Conference 2015 - Robert C. Seacord
https://www.youtube.com/watch?v=1ew0GvB3NpE
Blue box with code in it:
char *copy (size_t n, const char *a) {
if (n == 0) return NULL;
if (a == NULL) return NULL;
char *p = (char *)malloc(n);
if (p == NULL) return NULL;
for (int i = 0; i < n; i++) p[i] = *a++;
return p;
}
White text over-laying the blue box: Spot the defect

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<aside>Hello everyone!</aside>
# Welcome to CMPT 295
## Introduction to Computer Systems
My name is Anne Lavergne
Lecture 1 – Course Overview + Activity
<aside>How does it feel to be back on campus?</aside>
## Today’s Menu [1]
* COVID Protocol
* What is CMPT 295?
* What shall we learn in CMPT 295?
* What should we already know?
* Which resources do we have to help us learn all this?
* Activity
* Questions
## COVID protocol – About masks! [2]
Here is a message from Elizabeth Elle, SFU Vice Provost Learning & Teaching, based on the public health order:
* Unless we have an approved exemption, we are required to wear a mask in all indoor common and learning spaces, including classrooms. Please come to campus prepared with a non-medical mask.
* If we forget our mask, disposable masks are available from Student Central in Burnaby and at the information desks in Vancouver and Surrey.
* If we require a mask exemption in the classroom for medical reasons, please contact the Centre for Accessible Learning at cal_admin@sfu.ca for assistance.
* If we are requesting mask exemptions on other protected grounds, such as religion, we can contact the Office of Student Support, Rights and Responsibilities at student_support@sfu.ca.
* And please remember to be kind to each other. If we see someone not wearing a mask, do not make assumptions or judgments as that person may be exempt.
## What is CMPT 295? [3]
* The goal of this course is to give us, software developers, a look “under the hood” of a computer, i.e., to learn about Computer Systems => microprocessor, memory, … <img src="" alt="a car with its hood up.">
* This knowledge will allow us to become more efficient software developers
## The big picture: [4]
In CMPT 295, we shall learn …
* C programs (.c) -- How our code and data are represented in memory
* Assembly programs (.s) -- How a compiler transforms our code into machine executable code in several steps
* Object (.o) file an executable -- How a compiler optimizes (or not) our code
* Computer executes it -- How a microprocessor is designed and how it executes our code
* CPU, Memory -- How memory is designed
How all of this can impact the execution of our code How to write more efficient and reliable code:
* Be able to find and eliminate bugs
more efficiently
* Be able to ascertain program performance and tune it by optimizing our code
## What should we already know? [5]
* Write correct C programs
* C constructs (variables, data types, pointers, if/else, switch/case, for/while/do while, function calls, arrays, …)
* What a stack is and how it works
* Binary/decimal/hexadecimal numeral systems
* How to convert from one numeral system to the others
* Basic arithmetic
* Perform Boolean algebra using and, or, not, xor
## Which resources do we have? [6]
* Course web site
https://www2.cs.sfu.ca/CourseCentral/295/alavergn/index.html
* Textbook
* Computer Systems: A Programmer's Perspective, 3/E, Randal E. Bryant, David R. O'Hallaron, Pearson, 2016
* Labs in CSIL (Computing Science Instructional Lab)
* Target Machine: CSIL workstation
* Linux platform (or OS)
* C programming language
* x86-64 assembly language
* gcc compiler
* Instructor and TAs - Office hours
## Activity - Discover our resources [7]
Instructions:
1. Form teams of 3 to 4
2. Do Lecture 1 Activity on CourSys
3. Time: about 30 minutes
## Question? [8]
Blank page.
## Summary [9]
* COVID Protocol
* What is CMPT 295?
* What shall we learn in CMPT 295?
* What should we already know?
* Which resources do we have to help us learn all this?
* Activity
* Questions
## Next Lecture [10]
* Data Representation
* Representing information as bits
* To get ready for our next lecture:
* Optional: Read Chapter 1 of textbook
* Not so optional: Read Section 2.1 of Chapter 2
* Download the partial lecture notes found under the column Lecture in the table on our course web site

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<p><aside>Hello everyone!</aside></p>
<h1>Welcome to CMPT 295</h1>
<h2>Introduction to Computer Systems</h2>
<p>My name is Anne Lavergne</p>
<p>Lecture 1 – Course Overview + Activity</p>
<p><aside>How does it feel to be back on campus?</aside></p>
<h2>Today’s Menu [1]</h2>
<ul>
<li>COVID Protocol</li>
<li>What is CMPT 295?
<ul>
<li>What shall we learn in CMPT 295?</li>
<li>What should we already know?</li>
<li>Which resources do we have to help us learn all this?</li>
</ul>
</li>
<li>Activity</li>
<li>Questions</li>
</ul>
<h2>COVID protocol – About masks! [2]</h2>
<p>Here is a message from Elizabeth Elle, SFU Vice Provost Learning &amp; Teaching, based on the public health order:</p>
<ul>
<li>Unless we have an approved exemption, we are required to wear a mask in all indoor common and learning spaces, including classrooms. Please come to campus prepared with a non-medical mask.
<ul>
<li>If we forget our mask, disposable masks are available from Student Central in Burnaby and at the information desks in Vancouver and Surrey.</li>
<li>If we require a mask exemption in the classroom for medical reasons, please contact the Centre for Accessible Learning at cal_admin@sfu.ca for assistance.</li>
<li>If we are requesting mask exemptions on other protected grounds, such as religion, we can contact the Office of Student Support, Rights and Responsibilities at student_support@sfu.ca.</li>
</ul>
</li>
<li>And please remember to be kind to each other. If we see someone not wearing a mask, do not make assumptions or judgments as that person may be exempt.</li>
</ul>
<h2>What is CMPT 295? [3]</h2>
<ul>
<li>The goal of this course is to give us, software developers, a look “under the hood” of a computer, i.e., to learn about Computer Systems => microprocessor, memory, … <img src="" alt="a car with its hood up."></li>
<li>This knowledge will allow us to become more efficient software developers</li>
</ul>
<h2>The big picture: [4]</h2>
<p>In CMPT 295, we shall learn …</p>
<p>C programs (.c) &ndash; How our code and data are represented in memory</p>
<p>Assembly programs (.s) &ndash; How a compiler transforms our code into machine executable code in several steps</p>
<p>Object (.o) file an executable &ndash; How a compiler optimizes (or not) our code</p>
<p>Computer executes it &ndash; How a microprocessor is designed and how it executes our code</p>
<p>CPU, Memory &ndash; How memory is designed</p>
<p>How all of this can impact the execution of our code How to write more efficient and reliable code:
* Be able to find and eliminate bugs
more efficiently
* Be able to ascertain program performance and tune it by optimizing our code</p>
<h2>What should we already know? [5]</h2>
<ul>
<li>Write correct C programs
<ul>
<li>C constructs (variables, data types, pointers, if/else, switch/case, for/while/do while, function calls, arrays, …)</li>
</ul>
</li>
<li>What a stack is and how it works</li>
<li>Binary/decimal/hexadecimal numeral systems
<ul>
<li>How to convert from one numeral system to the others</li>
<li>Basic arithmetic</li>
</ul>
</li>
<li>Perform Boolean algebra using and, or, not, xor</li>
</ul>
<h2>Which resources do we have? [6]</h2>
<ul>
<li>Course web site
https://www2.cs.sfu.ca/CourseCentral/295/alavergn/index.html</li>
<li>Textbook
<ul>
<li>Computer Systems: A Programmer&rsquo;s Perspective, 3/E, Randal E. Bryant, David R. O'Hallaron, Pearson, 2016</li>
</ul>
</li>
<li>Labs in CSIL (Computing Science Instructional Lab)
<ul>
<li>Target Machine: CSIL workstation
<ul>
<li>Linux platform (or OS)</li>
<li>C programming language</li>
<li>x86-64 assembly language</li>
<li>gcc compiler</li>
</ul>
</li>
</ul>
</li>
<li>Instructor and TAs - Office hours</li>
</ul>
<h2>Activity - Discover our resources [7]</h2>
<p>Instructions:
1. Form teams of 3 to 4
2. Do Lecture 1 Activity on CourSys
3. Time: about 30 minutes</p>
<h2>Question? [8]</h2>
<p>Blank page.</p>
<h2>Summary [9]</h2>
<ul>
<li>COVID Protocol</li>
<li>What is CMPT 295?
<ul>
<li>What shall we learn in CMPT 295?</li>
<li>What should we already know?</li>
<li>Which resources do we have to help us learn all this?</li>
</ul>
</li>
<li>Activity</li>
<li>Questions</li>
</ul>
<h2>Next Lecture [10]</h2>
<ul>
<li><p>Data Representation</p>
<ul>
<li>Representing information as bits</li>
</ul>
</li>
<li><p>To get ready for our next lecture:</p>
<ul>
<li>Optional: Read Chapter 1 of textbook</li>
<li>Not so optional: Read Section 2.1 of Chapter 2</li>
<li>Download the partial lecture notes found under the column Lecture in the table on our course web site</li>
</ul>
</li>
</ul>

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CMPT 295
Unit - Data Representation
Lecture 3 – Representing integral numbers in memory - unsigned and signed
1
Last Lecture
 Von Neumann architecture
 Architecture of most computers
 Its components: CPU, memory, input and ouput, bus
 One of its characteristics: Data and code (programs) both stored in memory
 A look at memory: defined byte-addressable memory, diagram of (compressed) memory
 Word size (w): size of a series of bits (or bit vector) we manipulate, also size of machine words
(see Section 2.1.2)
 A look at bits in memory
 Why binary numeral system (0 and 1 -> two values) is used to represent information in memory
 Algorithm for converting binary to hexadecimal (hex)
1. Partition bit vector into groups of 4 bits, starting from right, i.e., least significant byte (LSB)
 If most significant “byte” (MSB) does not have 8 bits, pad it: add 0’s to its left
2. Translate each group of 4 bits into its hex value
 What do bits represent? Encoding scheme gives meaning to bits
 Order of bytes in memory: little endian versus big endian
 Bit manipulation – regardless of what bit vectors represent
 Boolean algebra: bitwise operations => AND (&), OR (|), XOR (^), NOT (~)
 Shift operations: left shift, right logical shift and right arithmetic shift
2
 Logical shift: Fill x with y 0’s on left
 Arithmetic shift: Fill x with y copies of x‘s sign bit on left
 Sign bit: Most significant bit (MSb) before shifting occurred
NOTE:
C logical operators
and C bitwise (bit-level)
operators behave
differently!
Watch out for && versus
&, || versus |, …
Today’s Menu
 Representing data in memory – Most of this is review
 “Under the Hood” - Von Neumann architecture
 Bits and bytes in memory
 How to diagram memory -> Used in this course and other references
 How to represent series of bits -> In binary, in hexadecimal (conversion)
 What kind of information (data) do series of bits represent -> Encoding scheme
 Order of bytes in memory -> Endian
 Bit manipulation – bitwise operations
 Boolean algebra + Shifting
 Representing integral numbers in memory
 Unsigned and signed
 Converting, expanding and truncating
 Arithmetic operations
 Representing real numbers in memory
3
 IEEE floating point representation
 Floating point in C – casting, rounding, addition, …
Warm up exercise!
As a warm up exercise, fill in the blanks!
 If the context is C (on our target machine)
 char
=> _____ bits/ _____ byte
 short => _____ bits/ _____ bytes
 int
=> _____ bits/ _____ bytes
 long
=> _____ bits/ _____ bytes
 float => _____ bits/ _____ bytes
 double=> _____ bits/ _____ bytes
 pointer (e.g. char *)
4
=> _____ bits/ _____ bytes
Remember:
Unsigned integral numbers
 What if the byte at M[0x0002] represented an unsigned integral
A series of bits
number, what would be its value?
=> bit vector
w =>width of
 X = 011010012
the bit vector
w=8
 Let’s apply the encoding scheme:
B2U(X) 
w1
 xi 2
i
i0
0 x 27 + 1 x 26 + 1 x 25 + 0 x 24 + 1 x 23 + 0 x 22 + 0 x 21 + 1 x 20 =
5
 For w = 8, range of possible unsigned values: [
]
 For any w, range of possible unsigned values: [
]
 Conclusion: w bits can only represent a fixed # of possible values,
but these w bits represent these values exactly
B2U(X) Conversion (Encoding scheme)
 Positional notation: expand and sum all terms
•••
10i
2i
10i-1
2i-1
100
10
1
4
2
1
•••
di di-1 ••• d2 d1 d0
Example: 24610 = 2 x 102 + 4 x 101 + 6 x 100
6
1’s = 100
10’s = 101
100’s = 102
B2U(X ) 
w1
 xi 2
i0
i
Range of possible values?
 If the context is C (on our target machine)
unsigned char?
unsigned short?
unsigned int?
unsigned long?
7
Examples of “Show your work”
U2B(X) Conversion (into 8-bit binary # => w = 8)
8
Method 1 - Using subtraction:
subtracting decreasing
power of 2 until reach 0
246 => 246 – 128 = 118 ->128 = 1 x 27
118 – 64 = 54
-> 64 = 1 x 26
54 – 32 = 22
-> 32 = 1 x 25
22 – 16 = 6
-> 16 = 1 x 24
6 – 8 = nop! -> 8 = 0 x 23
6 – 4 =2
-> 4 = 1 x 22
2– 2=0
-> 2 = 1 x 21
0 – 1 = nop! -> 1 = 0 x 20
Method 2 - Using division:
dividing by 2
until reach 0
246 => 246 / 2 = 123 -> R = 0
123 / 2 = 61 -> R = 1
61 / 2 = 30 -> R = 1
30 / 2 = 15 -> R = 0
15 / 2 = 7
-> R = 1
7/2 =3
-> R = 1
3/2 =1
-> R = 1
1/2 =0
-> R = 1
246 => 1 1 1 1 0 1 1 02
246 => 1 1 1 1 0 1 1 02
U2B(X) Conversion – A few tricks
 Decimal -> binary
 Trick: When decimal number is 2n, then its binary representation is 1 followed
by n zero’s
 Let’s try: if X = 32 => X = 25, then n = 5 => 100002 (w = 5)
What if w = 8?
Check: 1 x 24 = 32
 Decimal -> hex
 Trick: When decimal number is 2n, then its hexadecimal representation is 2i
followed by j zero’s, where n = i + 4j and 0 <= i <=3
 Let try: if X = 8192 => X = 213, then n = 13 and 13 = i + 4j => 1 + 4 x 3
=> 0x2000
9
Convert 0x2000 into a binary number:
Check: 2 x 163 = 2 x 4096 = 8192
Remember:
Signed integral numbers
 What if the byte at M[0x0001] represented a signed integral
number, what would be its value? T => Two’s Complement w =>width of
the bit vector
 X = 111101002 w = 8
w2
w1
i
B2T
(X
)
x
2
x
2
w1
i
 Let’s apply the encoding scheme:
Sign bit
i0
-1 x 27 + 1 x 26 + 1 x 25 + 1 x 24 + 0 x 23 + 1 x 22 + 0 x 21 + 0 x 20 =
 What would be the bit pattern of the …
 Most negative value:
 Most positive value:
10
 For w = 8, range of possible signed values: [
]
 For any w, range of possible signed values: [
]
 Conclusion: same as for unsigned integral numbers
Examples of “Show your work”
T2B(X) Conversion -> Two’s Complement
w=8
Method 1 If X < 0, (~(U2B(|X|)))+1
Method 2
If X = -14 (and 8 bit binary #s)
If X = -14 (and 8 bit binary #s)
1. |X| => |-14| =
1.
