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<h1 id="cmpt-295">CMPT 295</h1>
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<p>Unit - Data Representation</p>
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<p>Lecture 3 – Representing integral numbers in memory - unsigned and signed</p>
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<h2 id="last-lecture">Last Lecture</h2>
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<ul>
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<li>Von Neumann architecture
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<ul>
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<li>Architecture of most computers</li>
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<li>Its components: CPU, memory, input and ouput, bus</li>
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<li>One of its characteristics: Data and code (programs) both stored in memory</li>
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</ul>
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</li>
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<li>A look at memory: defined byte-addressable memory, diagram of (compressed) memory
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<ul>
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<li>Word size (w): size of a series of bits (or bit vector) we manipulate, also size of machine words (see Section 2.1.2)</li>
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</ul>
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</li>
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<li>A look at bits in memory
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<ul>
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<li>Why binary numeral system (0 and 1 -> two values) is used to represent information in memory</li>
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<li>Algorithm for converting binary to hexadecimal (hex)
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<ol>
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<li>Partition bit vector into groups of 4 bits, starting from right, i.e., least significant byte (LSB)
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<ul>
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<li>If most significant “byte” (MSB) does not have 8 bits, pad it: add 0’s to its left</li>
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</ul>
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</li>
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<li>Translate each group of 4 bits into its hex value</li>
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</ol>
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</li>
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<li>What do bits represent? Encoding scheme gives meaning to bits</li>
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<li>Order of bytes in memory: little endian versus big endian</li>
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</ul>
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</li>
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<li>Bit manipulation – regardless of what bit vectors represent
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<ul>
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<li>Boolean algebra: bitwise operations => AND (&), OR (|), XOR (^), NOT (~)</li>
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<li>Shift operations: left shift, right logical shift and right arithmetic shift
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<ul>
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<li>Logical shift: Fill x with y 0’s on left</li>
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<li>Arithmetic shift: Fill x with y copies of x‘s sign bit on left</li>
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<li>Sign bit: Most significant bit (MSb) before shifting occurred</li>
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</ul>
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</li>
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</ul>
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</li>
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</ul>
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<p>NOTE: C logical operators and C bitwise (bit-level) operators behave differently! Watch out for && versus &, || versus |, …</p>
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<h2 id="todays-menu">Today’s Menu</h2>
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<ul>
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<li>Representing data in memory – Most of this is review
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<ul>
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<li>“Under the Hood” - Von Neumann architecture</li>
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<li>Bits and bytes in memory
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<ul>
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<li>How to diagram memory -> Used in this course and other references</li>
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<li>How to represent series of bits -> In binary, in hexadecimal (conversion)</li>
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<li>What kind of information (data) do series of bits represent -> Encoding scheme</li>
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<li>Order of bytes in memory -> Endian</li>
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</ul>
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</li>
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<li>Bit manipulation – bitwise operations
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<ul>
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<li>Boolean algebra + Shifting</li>
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</ul>
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</li>
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</ul>
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</li>
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<li>Representing integral numbers in memory
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<ul>
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<li>Unsigned and signed</li>
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<li>Converting, expanding and truncating</li>
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<li>Arithmetic operations</li>
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</ul>
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</li>
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<li>Representing real numbers in memory
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<ul>
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<li>IEEE floating point representation</li>
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<li>Floating point in C – casting, rounding, addition, …</li>
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</ul>
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</li>
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</ul>
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<h2 id="warm-up-exercise">Warm up exercise!</h2>
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<p>As a warm up exercise, fill in the blanks!</p>
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<ul>
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<li>If the context is C (on our target machine)
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<ul>
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<li>char => _____ bits/ _____ byte</li>
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<li>short => _____ bits/ _____ bytes</li>
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<li>int => _____ bits/ _____ bytes</li>
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<li>long => _____ bits/ _____ bytes</li>
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<li>float => _____ bits/ _____ bytes</li>
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<li>double => _____ bits/ _____ bytes</li>
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<li>pointer (e.g. char *) => _____ bits/ _____ bytes</li>
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</ul>
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</li>
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</ul>
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<h2 id="unsigned-integral-numbers">Unsigned integral numbers</h2>
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<p>Remember:</p>
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<table>
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<tbody>
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<tr>
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<td>Address</td>
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<td>M[]</td>
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</tr>
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<tr>
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<td>size-1</td>
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<td> </td>
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</tr>
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<tr>
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<td>…</td>
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<td> </td>
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</tr>
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<tr>
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<td>0x0003</td>
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<td>01000010</td>
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</tr>
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<tr>
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<td>0x0002</td>
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<td>01101001</td>
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</tr>
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<tr>
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<td>0x0001</td>
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<td>01110100</td>
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</tr>
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<tr>
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<td>0x0000</td>
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<td>01110011</td>
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</tr>
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</tbody>
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</table>
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<ul>
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<li>What if the byte at M[0x0002] represented an unsigned integral number, what would be its value?
