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<main>
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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 4 – Representing integral numbers in memory – Arithmetic operations</p>
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<h2 id="warm-up-question">Warm up question</h2>
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<ul>
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<li>What is the value of …
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<ul>
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<li>TMin (in hex) for signed char in C: _________________</li>
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<li>TMax (in hex) for signed int in C: _________________</li>
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<li>TMin (in hex) for signed short in C: ________________</li>
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</ul>
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</li>
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</ul>
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<h2 id="last-lecture">Last Lecture</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 2w</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 (for signed, e.g., int to short and for unsigned, 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="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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<p>Let’s first illustrate what we covered last lecture with a demo!</p>
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<h2 id="demo--looking-at-size-and-sign-conversions-in-c">Demo – Looking at size and sign conversions in C</h2>
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<ul>
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<li>What does the demo illustrate?
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<ul>
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<li>Size conversion:
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<ul>
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<li>Converting to a larger (wider) data type -> Converting short to int</li>
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<li>Converting to a smaller (narrower) data type -> Converting short to char</li>
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</ul>
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</li>
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<li>Sign conversion:
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<ul>
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<li>Converting from signed to unsigned -> Converting short to unsigned short</li>
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<li>Converting from unsigned to signed -> Converting unsigned short to short</li>
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</ul>
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</li>
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<li>Size and Sign conversion:
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<ul>
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<li>Converting from signed to unsigned larger (wider) data type -> Converting short to unsigned int</li>
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<li>Converting from signed to unsigned smaller (narrower) data type ->
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Converting short to unsigned char</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>This demo (code and results) posted on our course web site</li>
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</ul>
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<h2 id="integer-addition-unlimited-space">Integer addition (unlimited space)</h2>
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<ul>
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<li>
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<p>What happens when we add two decimal numbers? (Highlights show carring the one to complete the equation)
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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><msub><mtext>107</mtext><mn>10</mn></msub><mo>+</mo><msub><mtext>938</mtext><mn>10</mn></msub><mo>=</mo><msub><mtext>1045</mtext><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">
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\text{107}_{10} + \text{938}_{10} = \text{1045}_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">107</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">938</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">1045</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p>
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</li>
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<li>
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<p>Same thing happens when we add two binary numbers:</p>
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</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><msub><mtext>101100</mtext><mn>2</mn></msub><mo>+</mo><msub><mtext>101110</mtext><mn>2</mn></msub><mo>=</mo><msub><mtext>1011010</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">
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\text{101100}_{2} + \text{101110}_{2} = \text{1011010}_{2}
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</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">101100</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">101110</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span 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">1011010</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></p>
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<h2 id="unsigned-addition-limited-space-ie-fixed-size-in-memory">Unsigned addition (limited space, i.e., fixed size in memory)</h2>
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<p>What happens when we add two unsigned values?</p>
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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>w</mi><mo>=</mo><mn>8</mn></mrow><annotation encoding="application/x-tex">w=8</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">8</span></span></span></span></span>
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<p>a)</p>
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<p>Binary:</p>
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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><msub><mtext>00111011</mtext><mn>2</mn></msub><mo>+</mo><msub><mtext>01011010</mtext><mn>2</mn></msub><mo>=</mo><mo stretchy="false">?</mo></mrow><annotation encoding="application/x-tex">
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\text{00111011}_{2} + \text{01011010}_{2} = ?
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</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">00111011</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">01011010</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mclose">?</span></span></span></span></span></p>
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<p>Decimal:</p>
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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><msub><mtext>59</mtext><mn>10</mn></msub><mo>+</mo><msub><mtext>90</mtext><mn>10</mn></msub><mo>=</mo><msub><mtext>149</mtext><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">
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\text{59}_{10} + \text{90}_{10} = \text{149}_{10}
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</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">59</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">90</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">149</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p>
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<p>b)</p>
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<p>Binary:</p>
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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><msub><mtext>10101110</mtext><mn>2</mn></msub><mo>+</mo><msub><mtext>11001011</mtext><mn>2</mn></msub><mo>=</mo><mo stretchy="false">?</mo></mrow><annotation encoding="application/x-tex">
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\text{10101110}_{2} + \text{11001011}_{2} = ?