2. U2B(14) =>
2. U2B(242) =>
3. ~(000011102) =>
4. (111100012)+1 =>
Binary addition:
11110001
+ 00000001
11
Check:
If X < 0, U2B(X + 2w)
X + 2w => -14 +
Using subtraction:
242 – 128 = 114 -> 1 x 27
114 – 64 = 50 -> 1 x 26
50 – 32 = 18
-> 1 x 25
18 – 16 = 2
-> 1 x 24
2 – 8 -> nop! -> 0 x 23
2 – 4 -> nop! -> 0 x 22
2–2=0
-> 1 x 21
0 – 1 -> nop! -> 0 x 20
Properties of unsigned & signed conversions
w=4
12
X
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
B2U(X)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
B2T(X)
0
1
2
3
4
5
6
7
–8
–7
–6
–5
–4
–3
–2
–1
 Equivalence
 Both encoding schemes (B2U
and B2T ) produce the same bit
patterns for nonnegative values
 Uniqueness
Every bit pattern produced by
these encoding schemes (B2U
and B2T ) represents a unique
(and exact) integer value
 Each representable integer has
unique bit pattern
Converting between signed & unsigned
of same size (same data type)
Unsigned
w=8
ux
If ux = 12910
Signed (Two’s Complement)
w=4
13
x
If x = -510
U2T
U2B X
B2T
Maintain Same Bit Pattern
then x =
Unsigned
T2U
T2B X
Signed (Two’s Complement)
x
B2U
Maintain Same Bit Pattern
ux
then ux =
 Conclusion - Converting between unsigned and signed numbers:
Both have same bit pattern, however, this bit pattern may be interpreted
differently, i.e., producing a different value
Converting signed  unsigned with w = 4
Signed
Bits
Unsigned
0
0000
0
1
0001
2
0010
2
3
0011
3
0100
4
5
0101
5
6
0110
6
7
0111
7
-8
1000
8
-7
1001
9
-6
1010
10
4
-5
U2T(X)
+ 16 (+24)
-4
-3
14
14
-2
-1
U2T(X)
T2U(X)
1
1011
T2U(X)
11
1100
- 16 (+24)
12
1101
13
1110
14
1111
15
Visualizing the relationship between
signed & unsigned
If w = 4,
24
UMax
UMax – 1
= 16
TMax
Signed
(2’s Complement)
Range
15
0
–1
–2
TMin
TMax + 1
TMax
0
Unsigned
Range
Sign extension
 Converting unsigned (or signed) of different sizes (different data types)
1. Small data type -> larger
Sign bit
 Sign extension
X
Unsigned: zero extension
•••
Signed: sign bit extension
 Conclusion: Value unchanged
•••
X
 Let’s try:
•••
•••
 Going from a data type that has a width of 3 bits (w = 3) to a data type
that has a width of 5 bits (w = 5)
 Unsigned: X = 3 =>
new X =
16
 Signed:
0112 w = 3
<=
w=5
X = 3 =>
0112 w = 3
new X =
<=
w=5
X = 4 =>
new X =
1002 w = 3
<=
w=5
X = -3 =>
1012 w = 3
new X =
<=
w=5
Truncation
 Converting unsigned (or signed) of different sizes(different data types)
2. Large data type -> smaller
•••
X
 Truncation
•••
 Conclusion: Value may be altered
A form of overflow
 Let’s try:
X
•••
 Going from a data type that has a width of 5 bits (w = 5) to a data type
that has a width of 3 bits (w = 3)
 Unsigned: X = 27 => 110112 w = 5
new X =
 Signed:
17
<=
w=3
X = -15 => 100012 w = 5
new X =
<=
w=3
X = -1 => 111112 w = 5
new X =
<=
w=3
Summary
 Interpretation of bit pattern B into either unsigned value U or signed value T
 B2U(X) and U2B(X) encoding schemes (conversion)
 B2T(X) and T2B(X) encoding schemes (conversion)
 Signed value expressed as two’s complement => T
 Conversions from unsigned <-> signed values
 U2T(X) and T2U(X) => adding or subtracting 2w
 Implication in C: when converting (implicitly via promotion and explicitly via casting):
 Sign:
 Unsigned <-> signed (of same size) -> Both have same bit pattern, however, this bit pattern may
be interpreted differently
 Can have unexpected effects -> producing a different value
 Size:
 Small -> large (for signed, e.g., short to int and for unsigned, e.g., unsigned short to unsigned int)
 sign extension: For unsigned -> zeros extension, for signed -> sign bit extension
 Both yield expected result –> resulting value unchanged
 Large -> small (e.g., unsigned int to unsigned short)
 truncation: Unsigned/signed -> most significant bits are truncated (discarded)
 May not yield expected results -> original value may be altered
18
 Both (sign and size): 1) size conversion is first done then 2) sign conversion is done
Next Lecture
 Representing data in memory – Most of this is review
 “Under the Hood” - Von Neumann architecture
 Bits and bytes in memory
 How to diagram memory -> Used in this course and other references
 How to represent series of bits -> In binary, in hexadecimal (conversion)
 What kind of information (data) do series of bits represent -> Encoding scheme
 Order of bytes in memory -> Endian
 Bit manipulation – bitwise operations
 Boolean algebra + Shifting
 Representing integral numbers in memory
 Unsigned and signed
 Converting, expanding and truncating
 Arithmetic operations
 Representing real numbers in memory
19
 IEEE floating point representation
 Floating point in C – casting, rounding, addition, …

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<h1 id="cmpt-295">CMPT 295</h1>
<p>Unit - Data Representation</p>
<p>Lecture 4 – Representing integral numbers in memory – Arithmetic operations</p>
<h2 id="warm-up-question">Warm up question</h2>
<ul>
<li>What is the value of …
<ul>
<li>TMin (in hex) for signed char in C: _________________</li>
<li>TMax (in hex) for signed int in C: _________________</li>
<li>TMin (in hex) for signed short in C: ________________</li>
</ul>
</li>
</ul>
<h2 id="last-lecture">Last Lecture</h2>
<ul>
<li>Interpretation of bit pattern B into either unsigned value U or signed value T
<ul>
<li>B2U(X) and U2B(X) encoding schemes (conversion)</li>
<li>B2T(X) and T2B(X) encoding schemes (conversion)
<ul>
<li>Signed value expressed as two’s complement =&gt; T</li>
</ul>
</li>
</ul>
</li>
<li>Conversions from unsigned &lt;-&gt; signed values
<ul>
<li>U2T(X) and T2U(X) =&gt; adding or subtracting 2w</li>
</ul>
</li>
<li>Implication in C: when converting (implicitly via promotion and explicitly via casting):
<ul>
<li>Sign:
<ul>
<li>Unsigned &lt;-&gt; signed (of same size) -&gt; Both have same bit pattern, however, this bit pattern may be interpreted differently
<ul>
<li>Can have unexpected effects -&gt; producing a different value</li>
</ul>
</li>
</ul>
</li>
<li>Size:
<ul>
<li>Small -&gt; large (for signed, e.g., short to int and for unsigned, e.g., unsigned short to unsigned int)
<ul>
<li>sign extension: For unsigned -&gt; zeros extension, for signed -&gt; sign bit extension</li>
<li>Both yield expected result –&gt; resulting value unchanged</li>
</ul>
</li>
<li>Large -&gt; small (for signed, e.g., int to short and for unsigned, e.g., unsigned int to unsigned short)
<ul>
<li>truncation: Unsigned/signed -&gt; most significant bits are truncated (discarded)</li>
<li>May not yield expected results -&gt; original value may be altered</li>
</ul>
</li>
</ul>
</li>
<li>Both (sign and size): 1) size conversion is first done then 2) sign conversion is done</li>
</ul>
</li>
</ul>
<h2 id="todays-menu">Today’s Menu</h2>
<ul>
<li>Representing data in memory – Most of this is review
<ul>
<li>“Under the Hood” - Von Neumann architecture</li>
<li>Bits and bytes in memory
<ul>
<li>How to diagram memory -&gt; Used in this course and other references</li>
<li>How to represent series of bits -&gt; In binary, in hexadecimal (conversion)</li>
<li>What kind of information (data) do series of bits represent -&gt; Encoding scheme</li>
<li>Order of bytes in memory -&gt; Endian</li>
</ul>
</li>
<li>Bit manipulation – bitwise operations
<ul>
<li>Boolean algebra + Shifting</li>
</ul>
</li>
</ul>
</li>
<li>Representing integral numbers in memory
<ul>
<li>Unsigned and signed</li>
<li>Converting, expanding and truncating</li>
<li>Arithmetic operations</li>
</ul>
</li>
<li>Representing real numbers in memory
<ul>
<li>IEEE floating point representation</li>
<li>Floating point in C – casting, rounding, addition, …</li>
</ul>
</li>
</ul>
<p>Let’s first illustrate what we covered last lecture with a demo!</p>
<h2 id="demo--looking-at-size-and-sign-conversions-in-c">Demo – Looking at size and sign conversions in C</h2>
<ul>
<li>What does the demo illustrate?
<ul>
<li>Size conversion:
<ul>
<li>Converting to a larger (wider) data type -&gt; Converting short to int</li>
<li>Converting to a smaller (narrower) data type -&gt; Converting short to char</li>
</ul>
</li>
<li>Sign conversion:
<ul>
<li>Converting from signed to unsigned -&gt; Converting short to unsigned short</li>
<li>Converting from unsigned to signed -&gt; Converting unsigned short to short</li>
</ul>
</li>
<li>Size and Sign conversion:
<ul>
<li>Converting from signed to unsigned larger (wider) data type -&gt; Converting short to unsigned int</li>
<li>Converting from signed to unsigned smaller (narrower) data type -&gt;
Converting short to unsigned char</li>
</ul>
</li>
</ul>
</li>
<li>This demo (code and results) posted on our course web site</li>
</ul>
<h2 id="integer-addition-unlimited-space">Integer addition (unlimited space)</h2>
<ul>
<li>
<p>What happens when we add two decimal numbers? (Highlights show carring the one to complete the equation)
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn><msub><mn>7</mn><mn>10</mn></msub><mo>+</mo><mn>93</mn><msub><mn>8</mn><mn>10</mn></msub><mo>=</mo><mn>104</mn><msub><mn>5</mn><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">
107_{10} + 938_{10} = 1045_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1</span><span class="mord">0</span><span class="mord"><span class="mord">7</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">9</span><span class="mord">3</span><span class="mord"><span class="mord">8</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1</span><span class="mord">0</span><span class="mord">4</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p>
</li>
<li>
<p>Same thing happens when we add two binary numbers:</p>
</li>
</ul>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10110</mn><msub><mn>0</mn><mn>2</mn></msub><mo>+</mo><mn>10111</mn><msub><mn>0</mn><mn>2</mn></msub><mo>=</mo><mn>101101</mn><msub><mn>0</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">
101100_{2} + 101110_{2} = 1011010_{2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1</span><span class="mord">0</span><span class="mord">1</span><span class="mord">1</span><span class="mord">0</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1</span><span class="mord">0</span><span class="mord">1</span><span class="mord">1</span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1</span><span class="mord">0</span><span class="mord">1</span><span class="mord">1</span><span class="mord">0</span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p>
<h2 id="unsigned-addition-limited-space-ie-fixed-size-in-memory">Unsigned addition (limited space, i.e., fixed size in memory)</h2>
<p>What happens when we add two unsigned values?</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>w</mi><mo>=</mo><mn>8</mn></mrow><annotation encoding="application/x-tex">w=8</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">8</span></span></span></span></span>
<p>a)</p>
<p>Binary:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mn mathvariant="italic">00111011</mn><mn>2</mn></msub><mo>+</mo><msub><mn mathvariant="italic">01011010</mn><mn>2</mn></msub><mo>=</mo><mo stretchy="false">?</mo></mrow><annotation encoding="application/x-tex">
\mathit{00111011}_{2} + \mathit{01011010}_{2} = ?
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathit">0</span><span class="mord mathit">0</span><span class="mord mathit">1</span><span class="mord mathit">1</span><span class="mord mathit">1</span><span class="mord mathit">0</span><span class="mord mathit">1</span><span class="mord mathit">1</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathit">0</span><span class="mord mathit">1</span><span class="mord mathit">0</span><span class="mord mathit">1</span><span class="mord mathit">1</span><span class="mord mathit">0</span><span class="mord mathit">1</span><span class="mord mathit">0</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mclose">?</span></span></span></span></span></p>
<p>Decimal:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mn mathvariant="italic">59</mn><mn>10</mn></msub><mo>+</mo><msub><mn>90</mn><mn>10</mn></msub><mo>=</mo><msub><mn>149</mn><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">
\mathit{59}_{10} + {90}_{10} = {149}_{10}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathit">5</span><span class="mord mathit">9</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord">9</span><span class="mord">0</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord">1</span><span class="mord">4</span><span class="mord">9</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p>
<p>b)</p>
<p>Binary:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mn mathvariant="italic">10101110</mn><mn>2</mn></msub><mo>+</mo><msub><mn mathvariant="italic">11001011</mn><mn>2</mn></msub><mo>=</mo><mo stretchy="false">?</mo></mrow><annotation encoding="application/x-tex">
\mathit{10101110}_{2} + \mathit{11001011}_{2} = ?