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x = a series of bits = bit vector
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w = width of bit vector</li>
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<li>
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<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>X</mi><mo>=</mo><msub><mtext>01101001</mtext><mn>2</mn></msub><mo separator="true">,</mo><mi>w</mi><mo>=</mo><mn>8</mn></mrow><annotation encoding="application/x-tex">X = \text{01101001}_{2}, w=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:0.07847em;">X</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:0.8388800000000001em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord text"><span class="mord">01101001</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="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></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>
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</li>
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<li>Let’s apply the encoding scheme:</li>
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</ul>
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<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>B2U</mtext><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="italic">0</mn></mrow><mrow><mi>w</mi><mo>−</mo><mn mathvariant="italic">1</mn></mrow></munderover><msub><mi>X</mi><mi>i</mi></msub><mo>×</mo><msup><mn mathvariant="italic">2</mn><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">
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\it \text{B2U}(X) = \sum_{i=0}^{w-1} X_{i} \times 2^i
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0758700000000005em;vertical-align:-1.274757em;"></span><span class="mord"><span class="mord text"><span class="mord">B2U</span></span><span class="mopen">(</span><span class="mord mathit">X</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 class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8011130000000006em;"><span style="top:-1.875243em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight">i</span><span class="mrel mtight">=</span><span class="mord mathit mtight">0</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight">w</span><span class="mbin mtight">−</span><span class="mord mathit mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.274757em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathit">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.308752em;"><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 mathit mtight">i</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 class="mord"><span class="mord mathit">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.871752em;"><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 mathit mtight">i</span></span></span></span></span></span></span></span></span></span></span></span></span></p>
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<p>Example: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>×</mo><msup><mn>2</mn><mn>7</mn></msup><mo>+</mo><mn>1</mn><mo>×</mo><msup><mn>2</mn><mn>6</mn></msup><mo>+</mo><mn>1</mn><mo>×</mo><msup><mn>2</mn><mn>5</mn></msup><mo>+</mo><mn>0</mn><mo>×</mo><msup><mn>2</mn><mn>4</mn></msup><mo>+</mo><mn>1</mn><mo>×</mo><msup><mn>2</mn><mn>3</mn></msup><mo>+</mo><mn>0</mn><mo>×</mo><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>×</mo><msup><mn>2</mn><mn>0</mn></msup><mo>=</mo></mrow><annotation encoding="application/x-tex">0 \times 2^7 + 1 \times 2^6 + 1 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^0 =</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">0</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:0.897438em;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">7</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:0.72777em;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:0.897438em;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">6</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:0.72777em;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:0.897438em;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">5</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:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">0</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:0.897438em;ve
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<ul>
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<li>For w = 8, range of possible unsigned values: [ <em>blank</em> ]</li>
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<li>For any w, range of possible unsigned values: [ <em>blank</em> ]</li>
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<li>Conclusion: w bits can only represent a fixed # of possible values, but these w bits represent these values exactly</li>
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</ul>
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<h2 id="b2ux-conversion-encoding-scheme">B2U(X) Conversion (Encoding scheme)</h2>
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<ul>
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<li>Positional notation: expand and sum all terms</li>
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</ul>
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<p>Decimal:</p>
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<table>
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<tbody>
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<tr>
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<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">d_{i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></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:0.31166399999999994em;"><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 mathnormal mtight">i</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></td>
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<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mtext>10</mtext><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">\text{10}^{i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.859004em;vertical-align:0em;"></span><span class="mord"><span class="mord text"><span class="mord">10</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.859004em;"><span style="top:-3.09734em;margin-right: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">i</span></span></span></span></span></span></span></span></span></span></span></span></td>
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</tr>
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<tr>
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<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">d_{i-1}</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:0.311664em;"><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 mathnormal mtight">i</span><span class="mbin 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></span></span></td>
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<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>10</mtext><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\text{10}_{i-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.852771em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord text"><span class="mord">10</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><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">i</span><span class="mbin 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></span></span></td>
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</tr>
|
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<tr>
|
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|
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|
<td>…</td>
|
|
|
|
|
|
<td>…</td>
|
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|
|
</tr>
|
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<tr>
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<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">d_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></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">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></td>
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<td>100</td>
|
|
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|
|
|
</tr>
|
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|
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|
|
<tr>
|
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|
|
<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">d_{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></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">1</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></td>
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|
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<td>10</td>
|
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</tr>
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<tr>
|
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<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></mrow><annotation encoding="application/x-tex">d_{0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></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></span></span></td>
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<td>1</td>
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</tr>
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</tbody>
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</table>
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<p>Binary:</p>
|
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|
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|
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|
<table>
|
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<tbody>
|
|
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<tr>
|
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<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>b</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">b_{i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><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 mathnormal mtight">i</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></td>
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<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">2^{i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.824664em;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:0.824664em;"><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">i</span></span></span></span></span></span></span></span></span></span></span></span></td>
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</tr>
|
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<tr>
|
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<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>b</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">b_{i-1}</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">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><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 mathnormal mtight">i</span><span class="mbin 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></span></span></td>
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<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">2^{i-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.824664em;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:0.824664em;"><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">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span></td>
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</tr>
|
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<tr>
|
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|
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|
<td>…</td>
|
|
|
|
|
|
<td>…</td>
|
|
|
|
|
|
</tr>
|
|
|
|
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<tr>
|
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<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>b</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">b_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">b</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></td>
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<td>4</td>
|
|
|
|
|
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</tr>
|
|
|
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|
|
<tr>
|
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<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>b</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">b_{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">b</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></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></td>
|
|
|
|
|
|
<td>2</td>
|
|
|
|
|
|
</tr>
|
|
|
|
|
|
<tr>
|
|
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<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>b</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">b_{0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">b</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></span></span></td>