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</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">10101110</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">11001011</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mclose">?</span></span></span></span></span></p>
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<p>Decimal:</p>
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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><msub><mtext>174</mtext><mn>10</mn></msub><mo>+</mo><msub><mtext>203</mtext><mn>10</mn></msub><mo>=</mo><msub><mtext>377</mtext><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">
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\text{174}_{10} + \text{203}_{10} = \text{377}_{10}
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</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">174</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">203</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">377</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p>
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<h2 id="unsigned-addition-u_w-and-overflow">Unsigned addition (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>+</mo><mi>w</mi><mi>u</mi></msubsup></mrow><annotation encoding="application/x-tex">+^{u}_{w}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.911392em;vertical-align:-0.247em;"></span><span class="mord"><span class="mbin">+</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>) and overflow</h2>
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<ul>
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<li>True/expected sum is the result of integer addition with unlimited space.</li>
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<li>Actual sum is the result of unsigned addition with limited space. Discarding the carry out bit.</li>
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<li>Discarding carry out bit has same effect as applying modular arithmetic</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><mi>s</mi><mo>=</mo><mi>U</mi><msubsup><mo>+</mo><mi>w</mi><mi>u</mi></msubsup><mi>v</mi><mo>=</mo><mo stretchy="false">(</mo><mi>u</mi><mo>+</mo><mi>v</mi><mo stretchy="false">)</mo><mspace></mspace><mspace width="1em"></mspace><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><msup><mn>2</mn><mi>w</mi></msup></mrow><annotation encoding="application/x-tex">
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s = U +^{u}_{w}v = (u + v) \mod{2^{w}}
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">s</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.9613919999999999em;vertical-align:-0.247em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin"><span class="mbin">+</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02691em;">w</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:1em;"></span></span><span class="base"><span class="strut" style="height:0.7143919999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">m</span><span class="mord mathrm">o</span><span class="mord mathrm">d</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02691em;">w</span></span></span></span></span></span></span></span></span></span></span></span></span></p>
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<ul>
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<li>Operands: w bits</li>
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<li>True Sum: w+1 bits</li>
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</ul>
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<h2 id="closer-look-at-unsigned-addition-overflow">Closer look at unsigned addition overflow</h2>
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<ul>
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<li>w = 8 -> [0..255]</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>255</mtext><mn>10</mn></msub><mo>=</mo><msub><mtext>11111111</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{255}_{10} = \text{11111111}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">255</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">11111111</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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<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>90</mtext><mn>10</mn></msub><mo>=</mo><msub><mtext>01011010</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{90}_{10} = \text{01011010}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">90</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">01011010</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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<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>45</mtext><mn>10</mn></msub><mo>=</mo><msub><mtext>00101101</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{45}_{10} = \text{00101101}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">45</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">00101101</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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<h3 id="example-1">Example 1:</h3>
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<p>Decimal (carry out the 1 in 135):</p>
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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><msub><mtext>90</mtext><mn>10</mn></msub><mo>+</mo><msub><mtext>45</mtext><mn>10</mn></msub><mo>=</mo><msub><mtext>135</mtext><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">
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\text{90}_{10} + \text{45}_{10} = \text{135}_{10}
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</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">90</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">45</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">135</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p>
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<p>Binary (carry out the 1 in <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>10000111</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{10000111}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">10000111</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>):</p>
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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><msub><mtext>01011010</mtext><mn>2</mn></msub><mo>+</mo><msub><mtext>00101101</mtext><mn>2</mn></msub><mo>=</mo><msub><mtext>10000111</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">
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\text{01011010}_{2} + \text{00101101}_{2} = \text{10000111}_{2}
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</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">01011010</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">00101101</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span 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">10000111</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></p>
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<ul>
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<li>True sum, w=8: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>10000111</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{10000111}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">10000111</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></li>
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<li>Actual (overflowed) sum, w=8: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>10000111</mtext><mn>2</mn></msub><mo>=</mo><msub><mtext>135</mtext><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">\text{10000111}_{2} = \text{135}_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">10000111</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span 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">135</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></span></span></li>
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</ul>
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<h3 id="example-2">Example 2:</h3>
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<p>Decimal (carry 1 to the 3 in 300):</p>
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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><msub><mtext>255</mtext><mn>10</mn></msub><mo>+</mo><msub><mtext>45</mtext><mn>10</mn></msub><mo>=</mo><msub><mtext>300</mtext><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">
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\text{255}_{10} + \text{45}_{10} = \text{300}_{10}
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</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">255</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">45</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">300</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p>
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<p>Binary (carry the 1 at the beginning of the result):</p>
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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><msub><mtext>11111111</mtext><mn>2</mn></msub><mo>+</mo><msub><mtext>00101101</mtext><mn>2</mn></msub><mo>=</mo><msub><mtext>100101100</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">
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\text{11111111}_{2} + \text{00101101}_{2} = \text{100101100}_{2}
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</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">11111111</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">00101101</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span 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">100101100</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></p>
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<ul>
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<li>True sum, w=9: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>100101100</mtext><mn>2</mn></msub><mo>=</mo><msub><mtext>300</mtext><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">\text{100101100}_{2} = \text{300}_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">100101100</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span 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">300</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></span></span></li>
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<li>Actual (overflowed) sum, w=8: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>00101100</mtext><mn>2</mn></msub><mo>=</mo><msub><mtext>44</mtext><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">\text{00101100}_{2} = \text{44}_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">00101100</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span 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">44</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></span></span></li>
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</ul>
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<h2 id="comparing-integer-addition-with-overflow-effect-of-unsigned-addition-w--4">Comparing integer addition with Overflow: Effect of unsigned addition (w = 4)</h2>
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<p>Annotation: with unlimited space:</p>
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<p>A 3d chart showing two numbers being added.