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathit">1</span><span class="mord mathit">0</span><span class="mord mathit">1</span><span class="mord mathit">0</span><span class="mord mathit">1</span><span class="mord mathit">1</span><span class="mord mathit">1</span><span class="mord mathit">0</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord"><span class="mord mathit">1</span><span class="mord mathit">1</span><span class="mord mathit">0</span><span class="mord mathit">0</span><span class="mord mathit">1</span><span class="mord mathit">0</span><span class="mord mathit">1</span><span class="mord mathit">1</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mclose">?</span></span></span></span></span></p>
<p>Decimal:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>17</mn><msub><mn>4</mn><mn>10</mn></msub><mo>+</mo><mn>20</mn><msub><mn>3</mn><mn>10</mn></msub><mo>=</mo><mn>37</mn><msub><mn>7</mn><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">
174_{10} + 203_{10} = 377_{10}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1</span><span class="mord">7</span><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">2</span><span class="mord">0</span><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">3</span><span class="mord">7</span><span class="mord"><span class="mord">7</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p>
<h2 id="unsigned-addition-u_w-and-overflow">Unsigned addition (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>+</mo><mi>w</mi><mi>u</mi></msubsup></mrow><annotation encoding="application/x-tex">+^{u}_{w}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.911392em;vertical-align:-0.247em;"></span><span class="mord"><span class="mbin">+</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>) and overflow</h2>
<ul>
<li>True/expected sum is the result of integer addition with unlimited space.</li>
<li>Actual sum is the result of unsigned addition with limited space. Discarding the carry out bit.</li>
<li>Discarding carry out bit has same effect as applying modular arithmetic</li>
</ul>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mo>=</mo><mi>U</mi><msubsup><mo>+</mo><mi>w</mi><mi>u</mi></msubsup><mi>v</mi><mo>=</mo><mo stretchy="false">(</mo><mi>u</mi><mo>+</mo><mi>v</mi><mo stretchy="false">)</mo><mspace></mspace><mspace width="1em"></mspace><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><msup><mn>2</mn><mi>w</mi></msup></mrow><annotation encoding="application/x-tex">
s = U+^{u}_{w}v = (u + v) \mod{2^{w}}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">s</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.9613919999999999em;vertical-align:-0.247em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin"><span class="mbin">+</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02691em;">w</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:1em;"></span></span><span class="base"><span class="strut" style="height:0.7143919999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">m</span><span class="mord mathrm">o</span><span class="mord mathrm">d</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02691em;">w</span></span></span></span></span></span></span></span></span></span></span></span></span></p>
<ul>
<li>Operands: w bits</li>
<li>True Sum: w+1 bits</li>
</ul>
<h2 id="closer-look-at-unsigned-addition-overflow">Closer look at unsigned addition overflow</h2>
<ul>
<li>w = 8 -&gt; [0..255]</li>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>25</mn><msub><mn>5</mn><mn>10</mn></msub><mo>=</mo><mn>1111111</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">255_{10} = 11111111_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">25</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1111111</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>
</li>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>9</mn><msub><mn>0</mn><mn>10</mn></msub><mo>=</mo><mn>0101101</mn><msub><mn>0</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">90_{10} = 01011010_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">9</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">0101101</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>
</li>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>4</mn><msub><mn>5</mn><mn>10</mn></msub><mo>=</mo><mn>0010110</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">45_{10} = 00101101_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">4</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">0010110</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>
</li>
</ul>
<h3 id="example-1">Example 1:</h3>
<p>Decimal (carry out the 1 in 135):</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>9</mn><msub><mn>0</mn><mn>10</mn></msub><mo>+</mo><mn>4</mn><msub><mn>5</mn><mn>10</mn></msub><mo>=</mo><mn>13</mn><msub><mn>5</mn><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">
90_{10} + 45_{10} = 135_{10}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">9</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">4</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1</span><span class="mord">3</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p>
<p>Binary (carry out the 1 in <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1000011</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">10000111_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1000011</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>):</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0101101</mn><msub><mn>0</mn><mn>2</mn></msub><mo>+</mo><mn>0010110</mn><msub><mn>1</mn><mn>2</mn></msub><mo>=</mo><mn>1000011</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">
01011010_{2} + 00101101_{2} = 10000111_{2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">0</span><span class="mord">1</span><span class="mord">0</span><span class="mord">1</span><span class="mord">1</span><span class="mord">0</span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">0</span><span class="mord">0</span><span class="mord">1</span><span class="mord">0</span><span class="mord">1</span><span class="mord">1</span><span class="mord">0</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1</span><span class="mord">0</span><span class="mord">0</span><span class="mord">0</span><span class="mord">0</span><span class="mord">1</span><span class="mord">1</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p>
<ul>
<li>True sum, w=8: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1000011</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">10000111_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1000011</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></li>
<li>Actual (overflowed) sum, w=8: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1000011</mn><msub><mn>1</mn><mn>2</mn></msub><mo>=</mo><mn>13</mn><msub><mn>5</mn><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">10000111_{2} = 135_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1000011</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">13</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></li>
</ul>
<h3 id="example-2">Example 2:</h3>
<p>Decimal (carry 1 to the 3 in 300):</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>25</mn><msub><mn>5</mn><mn>10</mn></msub><mo>+</mo><mn>4</mn><msub><mn>5</mn><mn>10</mn></msub><mo>=</mo><mn>30</mn><msub><mn>0</mn><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">
255_{10} + 45_{10} = 300_{10}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">2</span><span class="mord">5</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">4</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">3</span><span class="mord">0</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p>
<p>Binary (carry the 1 at the beginning of the result):</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1111111</mn><msub><mn>1</mn><mn>2</mn></msub><mo>+</mo><mn>0010110</mn><msub><mn>1</mn><mn>2</mn></msub><mo>=</mo><mn>10010110</mn><msub><mn>0</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">
11111111_{2} + 00101101_{2} = 100101100_{2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1</span><span class="mord">1</span><span class="mord">1</span><span class="mord">1</span><span class="mord">1</span><span class="mord">1</span><span class="mord">1</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">0</span><span class="mord">0</span><span class="mord">1</span><span class="mord">0</span><span class="mord">1</span><span class="mord">1</span><span class="mord">0</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1</span><span class="mord">0</span><span class="mord">0</span><span class="mord">1</span><span class="mord">0</span><span class="mord">1</span><span class="mord">1</span><span class="mord">0</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p>
<ul>
<li>True sum, w=9: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10010110</mn><msub><mn>0</mn><mn>2</mn></msub><mo>=</mo><mn>30</mn><msub><mn>0</mn><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">100101100_{2} = 300_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">10010110</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">30</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></li>
<li>Actual (overflowed) sum, w=8: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0010110</mn><msub><mn>0</mn><mn>2</mn></msub><mo>=</mo><mn>4</mn><msub><mn>4</mn><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">00101100_{2} = 44_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">0010110</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">4</span><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></li>
</ul>
<h2 id="comparing-integer-addition-with-overflow-effect-of-unsigned-addition-w--4">Comparing integer addition with Overflow: Effect of unsigned addition (w = 4)</h2>
<p>Annotation: with unlimited space:</p>
<p>A 3d chart showing two numbers being added.
a and b on the z and x axies, the sum on the y axis.
y goes to a maximum height of 32
(a = 15) + (b = 15) = (y = 30)
Annotation: With limited space (fixed-size memory):</p>
<p>A 3d chart showing two numbers being added.
a and b on the z and x axies, the sum on the y axis.
y goes to a maximum height of 15
(a = 15) + (b = 15) = (y = 14)</p>
<p>An overflow occurs when there is a carry out</p>
<p>For example: 15 (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>111</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">1111_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">111</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>) + 15 (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>111</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">1111_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">111</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>) = 30 (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1111</mn><msub><mn>0</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">11110_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1111</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>) as a true sum, and 14 (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1111</mn><msub><mn>0</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">11110_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1111</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>) as an actual sum.</p>
<h2 id="signed-addition-limited-space-ie-fixed-size-in-memory">Signed addition (limited space, i.e., fixed size in memory)</h2>
<p>What happens when we add two signed values:</p>
<p>w=8</p>
<p>a)</p>
<p>Binary:
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0011101</mn><msub><mn>1</mn><mn>2</mn></msub><mo>+</mo><mn>0101101</mn><msub><mn>0</mn><mn>2</mn></msub><mo>=</mo><mo stretchy="false">?</mo></mrow><annotation encoding="application/x-tex">00111011_{2} + 01011010_{2} = ?</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">0011101</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">0101101</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mclose">?</span></span></span></span></p>
<p>Decimal:
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>5</mn><msub><mn>9</mn><mn>10</mn></msub><mo>+</mo><mn>9</mn><msub><mn>0</mn><mn>10</mn></msub><mo>=</mo><mn>14</mn><msub><mn>9</mn><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">59_{10} + 90_{10} = 149_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">5</span><span class="mord"><span class="mord">9</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">9</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">14</span><span class="mord"><span class="mord">9</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p>
<p>b)</p>
<p>Binary: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1010111</mn><msub><mn>0</mn><mn>2</mn></msub><mo>+</mo><mn>1100101</mn><msub><mn>1</mn><mn>2</mn></msub><mo>=</mo><mo stretchy="false">?</mo></mrow><annotation encoding="application/x-tex">10101110_{2} + 11001011_{2} = ?</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1010111</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1100101</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mclose">?</span></span></span></span></p>
<p>Decimal: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>8</mn><msub><mn>2</mn><mn>10</mn></msub><mo>+</mo><mo>−</mo><mn>5</mn><msub><mn>3</mn><mn>10</mn></msub><mo>=</mo><mo>−</mo><mn>13</mn><msub><mn>5</mn><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">-82_{10} + -53_{10} = -135_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">−</span><span class="mord">8</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">−</span><span class="mord">5</span><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">−</span><span class="mord">13</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p>
<p>Observation: Unsigned and signed additions have identical behavior @ the bit level, i.e., their sum have the same bit-level representation, but their interpretation differs</p>
<h2 id="signed-addition-t_w-and-overflow">Signed addition (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>+</mo><mi>w</mi><mi>t</mi></msubsup></mrow><annotation encoding="application/x-tex">+^{t}_{w}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.040556em;vertical-align:-0.247em;"></span><span class="mord"><span class="mbin">+</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7935559999999999em;"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>) and overflow</h2>
<p>True sum would be the result of integer addition with unlimited space:
Actual sum is the result of signed addition with limited space:</p>
<p>Operands: w bits
True Sum: w+1 bits</p>
<p>After discarduing carry out bit: w bits (overflow)</p>
<ul>
<li>Discarding carry out bit has same effect as applying modular arithmetic</li>
</ul>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mo>=</mo><mi>u</mi><msubsup><mo>+</mo><mi>w</mi><mi>t</mi></msubsup><mi>v</mi><mo>=</mo><msub><mtext>U2T</mtext><mi>w</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>u</mi><mo>+</mo><mi>v</mi><mo stretchy="false">)</mo><mspace></mspace><mspace width="1em"></mspace><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><msup><mn>2</mn><mi>w</mi></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">
s = u +^{t}_{w} v = \text{U2T}_{w}[(u + v) \mod 2^{w}]
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">s</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0905559999999999em;vertical-align:-0.247em;"></span><span class="mord mathdefault">u</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin"><span class="mbin">+</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8435559999999999em;"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02691em;">w</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord text"><span class="mord">U2T</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02691em;">w</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">[</span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:1em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">m</span><span class="mord mathrm">o</span><span class="mord mathrm">d</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02691em;">w</span></span></span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></span></p>
<p>Negative overflow and positive overflows are possible.
Diagram showing negative overflows becoming positive, and positive overflows becoming negative.</p>
<h2 id="closer-look-at-signed-addition-overflow">Closer look at signed addition overflow</h2>
<ul>
<li>w = 8 -&gt; [-128..127]</li>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>9</mn><msub><mn>0</mn><mn>10</mn></msub><mo>=</mo><mn>0101101</mn><msub><mn>0</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">90_{10} = 01011010_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">9</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">0101101</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>
</li>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>4</mn><msub><mn>5</mn><mn>10</mn></msub><mo>=</mo><mn>0010110</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">45_{10} = 00101101_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">4</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">0010110</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>
</li>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo>−</mo><mn>4</mn><msub><mn>5</mn><mn>10</mn></msub><mo>=</mo><mn>1101001</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">-45_{10} = 11010011_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">−</span><span class="mord">4</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1101001</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>
</li>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo>−</mo><mn>9</mn><msub><mn>0</mn><mn>10</mn></msub><mo>=</mo><mn>1010011</mn><msub><mn>0</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">-90_{10} = 10100110_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">−</span><span class="mord">9</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1010011</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>
</li>
</ul>
<h3 id="example-1-1">Example 1:</h3>
<p>Decimal: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>9</mn><msub><mn>0</mn><mn>10</mn></msub><mo>+</mo><mn>4</mn><msub><mn>5</mn><mn>10</mn></msub><mo>=</mo><mn>13</mn><msub><mn>5</mn><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">90_{10} + 45_{10} = 135_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">9</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">4</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">13</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p>
<p>Binary: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0101101</mn><msub><mn>0</mn><mn>2</mn></msub><mo>+</mo><mn>0010110</mn><msub><mn>1</mn><mn>2</mn></msub><mo>=</mo><mn>01000011</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">01011010_{2} + 00101101_{2} = 010000111_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">0101101</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">0010110</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">01000011</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p>
<p>The binary result is -121</p>
<h3 id="example-2-1">Example 2:</h3>
<p>Decimal: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>9</mn><msub><mn>0</mn><mn>10</mn></msub><mo>+</mo><mo>−</mo><mn>4</mn><msub><mn>5</mn><mn>10</mn></msub><mo>=</mo><mo>−</mo><mn>13</mn><msub><mn>5</mn><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">-90_{10} + -45_{10} = -135_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">−</span><span class="mord">9</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">−</span><span class="mord">4</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">−</span><span class="mord">13</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p>
<p>Binary: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1010011</mn><msub><mn>0</mn><mn>2</mn></msub><mo>+</mo><mn>1101001</mn><msub><mn>1</mn><mn>2</mn></msub><mo>=</mo><mn>10111100</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">10100110_{2} + 11010011_{2} = 101111001_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1010011</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1101001</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">10111100</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p>
<p>Binary result is 121</p>
<h3 id="example-3">Example 3:</h3>
<p>Decimal: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>9</mn><msub><mn>0</mn><mn>10</mn></msub><mo>+</mo><mn>4</mn><msub><mn>5</mn><mn>10</mn></msub><mo>=</mo><mo>−</mo><mn>4</mn><msub><mn>5</mn><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">-90_{10} + 45_{10} = -45_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">−</span><span class="mord">9</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">4</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">−</span><span class="mord">4</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p>
<p>Binary: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1010011</mn><msub><mn>0</mn><mn>2</mn></msub><mo>+</mo><mn>0010110</mn><msub><mn>1</mn><mn>2</mn></msub><mo>=</mo><mn>01101001</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">10100110_{2} + 00101101_{2} = 011010011_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1010011</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">0010110</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">01101001</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p>
<h3 id="example-4">Example 4:</h3>
<p>Decimal: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>9</mn><msub><mn>0</mn><mn>10</mn></msub><mo>+</mo><mo>−</mo><mn>4</mn><msub><mn>5</mn><mn>10</mn></msub><mo>=</mo><mn>4</mn><msub><mn>5</mn><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">90_{10} + -45_{10} = 45_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">9</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">−</span><span class="mord">4</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">4</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p>
<p>Binary: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0101101</mn><msub><mn>0</mn><mn>2</mn></msub><mo>+</mo><mn>1101001</mn><msub><mn>1</mn><mn>2</mn></msub><mo>=</mo><mn>10010110</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">01011010_{2} + 11010011_{2} = 100101101_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">0101101</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1101001</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">10010110</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p>
<p>A chart showing the relationship between true sum and actual (overflowed) sum.
Actual sum has a possible value between 127 to -128.
A true sum, however has a range between 255 and -256.
As your true sum goes further down from -128, its actual sum becomes lower as well. Starting at -129 = 127.