|
|
|
|
|
|
<td>1</td>
|
|
|
|
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</tr>
|
|
|
|
|
|
</tbody>
|
|
|
|
|
|
</table>
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<p>Remember:
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<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>B2U</mtext><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="italic">0</mn></mrow><mrow><mi>w</mi><mo>−</mo><mn mathvariant="italic">1</mn></mrow></munderover><msub><mi>X</mi><mi>i</mi></msub><mo>×</mo><msup><mn mathvariant="italic">2</mn><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">
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\it \text{B2U}(X) = \sum_{i=0}^{w-1} X_{i} \times 2^i
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0758700000000005em;vertical-align:-1.274757em;"></span><span class="mord"><span class="mord text"><span class="mord">B2U</span></span><span class="mopen">(</span><span class="mord mathit">X</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 class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8011130000000006em;"><span style="top:-1.875243em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight">i</span><span class="mrel mtight">=</span><span class="mord mathit mtight">0</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight">w</span><span class="mbin mtight">−</span><span class="mord mathit mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.274757em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathit">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.308752em;"><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 mathit mtight">i</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 class="mord"><span class="mord mathit">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.871752em;"><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 mathit mtight">i</span></span></span></span></span></span></span></span></span></span></span></span></span></p>
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<p>Example: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>246</mtext><mn>10</mn></msub><mo>=</mo><mn>2</mn><mo>×</mo><msup><mtext>10</mtext><mn>2</mn></msup><mo>+</mo><mn>4</mn><mo>×</mo><msup><mtext>10</mtext><mn>1</mn></msup><mo>+</mo><mn>6</mn><mo>×</mo><msup><mtext>10</mtext><mn>0</mn></msup></mrow><annotation encoding="application/x-tex">\text{246}_{10} = 2 \times \text{10}^{2} + 4 \times \text{10}^{1} + 6 \times \text{10}^{0}</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 text"><span class="mord">246</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">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">2</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:0.931778em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord text"><span class="mord">10</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.848448em;"><span style="top:-3.09734em;margin-right: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></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:0.72777em;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:0.931778em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord text"><span class="mord">10</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.848448em;"><span style="top:-3.09734em;margin-right: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 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">6</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:0.848448em;vertical-align:0em;"></span><span class="mord"><span class="mord text"><span class="mord">10</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vl
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<h2 id="range-of-possible-values">Range of possible values?</h2>
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<ul>
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<li>If the context is C (on our target machine)
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<ul>
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<li>unsigned char?</li>
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<li>unsigned short?</li>
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<li>unsigned int?</li>
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<li>unsigned long?</li>
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</ul>
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</li>
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</ul>
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<h2 id="examples-of-show-your-work">Examples of “Show your work”</h2>
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<p>U2B(X) Conversion (into 8-bit binary # => w = 8)</p>
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<p>Method 1 - Using subtraction: subtracting decreasing power of 2 until reach 0:</p>
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<p>Starting number: 246</p>
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<ul>
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<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>246</mn><mtext>–</mtext><mn>128</mn><mo>=</mo><mn>118</mn></mrow><annotation encoding="application/x-tex">246 – 128 = 118</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">246–128</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">118</span></span></span></span> -> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>128</mn><mo>=</mo><mn>1</mn><mo>×</mo><msup><mn>2</mn><mn>7</mn></msup></mrow><annotation encoding="application/x-tex">128 = 1 \times 2^{7}</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">128</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:0.8141079999999999em;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></span></span></li>
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<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>118</mn><mtext>–</mtext><mn>64</mn><mo>=</mo><mn>54</mn></mrow><annotation encoding="application/x-tex">118 – 64 = 54</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">118–64</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">54</span></span></span></span> -> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>64</mn><mo>=</mo><mn>1</mn><mo>×</mo><msup><mn>2</mn><mn>6</mn></msup></mrow><annotation encoding="application/x-tex">64 = 1 \times 2^{6}</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">64</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:0.8141079999999999em;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">6</span></span></span></span></span></span></span></span></span></span></span></span></li>
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<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>54</mn><mtext>–</mtext><mn>32</mn><mo>=</mo><mn>22</mn></mrow><annotation encoding="application/x-tex">54 – 32 = 22</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">54–32</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">22</span></span></span></span> -> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>32</mn><mo>=</mo><mn>1</mn><mo>×</mo><msup><mn>2</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">32 = 1 \times 2^{5}</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">32</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:0.8141079999999999em;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></li>
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<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>22</mn><mtext>–</mtext><mn>16</mn><mo>=</mo><mn>6</mn></mrow><annotation encoding="application/x-tex">22 – 16 = 6</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">22–16</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">6</span></span></span></span> -> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>16</mn><mo>=</mo><mn>1</mn><mo>×</mo><msup><mn>2</mn><mn>4</mn></msup></mrow><annotation encoding="application/x-tex">16 = 1 \times 2^{4}</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">16</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:0.8141079999999999em;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">4</span></span></span></span></span></span></span></span></span></span></span></span></li>
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<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>6</mn><mtext>–</mtext><mn>8</mn><mo>=</mo><mtext>nop!</mtext></mrow><annotation encoding="application/x-tex">6 – 8 = \text{nop!}</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">6–8</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:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord">nop!</span></span></span></span></span> -> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>8</mn><mo>=</mo><mn>0</mn><mo>×</mo><msup><mn>2</mn><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">8 = 0 \times 2^{3}</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">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">0</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></li>
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<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>6</mn><mtext>–</mtext><mn>4</mn><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">6 – 4 = 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">6–4</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">2</span></span></span></span> -> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mo>=</mo><mn>1</mn><mo>×</mo><msup><mn>2</mn><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">4 = 1 \times 2^{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">4</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:0.8141079999999999em;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">2</span></span></span></span></span></span></span></span></span></span></span></span></li>
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<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mtext>–</mtext><mn>2</mn><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">2 – 2 = 0</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">2–2</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">0</span></span></span></span> -> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mo>=</mo><mn>1</mn><mo>×</mo><msup><mn>2</mn><mn>1</mn></msup></mrow><annotation encoding="application/x-tex">2 = 1 \times 2^{1}</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">2</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:0.8141079999999999em;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></li>
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<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mtext>–</mtext><mn>1</mn><mo>=</mo><mtext>nop!</mtext></mrow><annotation encoding="application/x-tex">0 – 1 = \text{nop!}</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">0–1</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:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord">nop!</span></span></span></span></span> -> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>=</mo><mn>0</mn><mo>×</mo><msup><mn>2</mn><mn>0</mn></msup></mrow><annotation encoding="application/x-tex">1 = 0 \times 2^{0}</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">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">0</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">0</span></span></span></span></span></span></span></span></span></span></span></span></li>
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<li>
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<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><mtext>246</mtext><mn>10</mn></msub><mo>=</mo><msub><mtext>11110110</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{246}_{10} = \text{11110110}_{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 text"><span class="mord">246</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">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 class="mord text"><span class="mord">11110110</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></span></span></span>
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</li>
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</ul>
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<p>Method 2 - Using division: dividing by 2 until reach 0</p>
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<p>Start with 246</p>
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<ul>
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<li>
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<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>246</mn><mo>÷</mo><mn>2</mn><mo>=</mo><mn>123</mn><mo separator="true">,</mo><mi>R</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">246 \div 2 = 123,R=0</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">246</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">2</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:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord">123</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</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">0</span></span></span></span></span>
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</li>
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<li>
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<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>123</mn><mo>÷</mo><mn>2</mn><mo>=</mo><mn>61</mn><mo separator="true">,</mo><mi>R</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">123 \div 2 = 61,R=1</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">123</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">2</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:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord">61</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</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">1</span></span></span></span></span>
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</li>
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<li>
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<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>61</mn><mo>÷</mo><mn>2</mn><mo>=</mo><mn>30</mn><mo separator="true">,</mo><mi>R</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">61 \div 2 = 30,R=1</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">61</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">2</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:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord">30</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</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">1</span></span></span></span></span>
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</li>
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<li>
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<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>30</mn><mo>÷</mo><mn>2</mn><mo>=</mo><mn>15</mn><mo separator="true">,</mo><mi>R</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">30 \div 2 = 15,R=0</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">30</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">2</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:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord">15</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</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">0</span></span></span></span></span>
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</li>
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<li>
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<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>15</mn><mo>÷</mo><mn>2</mn><mo>=</mo><mn>7</mn><mo separator="true">,</mo><mi>R</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">15 \div 2 = 7,R=1</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">15</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">2</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:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord">7</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</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">1</span></span></span></span></span>
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</li>
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<li>
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<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>7</mn><mo>÷</mo><mn>2</mn><mo>=</mo><mn>3</mn><mo separator="true">,</mo><mi>R</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">7 \div 2 = 3,R=1</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">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">2</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:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</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">1</span></span></span></span></span>
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</li>
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<li>
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<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>3</mn><mo>÷</mo><mn>2</mn><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mi>R</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">3 \div 2 = 1,R=1</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">3</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">2</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:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</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">1</span></span></span></span></span>
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</li>
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<li>
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<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>1</mn><mo>÷</mo><mn>2</mn><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mi>R</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1 \div 2 = 0,R=1</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="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">2</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:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</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">1</span></span></span></span></span>
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</li>
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<li>
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<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><mtext>246</mtext><mn>10</mn></msub><mo>=</mo><msub><mtext>11110110</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{246}_{10} = \text{11110110}_{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 text"><span class="mord">246</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">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 class="mord text"><span class="mord">11110110</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></span></span></span>
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</li>
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</ul>
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<h2 id="u2bx-conversion--a-few-tricks">U2B(X) Conversion – A few tricks</h2>
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<ul>
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<li>Decimal -> binary
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<ul>
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<li>Trick: When decimal number is 2n, then its binary representation is 1 followed by n zero’s</li>
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<li>Let’s try: <code class="language-plaintext highlighter-rouge">if X = 32 => X = 25, then n = 5 => 100002 (w = 5)</code> What if w = 8? Check: 1 x 24 = 32</li>
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</ul>
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</li>
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<li>Decimal -> hex
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<ul>
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<li>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</li>
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<li>Let try: if X = 8192 => X = 213, then n = 13 and 13 = i + 4j => 1 + 4 x 3
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=> 0x2000. Convert 0x2000 into a binary number: Check: 2 x 163 = 2 x 4096 = 8192</li>
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</ul>
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</li>
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</ul>
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<h2 id="signed-integral-numbers">Signed integral numbers</h2>
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<h3 id="remember">Remember:</h3>
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<table>
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<tbody>
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<tr>
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<td>Address</td>
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<td>M[]</td>
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</tr>
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<tr>
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<td>size-1</td>
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<td> </td>
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</tr>
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<tr>
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<td>…</td>
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<td> </td>
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</tr>
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<tr>
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<td>0x0003</td>
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<td>01000010</td>
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</tr>
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<tr>
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<td>0x0002</td>
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<td>01101001</td>
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</tr>
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<tr>
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<td>0x0001</td>
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<td>01110100</td>
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</tr>
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<tr>
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<td>0x0000</td>
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<td>01110011</td>
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</tr>
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</tbody>
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</table>
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<ul>
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<li>What if the byte at M[0x0001] represented a signed integral number, what would be its value?</li>
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<li>X = 111101002 w = 8</li>
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<li>T => Two’s Complement, w => width of the bit vector (annotation: first part of equaltion [everything before the plus sign] is the “sign bit”)</li>
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|
<li>Let’s apply the encoding scheme:</li>
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</ul>
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<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi>B</mi><mn mathvariant="italic">2</mn><mi>T</mi></mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><msub><mi>x</mi><mrow><mi>w</mi><mo>−</mo><mn mathvariant="italic">1</mn></mrow></msub><mo>×</mo><msup><mn mathvariant="italic">2</mn><mrow><mi>w</mi><mo>−</mo><mn mathvariant="italic">1</mn></mrow></msup><mo>+</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="italic">0</mn></mrow><mrow><mi>w</mi><mo>−</mo><mn mathvariant="italic">2</mn></mrow></munderover><msub><mi>x</mi><mi>i</mi></msub><mo>×</mo><msup><mn mathvariant="italic">2</mn><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">
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\it {B2T}(X) = -x_{w-1} \times 2^{w-1} + \sum_{i=0}^{w-2} x_{i} \times 2^{i}
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0758700000000005em;vertical-align:-1.274757em;"></span><span class="mord"><span class="mord"><span class="mord mathit">B</span><span class="mord mathit">2</span><span class="mord mathit">T</span></span><span class="mopen">(</span><span class="mord mathit">X</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 class="mord">−</span><span class="mord"><span class="mord mathit">x</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 mathit mtight">w</span><span class="mbin mtight">−</span><span class="mord mathit 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="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathit">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.864108em;"><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 mathit mtight">w</span><span class="mbin mtight">−</span><span class="mord mathit 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 class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8011130000000006em;"><span style="top:-1.875243em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight">i</span><span class="mrel mtight">=</span><span class="mord mathit mtight">0</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathit mtight">w</span><span class="mbin mtight">−</span><span class="mord mathit mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.274757em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathit">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.308752em;"><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 mathit mtight">i</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 class="mord"><span class="mor