As you true sum goes futher up from 127, the actual sum also rises, satrting with 128 = -128.</p>
<h2 id="visualizing-signed-addition-overflow-w--4">Visualizing signed addition overflow (w = 4)</h2>
<p>A 3D chart which has its x axis go from -8 to +7, its z axis go from -8 to +6, and a y axis which goes from :wq</p>
<p>Positive Overflow</p>
<p>For example: 7 (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>011</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">0111_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">011</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>) + 1 (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>000</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">0001_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">000</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>) = 8 (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>100</mn><msub><mn>0</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">1000_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">100</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>) is the true sum and = -8 (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>100</mn><msub><mn>0</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">1000_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">100</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>) is the actual sum</p>
<h2 id="what-about-subtraction---addition">What about subtraction? -&gt; Addition</h2>
<p>x + (-x) = 0</p>
<ul>
<li>Subtracting a number is equivalent to adding its additive inverse
<ul>
<li>Instead of subtracting a positive number, we could add its negative version:</li>
</ul>
</li>
</ul>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>107</mn><mo>−</mo><mn>118</mn><mo>=</mo><mo>−</mo><mn>11</mn></mrow><annotation encoding="application/x-tex">
107 - 118 = -11
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1</span><span class="mord">0</span><span class="mord">7</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">1</span><span class="mord">8</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mord">1</span></span></span></span></span></p>
<p>becomes:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>107</mn><mo>+</mo><mo stretchy="false">(</mo><mo>−</mo><mn>118</mn><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>18</mn></mrow><annotation encoding="application/x-tex">
107 + (-118) = -18
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1</span><span class="mord">0</span><span class="mord">7</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mord">1</span><span class="mord">8</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mord">8</span></span></span></span></span></p>
<ul>
<li>Let‘s try:</li>
</ul>
<p>Decimal:</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>10</mn><msub><mn>7</mn><mn>10</mn></msub><mo>−</mo><mn>11</mn><msub><mn>8</mn><mn>10</mn></msub><mo>=</mo><mo>−</mo><mn>11</mn></mrow><annotation encoding="application/x-tex">107_{10} - 118_{10} = -11</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">10</span><span class="mord"><span class="mord">7</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">11</span><span class="mord"><span class="mord">8</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">−</span><span class="mord">11</span></span></span></span></span>
<p>Binary:</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>0110101</mn><msub><mn>1</mn><mn>2</mn></msub><mo>−</mo><mn>0111011</mn><msub><mn>0</mn><mn>2</mn></msub><mo>=</mo></mrow><annotation encoding="application/x-tex">01101011_{2} - 01110110_{2} =</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">0110101</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">0111011</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span></span></span></span></span>
<p>Binary subtraction by addition:</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>0110101</mn><msub><mn>1</mn><mn>2</mn></msub><mo>+</mo><mn>1000101</mn><msub><mn>0</mn><mn>2</mn></msub><mo>=</mo><mn>1111010</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">01101011_{2} + 10001010_{2} = 11110101_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">0110101</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1000101</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1111010</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>
<p>Binary subtraction by addition is equal to -11</p>
<p>Check: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>128</mn><mo>+</mo><mn>64</mn><mo>+</mo><mn>32</mn><mo>+</mo><mn>16</mn><mo>+</mo><mn>4</mn><mo>+</mo><mn>1</mn><mo>=</mo><mo>−</mo><mn>1</mn><msub><mn>1</mn><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">-128 + 64 + 32 + 16 + 4 + 1 = -11_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">−</span><span class="mord">128</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">64</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">32</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">16</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p>
<p>T2B(X) conversion:</p>
<ol>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mtext> </mtext><mo stretchy="false">(</mo><mtext>U2B</mtext><mo stretchy="false">(</mo><mi mathvariant="normal">∣</mi><mi>X</mi><mi mathvariant="normal">∣</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">(~(\text{U2B}(|X|)))+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mspace nobreak"> </span><span class="mopen">(</span><span class="mord text"><span class="mord">U2B</span></span><span class="mopen">(</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">∣</span><span class="mclose">)))</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></span>
</li>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mtext> </mtext><mo stretchy="false">(</mo><mtext>U2B</mtext><mo stretchy="false">(</mo><mi mathvariant="normal">∣</mi><mo>−</mo><mn>118</mn><mi mathvariant="normal">∣</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">(~(\text{U2B}(|-118|)))+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mspace nobreak"> </span><span class="mopen">(</span><span class="mord text"><span class="mord">U2B</span></span><span class="mopen">(</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">118∣</span><span class="mclose">)))</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></span>
</li>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mtext> </mtext><mo stretchy="false">(</mo><mtext>U2B</mtext><mo stretchy="false">(</mo><mn>118</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">(~(\text{U2B}(118)))+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mspace nobreak"> </span><span class="mopen">(</span><span class="mord text"><span class="mord">U2B</span></span><span class="mopen">(</span><span class="mord">118</span><span class="mclose">)))</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></span>
</li>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mo>∼</mo><mo stretchy="false">(</mo><mn>0111011</mn><msub><mn>0</mn><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">(\sim(01110110_{2}))+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0111011</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></span>
</li>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mn>1000100</mn><msub><mn>1</mn><mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">(10001001_{2})+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1000100</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></span>
</li>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>1000101</mn><msub><mn>0</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">10001010_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1000101</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>
</li>
</ol>
<h2 id="multiplication-u_w-t_w-and-overflow">Multiplication (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∗</mo><mi>w</mi><mi>u</mi></msubsup><mo separator="true">,</mo><msubsup><mo>∗</mo><mi>w</mi><mi>t</mi></msubsup></mrow><annotation encoding="application/x-tex">*^{u}_{w}, *^{t}_{w}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.040556em;vertical-align:-0.247em;"></span><span class="mord"><span class="mbin">∗</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mbin">∗</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7935559999999999em;"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>) and overflow</h2>
<p>True product would be the result of integer multiplication with unlimited space: expected product
Actual product is the result of multiplication with limited space.</p>
<ul>
<li>Operands: w bits</li>
<li>True Product: 2w bits <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo>×</mo><mi>v</mi></mrow><annotation encoding="application/x-tex">u \times v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span></span></li>
<li>
<p>Discard: w bits</p>
</li>
<li>Discarding high order w bits has same effect as applying modular arithmetic</li>
</ul>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>p</mi><mo>=</mo><mi>u</mi><msubsup><mo>∗</mo><mi>w</mi><mi>u</mi></msubsup><mi>v</mi><mo>=</mo><mo stretchy="false">(</mo><mi>u</mi><mo>×</mo><mi>v</mi><mo stretchy="false">)</mo><mspace></mspace><mspace width="1em"/><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><msup><mn>2</mn><mi>w</mi></msup></mrow><annotation encoding="application/x-tex">p = u *^{u}_{w}v = (u \times v) \mod 2^{w}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.9613919999999999em;vertical-align:-0.247em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin"><span class="mbin">∗</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:1em;"></span></span><span class="base"><span class="strut" style="height:0.7143919999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span></span></span></span></span></span></span></span></span></span>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>p</mi><mo>=</mo><mi>u</mi><msubsup><mo>∗</mo><mi>w</mi><mi>t</mi></msubsup><mi>v</mi><mo>=</mo><msub><mtext>U2T</mtext><mi>w</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>u</mi><mo>×</mo><mi>v</mi><mo stretchy="false">)</mo><mspace></mspace><mspace width="1em"/><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><msup><mn>2</mn><mi>w</mi></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">p = u *^{t}_{w}v = \text{U2T}_{w}[(u \times v) \mod 2^{w}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0905559999999999em;vertical-align:-0.247em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin"><span class="mbin">∗</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8435559999999999em;"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord text"><span class="mord">U2T</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">[(</span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:1em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></span>
<ul>
<li>Example: w = 4</li>
</ul>
<p>Decimal:</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mn>5</mn><mn>10</mn></msub><mo>×</mo><msub><mn>5</mn><mn>10</mn></msub><mo>=</mo><mn>2</mn><msub><mn>5</mn><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">5_{10} \times 5_{10} = 25_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">2</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>
<p>Binary:</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>010</mn><msub><mn>1</mn><mn>2</mn></msub><mo>×</mo><mn>010</mn><msub><mn>1</mn><mn>2</mn></msub><mo>=</mo><menclose notation="horizontalstrike"><mn>001</mn></menclose><mn>100</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">0101_{2} \times 0101_{2} = \sout{001}1001_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">010</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">010</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.64444em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">001</span></span></span><span style="top:-3.2155em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy sout"></span></span></span></span></span></span><span class="mord">100</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>
<h2 id="multiplication-with-power-of-2-versus-shifting">Multiplication with power-of-2 versus shifting</h2>
<ul>
<li>If <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>×</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x \times y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><msup><mn>2</mn><mi>k</mi></msup></mrow><annotation encoding="application/x-tex">y = 2^{k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.849108em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span></span> then <code class="language-plaintext highlighter-rouge">x &lt;&lt; k</code>
<ul>
<li>For both signed and unsigned</li>
</ul>
</li>
<li>Example:
<ul>
<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>×</mo><mn>8</mn><mo>=</mo><mi>x</mi><mo>×</mo><msup><mn>2</mn><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">x \times 8 = x \times 2^{3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">8</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span></span> = <code class="language-plaintext highlighter-rouge">x &lt;&lt; 3</code></li>
<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>×</mo><mn>24</mn><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo>×</mo><mn>25</mn><mo stretchy="false">)</mo><mtext>–</mtext><mo stretchy="false">(</mo><mi>x</mi><mo>×</mo><mn>23</mn><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo>×</mo><mn>32</mn><mo stretchy="false">)</mo><mtext>–</mtext><mo stretchy="false">(</mo><mi>x</mi><mo>×</mo><mn>8</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \times 24 = (x \times 25) – (x \times 23) = (x \times 32) – (x \times 8)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">24</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">25</span><span class="mclose">)</span><span class="mord">–</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">23</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">32</span><span class="mclose">)</span><span class="mord">–</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">8</span><span class="mclose">)</span></span></span></span> = (x « 5) – (x « 3) (decompose 24 in powers of 2 =&gt; 32 – 8)</li>
</ul>
</li>
<li>Most machines shift and add faster than multiply
<ul>
<li>We’ll soon see that compiler generates this code automatically
17</li>
</ul>
</li>
</ul>
<h2 id="summary">Summary</h2>
<ul>
<li>Demo of size and sign conversion in C: code and results posted!</li>
<li>Addition:
<ul>
<li>Unsigned/signed:
<ul>
<li>Behave the same way at the bit level</li>
<li>Interpretation of resulting bit vector (sum) may differ</li>
</ul>
</li>
<li>Unsigned addition -&gt; may overflow, i.e., (w+1)th bit is set
<ul>
<li>If so, then actual sum obtained =&gt; <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo><mspace></mspace><mspace width="0.6666666666666666em"/><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><msup><mn>2</mn><mi>w</mi></msup></mrow><annotation encoding="application/x-tex">(x + y) \mod 2^{w}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.6666666666666666em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span></span></span></span></span></span></span></span></span></li>
</ul>
</li>
<li>Signed addition -&gt; may overflow, i.e., (w+1)th bit is set
<ul>
<li>If so, then true sum may be too +ve -&gt; positive overflow OR too -ve -&gt; negative overflow</li>
<li>Then actual sum obtained =&gt; <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>U2T</mtext><mi>w</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo><mspace></mspace><mspace width="0.6666666666666666em"/><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><msup><mn>2</mn><mi>w</mi></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\text{U2T}_{w}[(x + y) \mod 2^{w}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord text"><span class="mord">U2T</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">[(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.6666666666666666em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></li>
</ul>
</li>
</ul>
</li>
<li>Subtraction
<ul>
<li>Becomes an addition where negative operands are transformed into their additive inverse (in two’s complement)</li>
</ul>
</li>
<li>Multiplication:
<ul>
<li>Unsigned: actual product obtained -&gt; <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>×</mo><mi>y</mi><mo stretchy="false">)</mo><mspace></mspace><mspace width="0.6666666666666666em"/><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><msup><mn>2</mn><mi>w</mi></msup></mrow><annotation encoding="application/x-tex">(x \times y) \mod 2^{w}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.6666666666666666em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span></span></span></span></span></span></span></span></span></li>
<li>Signed: actual product obtained -&gt; $$\text{U2T}_{w}[(x \times y) \mod 2^{w}]</li>
<li>Can be replaced by additions and shifts</li>
</ul>
</li>
</ul>
<h2 id="next-lecture">Next lecture</h2>
<ul>
<li>Representing data in memory – Most of this is review
<ul>
<li>“Under the Hood” - Von Neumann architecture</li>
<li>Bits and bytes in memory
<ul>
<li>How to diagram memory -&gt; Used in this course and other references</li>
<li>How to represent series of bits -&gt; In binary, in hexadecimal (conversion)</li>
<li>What kind of information (data) do series of bits represent -&gt; Encoding scheme</li>
<li>Order of bytes in memory -&gt; Endian</li>
</ul>
</li>
<li>Bit manipulation – bitwise operations
<ul>
<li>Boolean algebra + Shifting</li>
</ul>
</li>
</ul>
</li>
<li>Representing integral numbers in memory
<ul>
<li>Unsigned and signed</li>
<li>Converting, expanding and truncating</li>
<li>Arithmetic operations</li>
</ul>
</li>
<li>Representing real numbers in memory
<ul>
<li>IEEE floating point representation</li>
<li>Floating point in C – casting, rounding, addition, …</li>
</ul>
</li>
</ul>
<p>We’ll illustrate what we covered today by having a demo!</p>
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</footer>
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#include <stdio.h>
#include <stdlib.h>
#include <limits.h>
typedef unsigned char *byte_pointer;
void show_bytes(byte_pointer start, size_t len) {
int i;
for ( i = len-1; i >= 0; i--)
printf(" %.2x", start[i]);
return;
}
int main(int argc, char *argv[]) {
if ( argc < 2 ) {
printf("Forgot the argment! Try again!\n");
return 1;
}
short aShort = atoi(argv[1]);
printf("\nAs a short data type, the variable aShort has the value %hd (hex: 0x", aShort);
show_bytes((byte_pointer) &aShort, sizeof(short));
printf(")\n\n");
/* Convert - size - implicitly */
printf("Converting SIZE implicitly: short -> integer *** Sign extension ***\n");
printf("This is done by issuing the statement: int anInt = aShort;\n");
int anInt = aShort;
printf("As a int data type, the variable anInt has the value %d (hex: 0x", anInt);
show_bytes((byte_pointer) &anInt, sizeof(int));
printf(")\n\n");
/* Convert - size - implicitly */
printf("Converting SIZE implicitly: short -> char *** Truncation ***\n");
printf("This is done by issuing the statement: signed char aChar = aShort;\n");
signed char aChar = aShort;
printf("As a char data type, the variable aChar has the value %hhi (hex: 0x", aChar);
show_bytes((byte_pointer) &aChar, sizeof(signed char));
printf(")\n\n");
/* Convert - sign - implicitly*/
printf("Converting SIGN implicitly: short -> unsigned short\n");
printf("This is done by issuing the statement: unsigned short aUShort = aShort;\n");
unsigned short aUShort = aShort;
printf("As an unsigned short data type, the variable aUShort has the value %hu (hex: 0x", aUShort);
show_bytes((byte_pointer) &aUShort, sizeof(unsigned short));
printf(")\n\n");
/* Convert - sign - implicitly*/
printf("Converting SIGN implicitly: unsigned short -> short\n");
printf("This is done by issuing the statement: short aShort1 = aUShort;\n");
short aShort1 = aUShort;
printf("As a signed short data type, the variable aShort1 has the value %hi (hex: 0x", (signed short) aShort1);
show_bytes((byte_pointer) &aShort1, sizeof(signed short));
printf(")\n\n");
/* Convert - both: 1) size, 2) sign */
printf("Converting both SIZE and SIGN: short -> unsigned int\n");
printf("This is done by issuing the statement: unsigned aUInt = aShort;\n");
unsigned aUInt = aShort;
printf("As an unsigned int data type, the variable aUInt has the value %u (hex: 0x", aUInt);
show_bytes((byte_pointer) &aUInt, sizeof(unsigned));
printf(")\n\n");
/* One step at a time */
printf("One step at a time - First conversion is SIZE: (int) aShort = %d\n", (int) aShort);
printf("One step at a time - Second conversion is SIGN: (unsigned) (int) aShort = %u\n\n", (unsigned) (int) aShort);
/* Reverse the process and see what happens ... */
printf("What if ... First conversion is SIGN: (unsigned short) aShort = %hu\n", (unsigned short) aShort);
printf("What if ... Second conversion is SIZE: (unsigned int) (unsigned short) aShort = %d\n\n", (unsigned int) (unsigned short) aShort);
/* Convert - both: 1) size, 2) sign */
printf("Converting both SIZE and SIGN: short -> unsigned char\n");
printf("This is done by issuing the statement: unsigned char anUChar = aShort;\n");
unsigned char anUChar = aShort;
printf("As an unsigned char data type, the variable anUChar has the value %hhu (hex: 0x", anUChar);
show_bytes((byte_pointer) &anUChar, sizeof(unsigned char));
printf(")\n\n");
/* One step at a time */
printf("One step at a time - First conversion is SIZE: (signed char) aShort = %hhi\n", (signed char) aShort);
printf("One step at a time - Second conversion is SIGN: (unsigned char) (signed char) aShort = %hhu\n\n", (unsigned char) (signed char) aShort);
/* Reverse the process and see what happens ... */
printf("What if ... First conversion is SIGN: (unsigned short) aShort = %hu\n", (unsigned short) aShort);
printf("What if ... Second conversion is SIZE: (unsigned char) (unsigned short) aShort = %hhu\n\n", (unsigned char) (unsigned short) aShort);
return 0;
}

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$ ./Demo 12345
As a short data type, the variable aShort has the value 12345 (hex: 0x 30 39)
Converting SIZE implicitly: short -> integer *** Sign extension ***
This is done by issuing the statement: int anInt = aShort;
As a int data type, the variable anInt has the value 12345 (hex: 0x 00 00 30 39)
Converting SIZE implicitly: short -> char *** Truncation ***
This is done by issuing the statement: signed char aChar = aShort;
As a char data type, the variable aChar has the value 57 (hex: 0x 39)
Converting SIGN implicitly: short -> unsigned short
This is done by issuing the statement: unsigned short aUShort = aShort;
As an unsigned short data type, the variable aUShort has the value 12345 (hex: 0x 30 39)
Converting SIGN implicitly: unsigned short -> short
This is done by issuing the statement: short aShort1 = aUShort;
As a signed short data type, the variable aShort1 has the value 12345 (hex: 0x 30 39)