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<p>Example: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>1</mn><mo>×</mo><msup><mn>2</mn><mn>7</mn></msup><mo>+</mo><mn>1</mn><mo>×</mo><msup><mn>2</mn><mn>6</mn></msup><mo>+</mo><mn>1</mn><mo>×</mo><msup><mn>2</mn><mn>5</mn></msup><mo>+</mo><mn>1</mn><mo>×</mo><msup><mn>2</mn><mn>4</mn></msup><mo>+</mo><mn>0</mn><mo>×</mo><msup><mn>2</mn><mn>3</mn></msup><mo>+</mo><mn>1</mn><mo>×</mo><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><mn>0</mn><mo>×</mo><msup><mn>2</mn><mn>1</mn></msup><mo>+</mo><mn>0</mn><mo>×</mo><msup><mn>2</mn><mn>0</mn></msup><mo>=</mo><mo stretchy="false">?</mo></mrow><annotation encoding="application/x-tex">-1 \times 2^{7} + 1 \times 2^{6} + 1 \times 2^{5} + 1 \times 2^{4} + 0 \times 2^{3} + 1 \times 2^{2} + 0 \times 2^{1} + 0 \times 2^{0} = ?</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">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:0.897438em;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:0.72777em;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:0.897438em;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">6</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:0.72777em;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:0.897438em;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">5</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:0.72777em;vertical-align:-0.08333em;"></span><span
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<ul>
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|
|
<li>What would be the bit pattern of the …
|
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|
|
<ul>
|
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|
|
<li>Most negative value:</li>
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|
|
<li>Most positive value:</li>
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|
</ul>
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|
</li>
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<li>For w = 8, range of possible signed values: [ <em>blank</em> ]</li>
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<li>For any w, range of possible signed values: [ <em>blank</em> ]</li>
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|
|
<li>Conclusion: same as for unsigned integral numbers</li>
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|
</ul>
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<h2 id="examples-of-show-your-work-1">Examples of “Show your work”</h2>
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<p>T2B(X) Conversion -> Two’s Complement</p>
|
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<p>Annotation: w = 8</p>
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<p>Method 1: If X < 0, (~(U2B(|X|)))+1</p>
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<p>If X = -14 (and 8 bit binary #s)</p>
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<ol>
|
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<li>
|
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<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><mi>X</mi><mo stretchy="false">∣</mo><mo>=</mo><mo>></mo><mo stretchy="false">∣</mo><mo>−</mo><mn>14</mn><mi mathvariant="normal">∣</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\lvert X\rvert => \lvert -14 \vert =</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" style="margin-right:0.07847em;">X</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">−</span><span class="mord">14∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span></span></span></span></span>
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</li>
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<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>U2B</mtext><mo stretchy="false">(</mo><mn>14</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{U2B}(14)</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 text"><span class="mord">U2B</span></span><span class="mopen">(</span><span class="mord">14</span><span class="mclose">)</span></span></span></span> =></li>
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<li>(first symbol is a tilde) <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∼</mo><mo stretchy="false">(</mo><msub><mtext>00001110</mtext><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\sim(\text{00001110}_{2})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;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:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord text"><span class="mord">00001110</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="mclose">)</span></span></span></span> =></li>
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<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mtext>11110001</mtext><mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">(\text{11110001}_{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"><span class="mord text"><span class="mord">11110001</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="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> =></li>
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</ol>
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<p>Binary addition:</p>
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<pre>
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11110001
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+ 00000001
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= ????????
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</pre>
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<p>Method 2: If X = -14 (and 8 bit binary #s)</p>
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<ol>
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<li>
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<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>X</mi><mo>+</mo><msup><mn>2</mn><mi>w</mi></msup><mo>=</mo><mo>></mo><mo>−</mo><mn>14</mn><mo>+</mo></mrow><annotation encoding="application/x-tex">X + 2^{w} => -14 +</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">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:0.7534919999999999em;vertical-align:-0.0391em;"></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="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">14</span><span class="mord">+</span></span></span></span></span>
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</li>
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<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>U2B</mtext><mo stretchy="false">(</mo><mn>242</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{U2B}(242)</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 text"><span class="mord">U2B</span></span><span class="mopen">(</span><span class="mord">242</span><span class="mclose">)</span></span></span></span> =></li>
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</ol>
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<p>Using subtraction:</p>
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<ol>
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<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>242</mn><mo>−</mo><mn>128</mn><mo>=</mo><mn>114</mn></mrow><annotation encoding="application/x-tex">242 - 128 = 114</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">242</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">128</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">114</span></span></span></span> -> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>×</mo><msup><mn>2</mn><mn>7</mn></msup></mrow><annotation encoding="application/x-tex">1 \times 2^{7}</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="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">7</span></span></span></span></span></span></span></span></span></span></span></span></li>
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<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>114</mn><mo>−</mo><mn>64</mn><mo>=</mo><mn>50</mn></mrow><annotation encoding="application/x-tex">114 - 64 = 50</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">114</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">64</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">50</span></span></span></span> -> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>×</mo><msup><mn>2</mn><mn>6</mn></msup></mrow><annotation encoding="application/x-tex">1 \times 2^{6}</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="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">6</span></span></span></span></span></span></span></span></span></span></span></span></li>
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<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>50</mn><mtext>–</mtext><mn>32</mn><mo>=</mo><mn>18</mn></mrow><annotation encoding="application/x-tex">50 – 32 = 18</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">50–32</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">18</span></span></span></span> -> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>×</mo><msup><mn>2</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">1 \times 2^{5}</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="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></li>
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<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>18</mn><mtext>–</mtext><mn>16</mn><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">18 – 16 = 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">18–16</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">2</span></span></span></span> -> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>×</mo><msup><mn>2</mn><mn>4</mn></msup></mrow><annotation encoding="application/x-tex">1 \times 2^{4}</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="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">4</span></span></span></span></span></span></span></span></span></span></span></span></li>
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<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mtext>–</mtext><mn>8</mn><mo>=</mo><mtext>nop!</mtext></mrow><annotation encoding="application/x-tex">2 – 8 = \text{nop!}</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">2–8</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:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord">nop!</span></span></span></span></span> -> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>×</mo><msup><mn>2</mn><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">0 \times 2^{3}</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">0</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></li>
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<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mtext>–</mtext><mn>4</mn><mo>=</mo><mtext>nop!</mtext></mrow><annotation encoding="application/x-tex">2 – 4 = \text{nop!}</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">2–4</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:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord">nop!</span></span></span></span></span> -> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>×</mo><msup><mn>2</mn><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">0 \times 2^{2}</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">0</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">2</span></span></span></span></span></span></span></span></span></span></span></span></li>
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<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mtext>–</mtext><mn>2</mn><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">2 – 2 = 0</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">2–2</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">0</span></span></span></span> -> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>×</mo><msup><mn>2</mn><mn>1</mn></msup></mrow><annotation encoding="application/x-tex">1 \times 2^{1}</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="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">1</span></span></span></span></span></span></span></span></span></span></span></span></li>
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<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mtext>–</mtext><mn>1</mn><mo>=</mo><mtext>nop!</mtext></mrow><annotation encoding="application/x-tex">0 – 1 = \text{nop!}</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">0–1</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:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord">nop!</span></span></span></span></span> -> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>×</mo><msup><mn>2</mn><mn>0</mn></msup></mrow><annotation encoding="application/x-tex">0 \times 2^{0}</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">0</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">0</span></span></span></span></span></span></span></span></span></span></span></span></li>
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</ol>
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<h2 id="properties-of-unsigned--signed-conversions">Properties of unsigned & signed conversions</h2>