Converting both SIZE and SIGN: short -> unsigned int
This is done by issuing the statement: unsigned aUInt = aShort;
As an unsigned int data type, the variable aUInt has the value 12345 (hex: 0x 00 00 30 39)
One step at a time - First conversion is SIZE: (int) aShort = 12345
One step at a time - Second conversion is SIGN: (unsigned) (int) aShort = 12345
What if ... First conversion is SIGN: (unsigned short) aShort = 12345
What if ... Second conversion is SIZE: (unsigned int) (unsigned short) aShort = 12345
Converting both SIZE and SIGN: short -> unsigned char
This is done by issuing the statement: unsigned char anUChar = aShort;
As an unsigned char data type, the variable anUChar has the value 57 (hex: 0x 39)
One step at a time - First conversion is SIZE: (signed char) aShort = 57
One step at a time - Second conversion is SIGN: (unsigned char) (signed char) aShort = 57
What if ... First conversion is SIGN: (unsigned short) aShort = 12345
What if ... Second conversion is SIZE: (unsigned char) (unsigned short) aShort = 57
----------------------------------------------------------------------------------------------
$ ./Demo -12345
As a short data type, the variable aShort has the value -12345 (hex: 0x cf c7)
Converting SIZE implicitly: short -> integer *** Sign extension ***
This is done by issuing the statement: int anInt = aShort;
As a int data type, the variable anInt has the value -12345 (hex: 0x ff ff cf c7)
Converting SIZE implicitly: short -> char *** Truncation ***
This is done by issuing the statement: signed char aChar = aShort;
As a char data type, the variable aChar has the value -57 (hex: 0x c7)
Converting SIGN implicitly: short -> unsigned short
This is done by issuing the statement: unsigned short aUShort = aShort;
As an unsigned short data type, the variable aUShort has the value 53191 (hex: 0x cf c7)
Converting SIGN implicitly: unsigned short -> short
This is done by issuing the statement: short aShort1 = aUShort;
As a signed short data type, the variable aShort1 has the value -12345 (hex: 0x cf c7)
Converting both SIZE and SIGN: short -> unsigned int
This is done by issuing the statement: unsigned aUInt = aShort;
As an unsigned int data type, the variable aUInt has the value 4294954951 (hex: 0x ff ff cf c7)
One step at a time - First conversion is SIZE: (int) aShort = -12345
One step at a time - Second conversion is SIGN: (unsigned) (int) aShort = 4294954951
What if ... First conversion is SIGN: (unsigned short) aShort = 53191
What if ... Second conversion is SIZE: (unsigned int) (unsigned short) aShort = 53191
Converting both SIZE and SIGN: short -> unsigned char
This is done by issuing the statement: unsigned char anUChar = aShort;
As an unsigned char data type, the variable anUChar has the value 199 (hex: 0x c7)
One step at a time - First conversion is SIZE: (signed char) aShort = -57
One step at a time - Second conversion is SIGN: (unsigned char) (signed char) aShort = 199
What if ... First conversion is SIGN: (unsigned short) aShort = 53191
What if ... Second conversion is SIZE: (unsigned char) (unsigned short) aShort = 199

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<!DOCTYPE html>
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<title> | tait.tech</title>
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<h1 id="cmpt-295">CMPT 295</h1>
<ul>
<li>Unit - Data Representation
<ul>
<li>Lecture 6 – Representing fractional numbers in memory</li>
<li>IEEE floating point representation – cont’d</li>
</ul>
</li>
</ul>
<h2 id="have-you-heard-of-that-new-band-1023-megabytes">Have you heard of that new band “1023 Megabytes”?</h2>
<p>They’re pretty good,
but they don’t have a gig just yet.
😭</p>
<h2 id="last-lecture">Last Lecture</h2>
<ul>
<li>Representing integral numbers in memory
<ul>
<li>Can encode a small range of values exactly (in 1, 2, 4, 8 bytes)
<ul>
<li>For example: We can represent the values -128 to 127 exactly in 1 byte using a signed char in C</li>
</ul>
</li>
</ul>
</li>
<li>Representing fractional numbers in memory
<ol>
<li>Positional notation has some advantages, but also disadvantages -&gt; so not used!</li>
<li>IEEE floating point representation: can encode a much larger range of e.g., single precision: [10-38..1038] values approximately (in 4 or 8 bytes)</li>
</ol>
</li>
<li>Overview of IEEE floating point representation
<ul>
<li>Precision options (float 32-bit, double 64-bit)</li>
<li>V = (-1)s x M x 2E</li>
<li>s –&gt; sign bit</li>
<li>exp encodes E (but != E)</li>
<li>frac encodes M (but != M)</li>
</ul>
</li>
</ul>
<p>We interpret the bit vector (expressed in IEEE floating point encoding) stored in memory using this equation.</p>
<h2 id="todays-menu">Today’s Menu</h2>
<ul>
<li>Representing data in memory – Most of this is review
<ul>
<li>“Under the Hood” - Von Neumann architecture</li>
<li>Bits and bytes in memory
<ul>
<li>How to diagram memory -&gt; Used in this course and other references</li>
<li>How to represent series of bits -&gt; In binary, in hexadecimal (conversion)</li>
<li>What kind of information (data) do series of bits represent -&gt; Encoding scheme</li>
<li>Order of bytes in memory -&gt; Endian</li>
</ul>
</li>
<li>Bit manipulation – bitwise operations
<ul>
<li>Boolean algebra + Shifting</li>
</ul>
</li>
</ul>
</li>
<li>Representing integral numbers in memory
<ul>
<li>Unsigned and signed</li>
<li>Converting, expanding and truncating</li>
<li>Arithmetic operations</li>
</ul>
</li>
<li>Representing real numbers in memory
4
<ul>
<li>IEEE floating point representation</li>
<li>Floating point in C – casting, rounding, addition, …</li>
</ul>
</li>
</ul>
<h2 id="ieee-floating-point-representation-three-kinds-of-values">IEEE Floating Point Representation Three “kinds” of values</h2>
<p>We interpret the bit vector
(expressed in IEEE floating point encoding) stored in memory using this equation:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>s</mi></msup><mi>M</mi><msup><mn>2</mn><mi>E</mi></msup></mrow><annotation encoding="application/x-tex">
V = (-1)^{s} M 2^{E}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.1413309999999999em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em;">E</span></span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>Bit breakdown–<code class="language-plaintext highlighter-rouge">exp</code> and <code class="language-plaintext highlighter-rouge">frac</code> interpreted as unsigned:</p>
<ul>
<li>s = 1 bit</li>
<li>exp = k bits
<ol>
<li>If <code class="language-plaintext highlighter-rouge">exp</code> != 0 and <code class="language-plaintext highlighter-rouge">exp</code> != 11…11 (exp range: [0000001…11111110]). Equations:
<ul>
<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>=</mo><mtext>exp</mtext><mo>−</mo><mtext>bias</mtext></mrow><annotation encoding="application/x-tex">E = \text{exp} - \text{bias}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.7777700000000001em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord">exp</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">bias</span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>bias</mtext><mo>=</mo><msup><mn>2</mn><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\text{bias} = 2^{k-1} - 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">bias</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.9324379999999999em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></li>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>M</mi><mo>=</mo><mn>1</mn><mo>+</mo><mtext>frac</mtext></mrow><annotation encoding="application/x-tex">M = 1 + \text{frac}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">frac</span></span></span></span></span></span>
</li>
</ul>
</li>
<li>If <code class="language-plaintext highlighter-rouge">exp</code> = 00…00 (all 0’s) =&gt; denormalized. Equations:
<ul>
<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>=</mo><mn>1</mn><mo>−</mo><mtext>bias</mtext></mrow><annotation encoding="application/x-tex">E = 1 - \text{bias}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">bias</span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>bias</mtext><mo>=</mo><msup><mn>2</mn><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\text{bias} = 2^{k-1} - 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">bias</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.9324379999999999em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></li>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>M</mi><mo>=</mo><mtext>frac</mtext></mrow><annotation encoding="application/x-tex">M = \text{frac}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">frac</span></span></span></span></span></span>
</li>
</ul>
</li>
<li>If <code class="language-plaintext highlighter-rouge">exp</code> 11…11 (all 1’s) =&gt; special cases.
<ul>
<li>Case 1: <code class="language-plaintext highlighter-rouge">frac</code> = 000…0</li>
<li>Case 2: <code class="language-plaintext highlighter-rouge">frac</code> != 000…0</li>
</ul>
</li>
</ol>
</li>
</ul>
<h2 id="ieee-floating-point-representation---normalized">IEEE floating point representation - normalized</h2>
<p>Numerical Form: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><mo stretchy="false">(</mo><mtext>–</mtext><mn>1</mn><msup><mo stretchy="false">)</mo><mi>s</mi></msup><mi>M</mi><msup><mn>2</mn><mi>E</mi></msup></mrow><annotation encoding="application/x-tex">V = (–1)^{s} M 2^{E}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">–1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">s</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>Bit breakdown:</p>
<ul>
<li>s = 1 bit</li>
<li><code class="language-plaintext highlighter-rouge">exp</code> = k bits
<ul>
<li>If <code class="language-plaintext highlighter-rouge">exp</code> != 0 and <code class="language-plaintext highlighter-rouge">exp</code> != 11…11 (<code class="language-plaintext highlighter-rouge">exp</code> range: [00000001…11111110]) =&gt; normalized. Equations:
<ul>
<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>=</mo><mtext>exp</mtext><mo>−</mo><mtext>bias</mtext></mrow><annotation encoding="application/x-tex">E = \text{exp} - \text{bias}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.7777700000000001em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord">exp</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">bias</span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>bias</mtext><mo>=</mo><msup><mn>2</mn><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\text{bias} = 2^{k-1} - 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">bias</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.9324379999999999em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></li>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>M</mi><mo>=</mo><mn>1</mn><mo>+</mo><mtext>frac</mtext></mrow><annotation encoding="application/x-tex">M = 1 + \text{frac}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">frac</span></span></span></span></span></span>
</li>
</ul>
</li>
</ul>
</li>
</ul>
<p>Why is <code class="language-plaintext highlighter-rouge">E</code> biased?</p>
<p>Using single precision as an example (s = 1 bit, exp = 8 bits, frac = 23 bits):</p>
<ul>
<li>(exp range: [00000001 .. 11111110]) =&gt; <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><msub><mn>1</mn><mn>10</mn></msub><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mn>25</mn><msub><mn>4</mn><mn>10</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[1_{10}...254_{10}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">...25</span><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></li>
<li>If <code class="language-plaintext highlighter-rouge">E</code> is not biased (i.e. E = exp), then <code class="language-plaintext highlighter-rouge">E</code> range <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><msub><mn>1</mn><mn>10</mn></msub><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mn>25</mn><msub><mn>4</mn><mn>10</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[1_{10} ... 254_{10}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">...25</span><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></li>
<li><code class="language-plaintext highlighter-rouge">V</code> range [<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>1</mn></msup></mrow><annotation encoding="application/x-tex">2^{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span>…<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>254</mn></msup></mrow><annotation encoding="application/x-tex">2^{254}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">254</span></span></span></span></span></span></span></span></span></span></span></span>] = [2…<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≈</mo><mn>2.89</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mn>76</mn></msup></mrow><annotation encoding="application/x-tex">\approx 2.89 \times 10^{76}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.48312em;vertical-align:0em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">2.89</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">76</span></span></span></span></span></span></span></span></span></span></span></span>] (so cannot express numbers &lt; 2)</li>
<li>By biasing <code class="language-plaintext highlighter-rouge">E</code> (i.e. E = exp - bias), then <code class="language-plaintext highlighter-rouge">E</code> range: [1-127…254-127] == [-126…127] (since k = 8, bias = <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mrow><mn>8</mn><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">2^{8-1} - 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.897438em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span> = 127)</li>
<li><code class="language-plaintext highlighter-rouge">V</code> range: [<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mrow><mo>−</mo><mn>126</mn></mrow></msup></mrow><annotation encoding="application/x-tex">2^{-126}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">126</span></span></span></span></span></span></span></span></span></span></span></span>…<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>127</mn></msup></mrow><annotation encoding="application/x-tex">2^{127}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">127</span></span></span></span></span></span></span></span></span></span></span></span>] = [<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≈</mo><mn>1.18</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mrow><mo>−</mo><mn>38</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\approx 1.18 \times 10^{-38}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.48312em;vertical-align:0em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1.18</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">38</span></span></span></span></span></span></span></span></span></span></span></span> … <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≈</mo><mn>1.7</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mn>38</mn></msup></mrow><annotation encoding="application/x-tex">\approx 1.7 \times 10^{38}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.48312em;vertical-align:0em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1.7</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">38</span></span></span></span></span></span></span></span></span></span></span></span> (so can now express very small (and very large) numbers)</li>
<li>Why adding 1 to <code class="language-plaintext highlighter-rouge">frac</code>? Because the number (or value) V is first normalized before it is converted.</li>
</ul>
<h2 id="review-scientific-notation-and-normalization">Review: Scientific Notation and normalization</h2>
<ul>
<li>From Wikipedia:
<ul>
<li>Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form (as they are long strings of digits).</li>
<li>In scientific notation, nonzero numbers are written in the form +/- M × 10n</li>
<li>In normalized notation, the exponent n is chosen such that the absolute value of the significand M is at least 1 (M = 1.0) but less than the base
<ul>
<li>M range for base 10 =&gt; [1.0 .. 10.0 – ε ]</li>
<li>M range for base 2 =&gt; [1.0 .. 2.0 – ε ]</li>
</ul>
</li>
</ul>
</li>
<li>
<p>Examples:</p>
<ul>
<li>A proton’s mass is 0.0000000000000000000000000016726 kg -&gt; <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1.6726</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mrow><mtext>−</mtext><mn>27</mn></mrow></msup></mrow><annotation encoding="application/x-tex">1.6726 \times 10^{−27}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1.6726</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−27</span></span></span></span></span></span></span></span></span></span></span></span> kg</li>
<li>Speed of light is 299,792,458 m/s -&gt; <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2.99792</mn><mo separator="true">,</mo><mn>458</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mn>8</mn></msup></mrow><annotation encoding="application/x-tex">2.99792,458 \times 10^{8}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em;"></span><span class="mord">2.99792</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">458</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span></span> m/s</li>
</ul>
</li>
</ul>
<p>Syntax of normalized notation:</p>
<table>
<thead>
<tr>
<th>Name</th>
<th>Notation</th>
</tr>
</thead>
<tbody>
<tr>
<td>Sign</td>
<td>+/-</td>
</tr>
<tr>
<td>Significant</td>
<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>d</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo separator="true">,</mo><msub><mi>d</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo separator="true">,</mo><msub><mi>d</mi><mrow><mo>−</mo><mn>3</mn></mrow></msub></mrow><annotation encoding="application/x-tex">d_{0}, d_{-1}, d_{-2}, d_{-3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.902771em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span>…<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mrow><mo>−</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">d_{-n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.902771em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.25833100000000003em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span></td>
</tr>
<tr>
<td>Base</td>
<td><code class="language-plaintext highlighter-rouge">b</code></td>
</tr>
<tr>
<td>Exponent</td>
<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow></mrow><mtext>exp</mtext></msup></mrow><annotation encoding="application/x-tex">^{\text{exp}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664392em;vertical-align:0em;"></span><span class="mord"><span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">exp</span></span></span></span></span></span></span></span></span></span></span></span></span></td>
</tr>
</tbody>
</table>
<ul>
<li>Let’s try: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>101011010.10</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">101011010.101_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">101011010.10</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> -&gt; ___</li>
</ul>
<h2 id="lets-try-normalizing-these-fractional-binary-numbers">Let’s try normalizing these fractional binary numbers!</h2>
<ol>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>101011010.10</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">101011010.101_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">101011010.10</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>
</li>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>0.00000000110</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">0.000000001101_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">0.00000000110</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>
</li>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>1100000011100</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">11000000111001_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1100000011100</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>
</li>
</ol>
<h2 id="ieee-floating-point-representation">IEEE floating point representation</h2>
<ul>
<li>Once V is normalized, we apply the equations
<ul>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>V</mi><mo>=</mo><mo stretchy="false">(</mo><mtext>–</mtext><mn>1</mn><msup><mo stretchy="false">)</mo><mi>s</mi></msup><mi>M</mi><msup><mn>2</mn><mi>E</mi></msup><mo>=</mo><mn>1.0101101010</mn><msub><mn>1</mn><mn>2</mn></msub><mo>×</mo><msup><mn>2</mn><mn>8</mn></msup></mrow><annotation encoding="application/x-tex">V = (–1)^{s} M 2^{E} = 1.01011010101_{2} \times 2^{8}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.1413309999999999em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">–1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">s</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1.0101101010</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8641079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span></span></span>
</li>
<li><code class="language-plaintext highlighter-rouge">s</code> = ???</li>
<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>=</mo><mtext>exp</mtext><mo>−</mo><mtext>bias</mtext></mrow><annotation encoding="application/x-tex">E = \text{exp} - \text{bias}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.7777700000000001em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord">exp</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">bias</span></span></span></span></span> where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>bias</mtext><mo>=</mo><msup><mn>2</mn><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>1</mn><mo>=</mo><msup><mn>2</mn><mn>7</mn></msup><mo>−</mo><mn>1</mn><mo>=</mo><mn>128</mn><mo>−</mo><mn>1</mn><mo>=</mo><mn>127</mn></mrow><annotation encoding="application/x-tex">\text{bias} = 2^{k-1} - 1 = 2^{7} - 1 = 128 - 1 = 127</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">bias</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.9324379999999999em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.897438em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">128</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">127</span></span></span></span></li>
<li><code class="language-plaintext highlighter-rouge">exp</code> = <code class="language-plaintext highlighter-rouge">E</code> + <code class="language-plaintext highlighter-rouge">bias</code> = ___</li>
<li><code class="language-plaintext highlighter-rouge">M</code> = 1 + <code class="language-plaintext highlighter-rouge">frac</code> = ___</li>
<li><code class="language-plaintext highlighter-rouge">s</code> = 1 bit, <code class="language-plaintext highlighter-rouge">exp</code> = k bits =&gt; 8 bits, <code class="language-plaintext highlighter-rouge">frac</code> n bits =&gt; 23 bits</li>
<li>bit vector in memory:</li>
</ul>
</li>
</ul>
<h2 id="why-adding-1-to-frac-or-subtracting-1-from-m">Why adding 1 to frac (or subtracting 1 from M)?</h2>
<ul>
<li>Because the number (or value) V is first normalized before it is converted.