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<p>Annotation: w = 4</p>
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<table>
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<tbody>
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<tr>
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<td>X</td>
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<td>B2U(X)</td>
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<td>B2T(X)</td>
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</tr>
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<tr>
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<td>0000</td>
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<td>0</td>
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<td>0</td>
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</tr>
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<tr>
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<td>0001</td>
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<td>1</td>
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<td>1</td>
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</tr>
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<tr>
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<td>0010</td>
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<td>2</td>
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<td>2</td>
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</tr>
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<tr>
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<td>0011</td>
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<td>3</td>
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<td>3</td>
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</tr>
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<tr>
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<td>0100</td>
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<td>4</td>
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<td>4</td>
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</tr>
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<tr>
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<td>0101</td>
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<td>5</td>
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<td>5</td>
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</tr>
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<tr>
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<td>0110</td>
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<td>6</td>
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<td>6</td>
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</tr>
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<tr>
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<td>0111</td>
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<td>7</td>
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<td>7</td>
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</tr>
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<tr>
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<td>1000</td>
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<td>8</td>
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<td>-8</td>
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</tr>
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<tr>
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<td>1001</td>
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<td>9</td>
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<td>-7</td>
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</tr>
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<tr>
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<td>1010</td>
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<td>10</td>
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<td>-6</td>
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</tr>
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<tr>
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<td>1011</td>
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<td>11</td>
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<td>-5</td>
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</tr>
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<tr>
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<td>1100</td>
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<td>12</td>
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<td>-4</td>
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</tr>
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<tr>
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<td>1101</td>
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<td>13</td>
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<td>-3</td>
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</tr>
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<tr>
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<td>1110</td>
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<td>14</td>
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<td>-2</td>
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</tr>
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<tr>
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<td>1111</td>
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<td>15</td>
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<td>-1</td>
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</tr>
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</tbody>
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</table>
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<ul>
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<li>Equivalence
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<ul>
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<li>Both encoding schemes (B2U and B2T ) produce the same bit patterns for nonnegative values</li>
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</ul>
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</li>
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<li>Uniqueness
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<ul>
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<li>Every bit pattern produced by these encoding schemes (B2U and B2T) represents a unique (and exact) integer value</li>
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<li>Each representable integer has unique bit pattern</li>
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</ul>
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</li>
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</ul>
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<h2 id="converting-between-signed--unsigned-of-same-size-same-data-type">Converting between signed & unsigned of same size (same data type)</h2>
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<ul>
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<li>Unsigned
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<ul>
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<li>w=8</li>
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<li>if unsigned <code class="language-plaintext highlighter-rouge">ux</code> = 129<sub>10</sub></li>
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<li>U2T(X) = B2T(U2B(X))</li>
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<li>then x = ???</li>
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<li>Maintain Same Bit Pattern</li>
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</ul>
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</li>
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<li>Signed (Two’s Complement)
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<ul>
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<li>w=4</li>
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<li>if signed (2’s C) <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mo>−</mo><msub><mn>5</mn><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">x = -5_{10}</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">x</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"><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>
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<li>T2U(X) = B2U(T2B(X))</li>
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<li>then unsigned <code class="language-plaintext highlighter-rouge">ux</code> = ???</li>
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<li>Maintain Same Bit Pattern</li>
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</ul>
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</li>
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<li>Conclusion - Converting between unsigned and signed numbers:
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Both have same bit pattern, however, this bit pattern may be interpreted differently, i.e., producing a different value</li>
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</ul>
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<h2 id="converting-signed-to-unsigned-and-back-with-w--4">Converting signed to unsigned (and back) with <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">w = 4</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">4</span></span></span></span></h2>
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<table>
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<tbody>
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<tr>
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<td>Signed</td>
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<td>Bits</td>
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<td>Unsigned</td>
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<td>Note</td>
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</tr>
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<tr>
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<td>0</td>
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<td>0000</td>
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<td>0</td>
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<td>All rows from 0-7 inclusive can be converted from signed to unsigned with T2U(X), and unsigned to signed with U2T(X).</td>
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</tr>
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<tr>
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<td>1</td>
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<td>0001</td>
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<td>1</td>
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<td> </td>
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</tr>
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<tr>
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<td>2</td>
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<td>0010</td>
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<td>2</td>
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<td> </td>
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</tr>
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<tr>
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<td>3</td>
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<td>0011</td>
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<td>3</td>
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<td> </td>
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</tr>
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<tr>
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<td>4</td>
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<td>0100</td>
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<td>4</td>
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<td> </td>
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</tr>
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<tr>
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<td>5</td>
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<td>0101</td>
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<td>5</td>
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<td> </td>
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</tr>
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<tr>
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<td>6</td>
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<td>0110</td>
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<td>6</td>
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<td> </td>
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</tr>
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<tr>
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<td>7</td>
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<td>0111</td>
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<td>7</td>
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<td> </td>
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</tr>
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<tr>