<ul>
<li>As part of this normalization process, we transform our binary number such that its significand M is within the range [1.0 .. 2.0 – ε ]</li>
<li>Remember: M range for base 2 =&gt; [1.0 … 2.0 – ε]</li>
<li>This implies that M is always at least 1.0, so its integral part always has the value 1</li>
<li>So since this bit is always part of M, IEEE 754 does not explicitly save it in its bit pattern (i.e., in memory)</li>
<li>Instead, this bit is implied!</li>
</ul>
</li>
</ul>
<h2 id="why-adding-1-to-frac-or-subtracting-1-from-m-1">Why adding 1 to frac (or subtracting 1 from M)?</h2>
<p>Implying this bit has the following effects:</p>
<p>We get the
leading bit
for free!</p>
<ol>
<li>We save 1 bit when we convert (represent) a fractional decimal number into a bit pattern using IEEE 754 floating point representation</li>
<li>We have to add this 1 bit back when we convert from a bit pattern (IEEE 754 floating point representation) back to a fractional decimal</li>
</ol>
<p>Example: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><mo stretchy="false">(</mo><mtext>–</mtext><mn>1</mn><msup><mo stretchy="false">)</mo><mi>s</mi></msup><mi>M</mi><msup><mn>2</mn><mi>E</mi></msup><mo>=</mo><mn>1.01011010101</mn><mo>×</mo><msup><mn>2</mn><mn>8</mn></msup></mrow><annotation encoding="application/x-tex">V = (–1)^{s} M 2^{E} = 1.01011010101 \times 2^{8}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">–1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">s</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1.01011010101</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>M = 1. 01011010101 =&gt; M = 1 + frac</p>
<p>This bit is implied hence not stored in the bit pattern produced
by the IEEE 754 floating point representation, and what we
store in the frac part of the IEEE 754 bit pattern is 01011010101</p>
<h2 id="ieee-floating-point-representation-single-precision">IEEE floating point representation (single precision)</h2>
<ul>
<li>What if the 4 bytes starting at M[0x0000] represented a fractional
decimal number (encoded as an IEEE floating point number) -&gt; value?
single precision</li>
</ul>
<p>Numerical Form: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><mo stretchy="false">(</mo><mtext>–</mtext><mn>1</mn><msup><mo stretchy="false">)</mo><mi>s</mi></msup><mi>M</mi><msup><mn>2</mn><mi>E</mi></msup></mrow><annotation encoding="application/x-tex">V = (–1)^{s} M 2^{E}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">–1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">s</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span></span></span></span></span></span></span></span></span></p>
<table>
<thead>
<tr>
<th>Value</th>
<th>Notes</th>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td> </td>
</tr>
<tr>
<td>10000111</td>
<td>k=8 bits, interpreted as unsigned</td>
</tr>
<tr>
<td>01011010101000000000000</td>
<td>n=23 bits, interpreted as unsigned</td>
</tr>
</tbody>
</table>
<ul>
<li>exp ≠ 0 and exp ≠ 111111112 -&gt; normalized</li>
<li>s = ___</li>
<li>E = exp – bias where bias = <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mtext>–</mtext><mn>1</mn><mo>=</mo><msup><mn>2</mn><mn>7</mn></msup><mtext>–</mtext><mn>1</mn><mo>=</mo><mn>128</mn><mtext>–</mtext><mn>1</mn><mo>=</mo><mn>127</mn></mrow><annotation encoding="application/x-tex">2^{k-1} – 1 = 2^{7} – 1 = 128 – 1 = 127</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord">–1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span><span class="mord">–1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">128–1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">127</span></span></span></span>$</li>
<li>E = ____ - 127 =</li>
<li>M = 1 + <code class="language-plaintext highlighter-rouge">frac</code> = 1 + ___</li>
<li>V = ____</li>
</ul>
<p>Little endian memory layout:</p>
<table>
<thead>
<tr>
<th>Address</th>
<th>M[]</th>
</tr>
</thead>
<tbody>
<tr>
<td>size-1</td>
<td> </td>
</tr>
<tr>
<td>…</td>
<td>…</td>
</tr>
<tr>
<td>0x0003</td>
<td>11000011</td>
</tr>
<tr>
<td>0x0002</td>
<td>10101101</td>
</tr>
<tr>
<td>0x0001</td>
<td>01010000</td>
</tr>
<tr>
<td>0x0000</td>
<td>00000000</td>
</tr>
</tbody>
</table>
<h2 id="lets-give-it-a-go">Let’s give it a go!</h2>
<ul>
<li>What if the 4 bytes starting at M[0x0000] represented a fractional
decimal number (encoded as an IEEE floating point number) -&gt; value?</li>
</ul>
<p>Numerical form: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>s</mi></msup><mi>M</mi><msup><mn>2</mn><mi>E</mi></msup></mrow><annotation encoding="application/x-tex">V = (-1)^{s} M 2^{E}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">s</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>single precision</p>
<table>
<thead>
<tr>
<th>Value</th>
<th>Notes</th>
</tr>
</thead>
<tbody>
<tr>
<td>0</td>
<td> </td>
</tr>
<tr>
<td>10001100</td>
<td>k=8 bits, interpreted as unsigned</td>
</tr>
<tr>
<td>11011011011000000000000</td>
<td>n=23 bits, interpreted as unsigned</td>
</tr>
</tbody>
</table>
<ul>
<li>exp ≠ 0 and exp ≠ 111111112 -&gt; normalized</li>
<li>s = ____</li>
<li>E = exp - bias where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>bias</mtext><mo>=</mo><msup><mn>2</mn><mn>7</mn></msup><mo>−</mo><mn>1</mn><mo>=</mo><mn>128</mn><mo>−</mo><mn>1</mn><mo>=</mo><mn>127</mn></mrow><annotation encoding="application/x-tex">\text{bias} = 2^{7} - 1 = 128 - 1 = 127</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">bias</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.897438em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">128</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">127</span></span></span></span></li>
<li>E = ____ - 127 = ___</li>
<li>M = 1 + frac = 1 + ____</li>
<li>V = ____</li>
</ul>
<p>Little endian memory map:</p>
<table>
<thead>
<tr>
<th>Address</th>
<th>M[]</th>
</tr>
</thead>
<tbody>
<tr>
<td>size-1</td>
<td> </td>
</tr>
<tr>
<td>…</td>
<td>…</td>
</tr>
<tr>
<td>0x0003</td>
<td>01000110</td>
</tr>
<tr>
<td>0x0002</td>
<td>01101101</td>
</tr>
<tr>
<td>0x0001</td>
<td>10110000</td>
</tr>
<tr>
<td>0x0000</td>
<td>00000000</td>
</tr>
</tbody>
</table>
<h2 id="ieee-floating-point-representation-single-precision-1">IEEE floating point representation (single precision)</h2>
<p>How would 47.21875 be encoded as IEEE floating point number?</p>
<ol>
<li>Convert 47.28 to binary (using the positional notation R2B(X)) =&gt;
<ul>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>47</mn><mo>=</mo><mn>10111</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">47 = 101111_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">47</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">10111</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>
</li>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">.</mi><mn>21875</mn><mo>=</mo><mi mathvariant="normal">.</mi><mn>0011</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">.21875 = .00111_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">.21875</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">.0011</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>
</li>
</ul>
</li>
<li>Normalize binary number:
101111.00111 =&gt; <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1.011110011</mn><msub><mn>1</mn><mn>2</mn></msub><mo>×</mo><msup><mn>2</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">1.0111100111_{2} \times 2^{5}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1.011110011</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></span>
<ul>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>V</mi><mo>=</mo><mo stretchy="false">(</mo><mtext>–</mtext><mn>1</mn><msup><mo stretchy="false">)</mo><mi>s</mi></msup><mi>M</mi><msup><mn>2</mn><mi>E</mi></msup></mrow><annotation encoding="application/x-tex">V = (–1)^{s} M 2^{E}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.1413309999999999em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">–1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">s</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span></span></span></span></span></span></span></span></span></span>
</li>
</ul>
</li>
<li>Determine …
<ul>
<li>s = 0</li>
<li>E = exp – bias where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>bias</mtext><mo>=</mo><msup><mn>2</mn><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mtext>–</mtext><mn>1</mn><mo>=</mo><msup><mn>2</mn><mn>7</mn></msup><mtext>–</mtext><mn>1</mn><mo>=</mo><mn>128</mn><mtext>–</mtext><mn>1</mn><mo>=</mo><mn>127</mn></mrow><annotation encoding="application/x-tex">\text{bias} = 2^{k-1} – 1 = 2^{7} – 1 = 128 – 1 = 127</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">bias</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8491079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord">–1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span><span class="mord">–1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">128–1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">127</span></span></span></span></li>
<li>exp = E + bias = 5 + 127 = 132 =&gt; U2B(132) =&gt; 10000100</li>
<li>M = 1 + frac -&gt; frac = M - 1 =&gt; <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1.011110011</mn><msub><mn>1</mn><mn>2</mn></msub><mo>−</mo><mn>1</mn><mo>=</mo><mi mathvariant="normal">.</mi><mn>011110011</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">1.0111100111_{2} - 1 = .0111100111_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1.011110011</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">.011110011</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></li>
</ul>
</li>
<li>32 bits organized in s|exp|frac: [0|10000100|01111001110000000000000}</li>
<li>0x423CE000</li>
</ol>
<h2 id="ieee-floating-point-representation-single-precision-2">IEEE floating point representation (single precision)</h2>
<p>How would 12345.75 be encoded as IEEE floating point number?</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>s</mi></msup><mi>M</mi><msup><mn>2</mn><mi>E</mi></msup></mrow><annotation encoding="application/x-tex">
V = (-1)^{s} M 2^{E}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.1413309999999999em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em;">E</span></span></span></span></span></span></span></span></span></span></span></span></span></p>
<ol>
<li>Convert 12345.75 to binary
<ul>
<li>12345 =&gt; ____ .75 =&gt; ____</li>
</ul>
</li>
<li>Normalize binary number:</li>
<li>Determine …
<ul>
<li>s = ____</li>
<li>E = exp – bias where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>bias</mtext><mo>=</mo><msup><mn>2</mn><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mtext>–</mtext><mn>1</mn><mo>=</mo><msup><mn>2</mn><mn>7</mn></msup><mtext>–</mtext><mn>1</mn><mo>=</mo><mn>128</mn><mtext>–</mtext><mn>1</mn><mo>=</mo><mn>127</mn></mrow><annotation encoding="application/x-tex">\text{bias} = 2^{k-1} – 1 = 2^{7} – 1 = 128 – 1 = 127</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">bias</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8491079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord">–1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span><span class="mord">–1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">128–1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">127</span></span></span></span></li>
<li>exp = E + bias = ____</li>
<li>M = 1 + frac -&gt; frac = M - 1</li>
</ul>
</li>
<li>[_|________|_______________________]</li>
<li>Express in hex:</li>
</ol>
<h2 id="summary">Summary</h2>
<ul>
<li>IEEE Floating Point Representation
<ol>
<li>Denormalized</li>
<li>Special cases</li>
<li>Normalized =&gt; exp ≠ 000…0 and exp ≠ 111…1
<ul>
<li>Single precision: bias = 127, exp: [1..254], E: [-126..127] =&gt; [10-38 … 1038]</li>
<li>Called “normalized” because binary numbers are normalized</li>
</ul>
<ul>
<li>Effect: “We get the leading bit for free”
<ul>
<li>Leading bit is always assumed (never part of bit pattern)</li>
</ul>
</li>
</ul>
</li>
</ol>
</li>
<li>IEEE floating point number as encoding scheme
<ul>
<li>Fractional decimal number  IEEE 754 (bit pattern)</li>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>V</mi><mo>=</mo><mo stretchy="false">(</mo><mtext>–</mtext><mn>1</mn><msup><mo stretchy="false">)</mo><mi>s</mi></msup><mi>M</mi><msup><mn>2</mn><mi>E</mi></msup></mrow><annotation encoding="application/x-tex">V = (–1)^{s} M 2^{E}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.1413309999999999em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">–1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">s</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span></span></span></span></span></span></span></span></span></span>
<ul>
<li>s is sign bit, M = 1 + frac, E = exp – bias, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>bias</mtext><mo>=</mo><msup><mn>2</mn><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mtext>–</mtext><mn>1</mn></mrow><annotation encoding="application/x-tex">\text{bias} = 2^{k-1} – 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">bias</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8491079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord">–1</span></span></span></span> and k is width of exp</li>
</ul>
</li>
</ul>
</li>
</ul>
<h2 id="next-lecture">Next Lecture</h2>
<ul>
<li>Representing data in memory – Most of this is review
<ul>
<li>“Under the Hood” - Von Neumann architecture</li>
<li>Bits and bytes in memory
<ul>
<li>How to diagram memory -&gt; Used in this course and other references</li>
<li>How to represent series of bits -&gt; In binary, in hexadecimal (conversion)</li>
<li>What kind of information (data) do series of bits represent -&gt; Encoding scheme</li>
<li>Order of bytes in memory -&gt; Endian</li>
</ul>
</li>
<li>Bit manipulation – bitwise operations
<ul>
<li>Boolean algebra + Shifting</li>
</ul>
</li>
</ul>
</li>
<li>Representing integral numbers in memory
<ul>
<li>Unsigned and signed</li>
<li>Converting, expanding and truncating</li>
<li>Arithmetic operations</li>
</ul>
</li>
<li>Representing real numbers in memory
<ul>
<li>IEEE floating point representation</li>
<li>Floating point in C – casting, rounding, addition, …</li>
</ul>
</li>
</ul>
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<h1 id="cmpt-295">CMPT 295</h1>
<ul>
<li>Unit - Data Representation
<ul>
<li>Lecture 6 – Representing fractional numbers in memory</li>
<li>IEEE floating point representation – cont’d</li>
</ul>
</li>
</ul>
<h2 id="have-you-heard-of-that-new-band-1023-megabytes">Have you heard of that new band “1023 Megabytes”?</h2>
<p>They’re pretty good,
but they don’t have a gig just yet.
😭</p>
<h2 id="last-lecture">Last Lecture</h2>
<ul>
<li>Representing integral numbers in memory
<ul>
<li>Can encode a small range of values exactly (in 1, 2, 4, 8 bytes)
<ul>
<li>For example: We can represent the values -128 to 127 exactly in 1 byte using a signed char in C</li>
</ul>
</li>
</ul>
</li>
<li>Representing fractional numbers in memory
<ol>
<li>Positional notation has some advantages, but also disadvantages -&gt; so not used!</li>
<li>IEEE floating point representation: can encode a much larger range of e.g., single precision: [10-38..1038] values approximately (in 4 or 8 bytes)</li>
</ol>
</li>
<li>Overview of IEEE floating point representation
<ul>
<li>Precision options (float 32-bit, double 64-bit)</li>
<li>V = (-1)s x M x 2E</li>
<li>s –&gt; sign bit</li>
<li>exp encodes E (but != E)</li>
<li>frac encodes M (but != M)</li>
</ul>
</li>
</ul>
<p>We interpret the bit vector (expressed in IEEE floating point encoding) stored in memory using this equation.</p>
<h2 id="todays-menu">Today’s Menu</h2>
<ul>
<li>Representing data in memory – Most of this is review
<ul>
<li>“Under the Hood” - Von Neumann architecture</li>
<li>Bits and bytes in memory
<ul>
<li>How to diagram memory -&gt; Used in this course and other references</li>
<li>How to represent series of bits -&gt; In binary, in hexadecimal (conversion)</li>
<li>What kind of information (data) do series of bits represent -&gt; Encoding scheme</li>
<li>Order of bytes in memory -&gt; Endian</li>
</ul>
</li>
<li>Bit manipulation – bitwise operations
<ul>
<li>Boolean algebra + Shifting</li>
</ul>
</li>
</ul>
</li>
<li>Representing integral numbers in memory
<ul>
<li>Unsigned and signed</li>
<li>Converting, expanding and truncating</li>
<li>Arithmetic operations</li>
</ul>
</li>
<li>Representing real numbers in memory
4
<ul>
<li>IEEE floating point representation</li>
<li>Floating point in C – casting, rounding, addition, …</li>
</ul>
</li>
</ul>
<h2 id="ieee-floating-point-representation-three-kinds-of-values">IEEE Floating Point Representation Three “kinds” of values</h2>
<p>We interpret the bit vector
(expressed in IEEE floating point encoding) stored in memory using this equation:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>s</mi></msup><mi>M</mi><msup><mn>2</mn><mi>E</mi></msup></mrow><annotation encoding="application/x-tex">
V = (-1)^{s} M 2^{E}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.1413309999999999em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em;">E</span></span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>Bit breakdown–<code class="language-plaintext highlighter-rouge">exp</code> and <code class="language-plaintext highlighter-rouge">frac</code> interpreted as unsigned:</p>
<ul>
<li>s = 1 bit</li>
<li>exp = k bits
<ol>
<li>If <code class="language-plaintext highlighter-rouge">exp</code> != 0 and <code class="language-plaintext highlighter-rouge">exp</code> != 11…11 (exp range: [0000001…11111110]). Equations:
<ul>
<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>=</mo><mtext>exp</mtext><mo>−</mo><mtext>bias</mtext></mrow><annotation encoding="application/x-tex">E = \text{exp} - \text{bias}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.7777700000000001em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord">exp</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">bias</span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>bias</mtext><mo>=</mo><msup><mn>2</mn><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\text{bias} = 2^{k-1} - 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">bias</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.9324379999999999em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></li>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>M</mi><mo>=</mo><mn>1</mn><mo>+</mo><mtext>frac</mtext></mrow><annotation encoding="application/x-tex">M = 1 + \text{frac}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">frac</span></span></span></span></span></span>
</li>
</ul>
</li>
<li>If <code class="language-plaintext highlighter-rouge">exp</code> = 00…00 (all 0’s) =&gt; denormalized. Equations:
<ul>
<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>=</mo><mn>1</mn><mo>−</mo><mtext>bias</mtext></mrow><annotation encoding="application/x-tex">E = 1 - \text{bias}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">bias</span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>bias</mtext><mo>=</mo><msup><mn>2</mn><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\text{bias} = 2^{k-1} - 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">bias</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.9324379999999999em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></li>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>M</mi><mo>=</mo><mtext>frac</mtext></mrow><annotation encoding="application/x-tex">M = \text{frac}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">frac</span></span></span></span></span></span>
</li>
</ul>
</li>
<li>If <code class="language-plaintext highlighter-rouge">exp</code> 11…11 (all 1’s) =&gt; special cases.
<ul>
<li>Case 1: <code class="language-plaintext highlighter-rouge">frac</code> = 000…0</li>
<li>Case 2: <code class="language-plaintext highlighter-rouge">frac</code> != 000…0</li>
</ul>
</li>
</ol>
</li>
</ul>
<h2 id="ieee-floating-point-representation---normalized">IEEE floating point representation - normalized</h2>
<p>Numerical Form: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><mo stretchy="false">(</mo><mtext>–</mtext><mn>1</mn><msup><mo stretchy="false">)</mo><mi>s</mi></msup><mi>M</mi><msup><mn>2</mn><mi>E</mi></msup></mrow><annotation encoding="application/x-tex">V = (–1)^{s} M 2^{E}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">–1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">s</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>Bit breakdown:</p>
<ul>
<li>s = 1 bit</li>
<li><code class="language-plaintext highlighter-rouge">exp</code> = k bits
<ul>
<li>If <code class="language-plaintext highlighter-rouge">exp</code> != 0 and <code class="language-plaintext highlighter-rouge">exp</code> != 11…11 (<code class="language-plaintext highlighter-rouge">exp</code> range: [00000001…11111110]) =&gt; normalized. Equations:
<ul>
<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>=</mo><mtext>exp</mtext><mo>−</mo><mtext>bias</mtext></mrow><annotation encoding="application/x-tex">E = \text{exp} - \text{bias}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.7777700000000001em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord">exp</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">bias</span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>bias</mtext><mo>=</mo><msup><mn>2</mn><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\text{bias} = 2^{k-1} - 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">bias</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.9324379999999999em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></li>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>M</mi><mo>=</mo><mn>1</mn><mo>+</mo><mtext>frac</mtext></mrow><annotation encoding="application/x-tex">M = 1 + \text{frac}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">frac</span></span></span></span></span></span>
</li>
</ul>
</li>
</ul>
</li>
</ul>
<p>Why is <code class="language-plaintext highlighter-rouge">E</code> biased?</p>
<p>Using single precision as an example (s = 1 bit, exp = 8 bits, frac = 23 bits):</p>
<ul>
<li>(exp range: [00000001 .. 11111110]) =&gt; <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><msub><mn>1</mn><mn>10</mn></msub><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mn>25</mn><msub><mn>4</mn><mn>10</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[1_{10}...254_{10}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">...25</span><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></li>
<li>If <code class="language-plaintext highlighter-rouge">E</code> is not biased (i.e. E = exp), then <code class="language-plaintext highlighter-rouge">E</code> range <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><msub><mn>1</mn><mn>10</mn></msub><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mn>25</mn><msub><mn>4</mn><mn>10</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[1_{10} ... 254_{10}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">...25</span><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></li>
<li><code class="language-plaintext highlighter-rouge">V</code> range [<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>1</mn></msup></mrow><annotation encoding="application/x-tex">2^{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span>…<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>254</mn></msup></mrow><annotation encoding="application/x-tex">2^{254}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">254</span></span></span></span></span></span></span></span></span></span></span></span>] = [2…<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≈</mo><mn>2.89</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mn>76</mn></msup></mrow><annotation encoding="application/x-tex">\approx 2.89 \times 10^{76}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.48312em;vertical-align:0em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">2.89</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">76</span></span></span></span></span></span></span></span></span></span></span></span>] (so cannot express numbers &lt; 2)</li>
<li>By biasing <code class="language-plaintext highlighter-rouge">E</code> (i.e. E = exp - bias), then <code class="language-plaintext highlighter-rouge">E</code> range: [1-127…254-127] == [-126…127] (since k = 8, bias = <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mrow><mn>8</mn><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">2^{8-1} - 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.897438em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span> = 127)</li>
<li><code class="language-plaintext highlighter-rouge">V</code> range: [<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mrow><mo>−</mo><mn>126</mn></mrow></msup></mrow><annotation encoding="application/x-tex">2^{-126}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">126</span></span></span></span></span></span></span></span></span></span></span></span>…<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>127</mn></msup></mrow><annotation encoding="application/x-tex">2^{127}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">127</span></span></span></span></span></span></span></span></span></span></span></span>] = [<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≈</mo><mn>1.18</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mrow><mo>−</mo><mn>38</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\approx 1.18 \times 10^{-38}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.48312em;vertical-align:0em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1.18</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">38</span></span></span></span></span></span></span></span></span></span></span></span> … <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≈</mo><mn>1.7</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mn>38</mn></msup></mrow><annotation encoding="application/x-tex">\approx 1.7 \times 10^{38}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.48312em;vertical-align:0em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1.7</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">38</span></span></span></span></span></span></span></span></span></span></span></span> (so can now express very small (and very large) numbers)</li>
<li>Why adding 1 to <code class="language-plaintext highlighter-rouge">frac</code>? Because the number (or value) V is first normalized before it is converted.</li>
</ul>
<h2 id="review-scientific-notation-and-normalization">Review: Scientific Notation and normalization</h2>
<ul>
<li>From Wikipedia:
<ul>
<li>Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form (as they are long strings of digits).</li>
<li>In scientific notation, nonzero numbers are written in the form +/- M × 10n</li>
<li>In normalized notation, the exponent n is chosen such that the absolute value of the significand M is at least 1 (M = 1.0) but less than the base
<ul>
<li>M range for base 10 =&gt; [1.0 .. 10.0 – ε ]</li>
<li>M range for base 2 =&gt; [1.0 .. 2.0 – ε ]</li>
</ul>
</li>
</ul>
</li>
<li>
<p>Examples:</p>
<ul>
<li>A proton’s mass is 0.0000000000000000000000000016726 kg -&gt; <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1.6726</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mrow><mtext>−</mtext><mn>27</mn></mrow></msup></mrow><annotation encoding="application/x-tex">1.6726 \times 10^{−27}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1.6726</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−27</span></span></span></span></span></span></span></span></span></span></span></span> kg</li>
<li>Speed of light is 299,792,458 m/s -&gt; <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2.99792</mn><mo separator="true">,</mo><mn>458</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mn>8</mn></msup></mrow><annotation encoding="application/x-tex">2.99792,458 \times 10^{8}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em;"></span><span class="mord">2.99792</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">458</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span></span> m/s</li>
</ul>
</li>
</ul>
<p>Syntax of normalized notation:</p>
<table>
<thead>
<tr>
<th>Name</th>
<th>Notation</th>
</tr>
</thead>
<tbody>
<tr>
<td>Sign</td>
<td>+/-</td>
</tr>
<tr>
<td>Significant</td>
<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>d</mi><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo separator="true">,</mo><msub><mi>d</mi><mrow><mo>−</mo><mn>2</mn></mrow></msub><mo separator="true">,</mo><msub><mi>d</mi><mrow><mo>−</mo><mn>3</mn></mrow></msub></mrow><annotation encoding="application/x-tex">d_{0}, d_{-1}, d_{-2}, d_{-3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.902771em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span>…<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mrow><mo>−</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">d_{-n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.902771em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.25833100000000003em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span></td>
</tr>
<tr>
<td>Base</td>
<td><code class="language-plaintext highlighter-rouge">b</code></td>
</tr>
<tr>
<td>Exponent</td>
<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow></mrow><mtext>exp</mtext></msup></mrow><annotation encoding="application/x-tex">^{\text{exp}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.664392em;vertical-align:0em;"></span><span class="mord"><span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">exp</span></span></span></span></span></span></span></span></span></span></span></span></span></td>
</tr>
</tbody>
</table>
<ul>
<li>Let’s try: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>101011010.10</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">101011010.101_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">101011010.10</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> -&gt; ___</li>
</ul>
<h2 id="lets-try-normalizing-these-fractional-binary-numbers">Let’s try normalizing these fractional binary numbers!</h2>
<ol>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>101011010.10</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">101011010.101_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">101011010.10</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>
</li>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>0.00000000110</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">0.000000001101_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">0.00000000110</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>
</li>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>1100000011100</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">11000000111001_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1100000011100</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>
</li>
</ol>
<h2 id="ieee-floating-point-representation">IEEE floating point representation</h2>
<ul>
<li>Once V is normalized, we apply the equations
<ul>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>V</mi><mo>=</mo><mo stretchy="false">(</mo><mtext>–</mtext><mn>1</mn><msup><mo stretchy="false">)</mo><mi>s</mi></msup><mi>M</mi><msup><mn>2</mn><mi>E</mi></msup><mo>=</mo><mn>1.0101101010</mn><msub><mn>1</mn><mn>2</mn></msub><mo>×</mo><msup><mn>2</mn><mn>8</mn></msup></mrow><annotation encoding="application/x-tex">V = (–1)^{s} M 2^{E} = 1.01011010101_{2} \times 2^{8}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.1413309999999999em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">–1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">s</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1.0101101010</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8641079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span></span></span>
</li>
<li><code class="language-plaintext highlighter-rouge">s</code> = ???</li>
<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>=</mo><mtext>exp</mtext><mo>−</mo><mtext>bias</mtext></mrow><annotation encoding="application/x-tex">E = \text{exp} - \text{bias}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.7777700000000001em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord">exp</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">bias</span></span></span></span></span> where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>bias</mtext><mo>=</mo><msup><mn>2</mn><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>1</mn><mo>=</mo><msup><mn>2</mn><mn>7</mn></msup><mo>−</mo><mn>1</mn><mo>=</mo><mn>128</mn><mo>−</mo><mn>1</mn><mo>=</mo><mn>127</mn></mrow><annotation encoding="application/x-tex">\text{bias} = 2^{k-1} - 1 = 2^{7} - 1 = 128 - 1 = 127</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">bias</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.9324379999999999em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.897438em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">128</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">127</span></span></span></span></li>
<li><code class="language-plaintext highlighter-rouge">exp</code> = <code class="language-plaintext highlighter-rouge">E</code> + <code class="language-plaintext highlighter-rouge">bias</code> = ___</li>
<li><code class="language-plaintext highlighter-rouge">M</code> = 1 + <code class="language-plaintext highlighter-rouge">frac</code> = ___</li>
<li><code class="language-plaintext highlighter-rouge">s</code> = 1 bit, <code class="language-plaintext highlighter-rouge">exp</code> = k bits =&gt; 8 bits, <code class="language-plaintext highlighter-rouge">frac</code> n bits =&gt; 23 bits</li>
<li>bit vector in memory:</li>
</ul>
</li>
</ul>
<h2 id="why-adding-1-to-frac-or-subtracting-1-from-m">Why adding 1 to frac (or subtracting 1 from M)?</h2>
<ul>
<li>Because the number (or value) V is first normalized before it is converted.