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<td>-8</td>
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<td>1000</td>
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<td>8</td>
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<td>All rows from here to 15 inclusive can be converted to the other like so: T2U(signed + 16/<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>4</mn></msup></mrow><annotation encoding="application/x-tex">2^{4}</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">4</span></span></span></span></span></span></span></span></span></span></span></span>) -> unsigned, U2T(unsigned - 16/<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>4</mn></msup></mrow><annotation encoding="application/x-tex">2^{4}</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">4</span></span></span></span></span></span></span></span></span></span></span></span>) -> signed.</td>
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</tr>
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<tr>
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<td>-7</td>
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<td>1001</td>
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<td>9</td>
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<td> </td>
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</tr>
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<tr>
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<td>-6</td>
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<td>1010</td>
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<td>10</td>
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<td> </td>
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</tr>
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<tr>
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<td>-5</td>
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<td>1011</td>
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<td>11</td>
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<td> </td>
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</tr>
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<tr>
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<td>-4</td>
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<td>1100</td>
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<td>12</td>
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<td> </td>
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</tr>
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<tr>
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<td>-3</td>
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<td>1101</td>
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<td>13</td>
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<td> </td>
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</tr>
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<tr>
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<td>-2</td>
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<td>1110</td>
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<td>14</td>
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<td> </td>
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</tr>
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<tr>
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<td>-1</td>
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<td>1111</td>
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<td>15</td>
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<td> </td>
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</tr>
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</tbody>
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</table>
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<h2 id="visualizing-the-relationship-between-signed--unsigned">Visualizing the relationship between signed & unsigned</h2>
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<p>If <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi><mo>=</mo><mn>4</mn><mo separator="true">,</mo><msup><mn>2</mn><mn>4</mn></msup><mo>=</mo><mn>16</mn></mrow><annotation encoding="application/x-tex">w = 4, 2^{4} = 16</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:1.008548em;vertical-align:-0.19444em;"></span><span class="mord">4</span><span class="mpunct">,</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:0.8141079999999999em;"><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">4</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:0.64444em;vertical-align:0em;"></span><span class="mord">16</span></span></span></span></p>
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<ul>
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<li>Signed (2’s Complement) Range: TMin to TMax (0 is the center)</li>
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<li>Unsigned range: 0 to UMax (TMax is the center)</li>
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</ul>
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<h2 id="sign-extension">Sign extension</h2>
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<ul>
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<li>Converting unsigned (or signed) of different sizes (different data types)
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<ol>
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<li>Small data type -> larger
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<ul>
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<li>Sign extension
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<ul>
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<li>Unsigned: zero extension</li>
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<li>Signed: sign bit extension</li>
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</ul>
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</li>
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</ul>
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</li>
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</ol>
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</li>
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<li>Conclusion: Value unchanged</li>
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<li>Let’s try:
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<ul>
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<li>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)</li>
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</ul>
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</li>
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<li>Unsigned: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><mn>3</mn><mo>=</mo><msub><mtext>011</mtext><mn>2</mn></msub><mo separator="true">,</mo><mi>w</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">X = 3 = \text{011}_{2},w=3</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:0.07847em;">X</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">3</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:0.8388800000000001em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord text"><span class="mord">011</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="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></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">3</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><mn>4</mn><mo>=</mo><msub><mtext>100</mtext><mn>2</mn></msub><mo separator="true">,</mo><mi>w</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">X = 4 = \text{100}_{2},w = 3</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:0.07847em;">X</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">4</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:0.8388800000000001em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord text"><span class="mord">100</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="mpunct">,</span><span class="mspace" style="margin-right:0.166
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<ul>
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<li>New: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><mo stretchy="false">?</mo><mo>=</mo><msub><mo stretchy="false">?</mo><mn>2</mn></msub><mo separator="true">,</mo><mi>w</mi><mo>=</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">X = ? = ?_{2},w=5</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:0.07847em;">X</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 class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mclose"><span class="mclose">?</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="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></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">5</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><mo stretchy="false">?</mo><mo>+</mo><msub><mo stretchy="false">?</mo><mn>2</mn></msub><mo separator="true">,</mo><mi>w</mi><mo>=</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">X = ? + ?_{2},w=5</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:0.07847em;">X</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mclose">?</span><span class="mord">+</span><span class="mclose"><span class="mclose">?</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="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></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">5</span></span></span></span></li>
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</ul>
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</li>
|
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<li>Signed: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><mn>3</mn><mo>=</mo><msub><mtext>011</mtext><mn>2</mn></msub><mo separator="true">,</mo><mi>w</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">X = 3 = \text{011}_{2},w=3</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:0.07847em;">X</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">3</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:0.8388800000000001em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord text"><span class="mord">011</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="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></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">3</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><mo>−</mo><mn>3</mn><mo>=</mo><msub><mtext>101</mtext><mn>2</mn></msub><mo separator="true">,</mo><mi>w</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">X=-3 = \text{101}_{2},w=3</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:0.07847em;">X</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">3</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:0.8388800000000001em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord text"><span class="mord">101</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="mpunct">,</span><sp
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<ul>
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<li>New: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><mo stretchy="false">?</mo><mo>=</mo><msub><mo stretchy="false">?</mo><mn>2</mn></msub><mo separator="true">,</mo><mi>w</mi><mo>=</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">X = ? = ?_{2},w=5</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:0.07847em;">X</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 class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mclose"><span class="mclose">?</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="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></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">5</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mo stretchy="false">?</mo><mo>=</mo><msub><mo stretchy="false">?</mo><mn>2</mn></msub><mo separator="true">,</mo><mi>w</mi><mo>=</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">x = ? = ?_{2}, w=5</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">x</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 class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mclose"><span class="mclose">?</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="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></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="st
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</ul>
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</li>
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</ul>
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<h2 id="truncation">Truncation</h2>
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|
<ul>
|
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|
|
|
|
<li>Converting unsigned (or signed) of different sizes(different data types)
|
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|
<ol>
|
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|
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|
|
<li>Large data type -> smaller
|
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<ul>
|
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|
<li>Truncation</li>
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</ul>
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</li>
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</ol>
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</li>
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<li>Conclusion: Value may be altered
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<ul>
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<li>A form of overflow</li>
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</ul>
|
|
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|
</li>
|
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|
<li>Let’s try:
|
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|
|
|
<ul>
|
|
|
|
|
|
<li>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)</li>
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|
|
</ul>
|
|
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|
</li>
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<li>Unsigned: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><mn>27</mn><mo>=</mo><msub><mtext>11011</mtext><mn>2</mn></msub><mo separator="true">,</mo><mi>w</mi><mo>=</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">X = 27 = \text{11011}_{2},w = 5</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:0.07847em;">X</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">27</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:0.8388800000000001em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord text"><span class="mord">11011</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="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></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">5</span></span></span></span>
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<ul>
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<li>New: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><mo stretchy="false">?</mo><mo>=</mo><msub><mo stretchy="false">?</mo><mn>2</mn></msub><mo separator="true">,</mo><mi>w</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">X = ? = ?_{2},w=3</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:0.07847em;">X</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 class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mclose"><span class="mclose">?</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="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></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">3</span></span></span></span></li>
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</ul>
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</li>
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<li>Signed: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><mo>−</mo><mn>15</mn><mo>=</mo><msub><mtext>10001</mtext><mn>2</mn></msub><mo separator="true">,</mo><mi>w</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">X = -15 = \text{10001}_{2},w=3</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:0.07847em;">X</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">15</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:0.8388800000000001em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord text"><span class="mord">10001</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="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></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">3</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><mo>−</mo><mn>1</mn><mo>=</mo><msub><mtext>11111</mtext><mn>2</mn></msub><mo separator="true">,</mo><mi>w</mi><mo>=</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">X = -1 = \text{11111}_{2}, w=5</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:0.07847em;">X</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="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:0.8388800000000001em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord text"><span class="mord">11111</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>
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<ul>
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<li>New: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><mo stretchy="false">?</mo><mo>=</mo><msub><mo stretchy="false">?</mo><mn>2</mn></msub><mo separator="true">,</mo><mi>w</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">X = ? = ?_{2}, w=3</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:0.07847em;">X</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 class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mclose"><span class="mclose">?</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="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></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">3</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><mo stretchy="false">?</mo><mo>=</mo><msub><mo stretchy="false">?</mo><mn>2</mn></msub><mo separator="true">,</mo><mi>w</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">X = ? = ?_{2}, w=3</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:0.07847em;">X</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 class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mclose"><span class="mclose">?</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="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></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><
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</ul>
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</li>
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</ul>
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<h2 id="summary">Summary</h2>
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<ul>
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<li>Interpretation of bit pattern B into either unsigned value U or signed value T
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<ul>
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<li>B2U(X) and U2B(X) encoding schemes (conversion)</li>
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<li>B2T(X) and T2B(X) encoding schemes (conversion)
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<ul>
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<li>Signed value expressed as two’s complement => T</li>
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</ul>
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</li>
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</ul>
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</li>
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<li>Conversions from unsigned <-> signed values
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<ul>
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<li>U2T(X) and T2U(X) => adding or subtracting <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>w</mi></msup></mrow><annotation encoding="application/x-tex">2^{w}</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 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>
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</ul>
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</li>
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<li>Implication in C: when converting (implicitly via promotion and explicitly via casting):
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<ul>
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<li>Sign:
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<ul>
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<li>Unsigned <-> signed (of same size) -> Both have same bit pattern, however, this bit pattern may be interpreted differently
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<ul>
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<li>Can have unexpected effects -> producing a different value</li>
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</ul>
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</li>
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</ul>
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</li>
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<li>Size:
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<ul>
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<li>Small -> large (for signed, e.g., short to int and for unsigned, e.g., unsigned short to unsigned int)
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<ul>
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<li>sign extension: For unsigned -> zeros extension, for signed -> sign bit extension</li>
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<li>Both yield expected result –> resulting value unchanged</li>
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</ul>
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</li>
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<li>Large -> small (e.g., unsigned int to unsigned short)
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<ul>
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<li>truncation: Unsigned/signed -> most significant bits are truncated (discarded)</li>
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<li>May not yield expected results -> original value may be altered</li>
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</ul>
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</li>
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</ul>
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</li>
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<li>Both (sign and size): 1) size conversion is first done then 2) sign conversion is done</li>
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</ul>
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</li>
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</ul>
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<h2 id="next-lecture">Next Lecture</h2>
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<ul>
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<li>Representing data in memory – Most of this is review
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<ul>
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<li>“Under the Hood” - Von Neumann architecture</li>
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<li>Bits and bytes in memory
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<ul>
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<li>How to diagram memory -> Used in this course and other references</li>
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<li>How to represent series of bits -> In binary, in hexadecimal (conversion)</li>
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<li>What kind of information (data) do series of bits represent -> Encoding scheme</li>
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<li>Order of bytes in memory -> Endian</li>
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</ul>
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</li>
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<li>Bit manipulation – bitwise operations
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<ul>
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<li>Boolean algebra + Shifting</li>
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</ul>
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</li>
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</ul>
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</li>
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<li>Representing integral numbers in memory
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<ul>
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<li>Unsigned and signed</li>
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<li>Converting, expanding and truncating</li>
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<li>Arithmetic operations</li>
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</ul>
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</li>
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<li>Representing real numbers in memory
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19
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<ul>
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<li>IEEE floating point representation</li>
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<li>Floating point in C – casting, rounding, addition, …</li>
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</ul>
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</li>
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</ul>
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</div>
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</main>
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<hr>
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<footer>
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</footer>
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</body>
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</html>
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