<ul>
<li>As part of this normalization process, we transform our binary number such that its significand M is within the range [1.0 .. 2.0 – ε ]</li>
<li>Remember: M range for base 2 =&gt; [1.0 … 2.0 – ε]</li>
<li>This implies that M is always at least 1.0, so its integral part always has the value 1</li>
<li>So since this bit is always part of M, IEEE 754 does not explicitly save it in its bit pattern (i.e., in memory)</li>
<li>Instead, this bit is implied!</li>
</ul>
</li>
</ul>
<h2 id="why-adding-1-to-frac-or-subtracting-1-from-m-1">Why adding 1 to frac (or subtracting 1 from M)?</h2>
<p>Implying this bit has the following effects:</p>
<p>We get the
leading bit
for free!</p>
<ol>
<li>We save 1 bit when we convert (represent) a fractional decimal number into a bit pattern using IEEE 754 floating point representation</li>
<li>We have to add this 1 bit back when we convert from a bit pattern (IEEE 754 floating point representation) back to a fractional decimal</li>
</ol>
<p>Example: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><mo stretchy="false">(</mo><mtext>–</mtext><mn>1</mn><msup><mo stretchy="false">)</mo><mi>s</mi></msup><mi>M</mi><msup><mn>2</mn><mi>E</mi></msup><mo>=</mo><mn>1.01011010101</mn><mo>×</mo><msup><mn>2</mn><mn>8</mn></msup></mrow><annotation encoding="application/x-tex">V = (–1)^{s} M 2^{E} = 1.01011010101 \times 2^{8}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">–1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">s</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1.01011010101</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>M = 1. 01011010101 =&gt; M = 1 + frac</p>
<p>This bit is implied hence not stored in the bit pattern produced
by the IEEE 754 floating point representation, and what we
store in the frac part of the IEEE 754 bit pattern is 01011010101</p>
<h2 id="ieee-floating-point-representation-single-precision">IEEE floating point representation (single precision)</h2>
<ul>
<li>What if the 4 bytes starting at M[0x0000] represented a fractional
decimal number (encoded as an IEEE floating point number) -&gt; value?
single precision</li>
</ul>
<p>Numerical Form: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><mo stretchy="false">(</mo><mtext>–</mtext><mn>1</mn><msup><mo stretchy="false">)</mo><mi>s</mi></msup><mi>M</mi><msup><mn>2</mn><mi>E</mi></msup></mrow><annotation encoding="application/x-tex">V = (–1)^{s} M 2^{E}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">–1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">s</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span></span></span></span></span></span></span></span></span></p>
<table>
<thead>
<tr>
<th>Value</th>
<th>Notes</th>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td> </td>
</tr>
<tr>
<td>10000111</td>
<td>k=8 bits, interpreted as unsigned</td>
</tr>
<tr>
<td>01011010101000000000000</td>
<td>n=23 bits, interpreted as unsigned</td>
</tr>
</tbody>
</table>
<ul>
<li>exp ≠ 0 and exp ≠ 111111112 -&gt; normalized</li>
<li>s = ___</li>
<li>E = exp – bias where bias = <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mtext>–</mtext><mn>1</mn><mo>=</mo><msup><mn>2</mn><mn>7</mn></msup><mtext>–</mtext><mn>1</mn><mo>=</mo><mn>128</mn><mtext>–</mtext><mn>1</mn><mo>=</mo><mn>127</mn></mrow><annotation encoding="application/x-tex">2^{k-1} – 1 = 2^{7} – 1 = 128 – 1 = 127</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord">–1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span><span class="mord">–1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">128–1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">127</span></span></span></span>$</li>
<li>E = ____ - 127 =</li>
<li>M = 1 + <code class="language-plaintext highlighter-rouge">frac</code> = 1 + ___</li>
<li>V = ____</li>
</ul>
<p>Little endian memory layout:</p>
<table>
<thead>
<tr>
<th>Address</th>
<th>M[]</th>
</tr>
</thead>
<tbody>
<tr>
<td>size-1</td>
<td> </td>
</tr>
<tr>
<td>…</td>
<td>…</td>
</tr>
<tr>
<td>0x0003</td>
<td>11000011</td>
</tr>
<tr>
<td>0x0002</td>
<td>10101101</td>
</tr>
<tr>
<td>0x0001</td>
<td>01010000</td>
</tr>
<tr>
<td>0x0000</td>
<td>00000000</td>
</tr>
</tbody>
</table>
<h2 id="lets-give-it-a-go">Let’s give it a go!</h2>
<ul>
<li>What if the 4 bytes starting at M[0x0000] represented a fractional
decimal number (encoded as an IEEE floating point number) -&gt; value?</li>
</ul>
<p>Numerical form: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>s</mi></msup><mi>M</mi><msup><mn>2</mn><mi>E</mi></msup></mrow><annotation encoding="application/x-tex">V = (-1)^{s} M 2^{E}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">s</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>single precision</p>
<table>
<thead>
<tr>
<th>Value</th>
<th>Notes</th>
</tr>
</thead>
<tbody>
<tr>
<td>0</td>
<td> </td>
</tr>
<tr>
<td>10001100</td>
<td>k=8 bits, interpreted as unsigned</td>
</tr>
<tr>
<td>11011011011000000000000</td>
<td>n=23 bits, interpreted as unsigned</td>
</tr>
</tbody>
</table>
<ul>
<li>exp ≠ 0 and exp ≠ 111111112 -&gt; normalized</li>
<li>s = ____</li>
<li>E = exp - bias where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>bias</mtext><mo>=</mo><msup><mn>2</mn><mn>7</mn></msup><mo>−</mo><mn>1</mn><mo>=</mo><mn>128</mn><mo>−</mo><mn>1</mn><mo>=</mo><mn>127</mn></mrow><annotation encoding="application/x-tex">\text{bias} = 2^{7} - 1 = 128 - 1 = 127</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">bias</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.897438em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">128</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">127</span></span></span></span></li>
<li>E = ____ - 127 = ___</li>
<li>M = 1 + frac = 1 + ____</li>
<li>V = ____</li>
</ul>
<p>Little endian memory map:</p>
<table>
<thead>
<tr>
<th>Address</th>
<th>M[]</th>
</tr>
</thead>
<tbody>
<tr>
<td>size-1</td>
<td> </td>
</tr>
<tr>
<td>…</td>
<td>…</td>
</tr>
<tr>
<td>0x0003</td>
<td>01000110</td>
</tr>
<tr>
<td>0x0002</td>
<td>01101101</td>
</tr>
<tr>
<td>0x0001</td>
<td>10110000</td>
</tr>
<tr>
<td>0x0000</td>
<td>00000000</td>
</tr>
</tbody>
</table>
<h2 id="ieee-floating-point-representation-single-precision-1">IEEE floating point representation (single precision)</h2>
<p>How would 47.21875 be encoded as IEEE floating point number?</p>
<ol>
<li>Convert 47.28 to binary (using the positional notation R2B(X)) =&gt;
<ul>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>47</mn><mo>=</mo><mn>10111</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">47 = 101111_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">47</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">10111</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>
</li>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">.</mi><mn>21875</mn><mo>=</mo><mi mathvariant="normal">.</mi><mn>0011</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">.21875 = .00111_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">.21875</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">.0011</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>
</li>
</ul>
</li>
<li>Normalize binary number:
101111.00111 =&gt; <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1.011110011</mn><msub><mn>1</mn><mn>2</mn></msub><mo>×</mo><msup><mn>2</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">1.0111100111_{2} \times 2^{5}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1.011110011</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></span>
<ul>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>V</mi><mo>=</mo><mo stretchy="false">(</mo><mtext>–</mtext><mn>1</mn><msup><mo stretchy="false">)</mo><mi>s</mi></msup><mi>M</mi><msup><mn>2</mn><mi>E</mi></msup></mrow><annotation encoding="application/x-tex">V = (–1)^{s} M 2^{E}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.1413309999999999em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">–1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">s</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span></span></span></span></span></span></span></span></span></span>
</li>
</ul>
</li>
<li>Determine …
<ul>
<li>s = 0</li>
<li>E = exp – bias where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>bias</mtext><mo>=</mo><msup><mn>2</mn><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mtext>–</mtext><mn>1</mn><mo>=</mo><msup><mn>2</mn><mn>7</mn></msup><mtext>–</mtext><mn>1</mn><mo>=</mo><mn>128</mn><mtext>–</mtext><mn>1</mn><mo>=</mo><mn>127</mn></mrow><annotation encoding="application/x-tex">\text{bias} = 2^{k-1} – 1 = 2^{7} – 1 = 128 – 1 = 127</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">bias</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8491079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord">–1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span><span class="mord">–1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">128–1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">127</span></span></span></span></li>
<li>exp = E + bias = 5 + 127 = 132 =&gt; U2B(132) =&gt; 10000100</li>
<li>M = 1 + frac -&gt; frac = M - 1 =&gt; <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1.011110011</mn><msub><mn>1</mn><mn>2</mn></msub><mo>−</mo><mn>1</mn><mo>=</mo><mi mathvariant="normal">.</mi><mn>011110011</mn><msub><mn>1</mn><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">1.0111100111_{2} - 1 = .0111100111_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">1.011110011</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord">.011110011</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></li>
</ul>
</li>
<li>32 bits organized in s|exp|frac: [0|10000100|01111001110000000000000}</li>
<li>0x423CE000</li>
</ol>
<h2 id="ieee-floating-point-representation-single-precision-2">IEEE floating point representation (single precision)</h2>
<p>How would 12345.75 be encoded as IEEE floating point number?</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>=</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>s</mi></msup><mi>M</mi><msup><mn>2</mn><mi>E</mi></msup></mrow><annotation encoding="application/x-tex">
V = (-1)^{s} M 2^{E}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.1413309999999999em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em;">E</span></span></span></span></span></span></span></span></span></span></span></span></span></p>
<ol>
<li>Convert 12345.75 to binary
<ul>
<li>12345 =&gt; ____ .75 =&gt; ____</li>
</ul>
</li>
<li>Normalize binary number:</li>
<li>Determine …
<ul>
<li>s = ____</li>
<li>E = exp – bias where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>bias</mtext><mo>=</mo><msup><mn>2</mn><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mtext>–</mtext><mn>1</mn><mo>=</mo><msup><mn>2</mn><mn>7</mn></msup><mtext>–</mtext><mn>1</mn><mo>=</mo><mn>128</mn><mtext>–</mtext><mn>1</mn><mo>=</mo><mn>127</mn></mrow><annotation encoding="application/x-tex">\text{bias} = 2^{k-1} – 1 = 2^{7} – 1 = 128 – 1 = 127</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">bias</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8491079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord">–1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span><span class="mord">–1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">128–1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">127</span></span></span></span></li>
<li>exp = E + bias = ____</li>
<li>M = 1 + frac -&gt; frac = M - 1</li>
</ul>
</li>
<li>[_|________|_______________________]</li>
<li>Express in hex:</li>
</ol>
<h2 id="summary">Summary</h2>
<ul>
<li>IEEE Floating Point Representation
<ol>
<li>Denormalized</li>
<li>Special cases</li>
<li>Normalized =&gt; exp ≠ 000…0 and exp ≠ 111…1
<ul>
<li>Single precision: bias = 127, exp: [1..254], E: [-126..127] =&gt; [10-38 … 1038]</li>
<li>Called “normalized” because binary numbers are normalized</li>
</ul>
<ul>
<li>Effect: “We get the leading bit for free”
<ul>
<li>Leading bit is always assumed (never part of bit pattern)</li>
</ul>
</li>
</ul>
</li>
</ol>
</li>
<li>IEEE floating point number as encoding scheme
<ul>
<li>Fractional decimal number  IEEE 754 (bit pattern)</li>
<li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>V</mi><mo>=</mo><mo stretchy="false">(</mo><mtext>–</mtext><mn>1</mn><msup><mo stretchy="false">)</mo><mi>s</mi></msup><mi>M</mi><msup><mn>2</mn><mi>E</mi></msup></mrow><annotation encoding="application/x-tex">V = (–1)^{s} M 2^{E}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.1413309999999999em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">–1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">s</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span></span></span></span></span></span></span></span></span></span>
<ul>
<li>s is sign bit, M = 1 + frac, E = exp – bias, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>bias</mtext><mo>=</mo><msup><mn>2</mn><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mtext>–</mtext><mn>1</mn></mrow><annotation encoding="application/x-tex">\text{bias} = 2^{k-1} – 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">bias</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8491079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord">–1</span></span></span></span> and k is width of exp</li>
</ul>
</li>
</ul>
</li>
</ul>
<h2 id="next-lecture">Next Lecture</h2>
<ul>
<li>Representing data in memory – Most of this is review
<ul>
<li>“Under the Hood” - Von Neumann architecture</li>
<li>Bits and bytes in memory
<ul>
<li>How to diagram memory -&gt; Used in this course and other references</li>
<li>How to represent series of bits -&gt; In binary, in hexadecimal (conversion)</li>
<li>What kind of information (data) do series of bits represent -&gt; Encoding scheme</li>
<li>Order of bytes in memory -&gt; Endian</li>
</ul>
</li>
<li>Bit manipulation – bitwise operations
<ul>
<li>Boolean algebra + Shifting</li>
</ul>
</li>
</ul>
</li>
<li>Representing integral numbers in memory
<ul>
<li>Unsigned and signed</li>
<li>Converting, expanding and truncating</li>
<li>Arithmetic operations</li>
</ul>
</li>
<li>Representing real numbers in memory
<ul>
<li>IEEE floating point representation</li>
<li>Floating point in C – casting, rounding, addition, …</li>
</ul>
</li>
</ul>
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