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a and b on the z and x axies, the sum on the y axis.
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y goes to a maximum height of 32
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(a = 15) + (b = 15) = (y = 30)
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Annotation: With limited space (fixed-size memory):</p>
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<p>A 3d chart showing two numbers being added.
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a and b on the z and x axies, the sum on the y axis.
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y goes to a maximum height of 15
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(a = 15) + (b = 15) = (y = 14)</p>
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<p>An overflow occurs when there is a carry out</p>
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<p>For example: 15 (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>1111</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{1111}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">1111</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>) + 15 (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>1111</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{1111}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">1111</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>) = 30 (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>11110</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{11110}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">11110</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>) as a true sum, and 14 (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>11110</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{11110}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">11110</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="heigh
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<h2 id="signed-addition-limited-space-ie-fixed-size-in-memory">Signed addition (limited space, i.e., fixed size in memory)</h2>
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<p>What happens when we add two signed values:</p>
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<p>w=8</p>
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<p>a)</p>
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<p>Binary:
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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>00111011</mtext><mn>2</mn></msub><mo>+</mo><msub><mtext>01011010</mtext><mn>2</mn></msub><mo>=</mo><mo stretchy="false">?</mo></mrow><annotation encoding="application/x-tex">\text{00111011}_{2} + \text{01011010}_{2} = ?</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">00111011</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">01011010</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mclose">?</span></span></span></span></p>
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<p>Decimal:
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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>59</mtext><mn>10</mn></msub><mo>+</mo><msub><mtext>90</mtext><mn>10</mn></msub><mo>=</mo><msub><mtext>149</mtext><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">\text{59}_{10} + \text{90}_{10} = \text{149}_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">59</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:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">90</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">149</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></span></span></p>
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<p>b)</p>
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<p>Binary: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>10101110</mtext><mn>2</mn></msub><mo>+</mo><msub><mtext>11001011</mtext><mn>2</mn></msub><mo>=</mo><mo stretchy="false">?</mo></mrow><annotation encoding="application/x-tex">\text{10101110}_{2} + \text{11001011}_{2} = ?</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">10101110</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">11001011</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mclose">?</span></span></span></span></p>
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<p>Decimal: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>-82</mtext><mn>10</mn></msub><mo>+</mo><msub><mtext>-53</mtext><mn>10</mn></msub><mo>=</mo><msub><mtext>-135</mtext><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">\text{-82}_{10} + \text{-53}_{10} = \text{-135}_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">-82</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:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">-53</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">-135</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></span></span></p>
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<p>Observation: Unsigned and signed additions have identical behavior @ the bit level, i.e., their sum have the same bit-level representation, but their interpretation differs</p>
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<h2 id="signed-addition-t_w-and-overflow">Signed addition (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>+</mo><mi>w</mi><mi>t</mi></msubsup></mrow><annotation encoding="application/x-tex">+^{t}_{w}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.040556em;vertical-align:-0.247em;"></span><span class="mord"><span class="mbin">+</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7935559999999999em;"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>) and overflow</h2>
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<p>True sum would be the result of integer addition with unlimited space:
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Actual sum is the result of signed addition with limited space:</p>
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<p>Operands: w bits
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True Sum: w+1 bits</p>
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<p>After discarduing carry out bit: w bits (overflow)</p>
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<ul>
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<li>Discarding carry out bit has same effect as applying modular arithmetic</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><mi>s</mi><mo>=</mo><mi>u</mi><msubsup><mo>+</mo><mi>w</mi><mi>t</mi></msubsup><mi>v</mi><mo>=</mo><msub><mtext>U2T</mtext><mi>w</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>u</mi><mo>+</mo><mi>v</mi><mo stretchy="false">)</mo><mspace></mspace><mspace width="1em"></mspace><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><msup><mn>2</mn><mi>w</mi></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">
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s = u +^{t}_{w} v = \text{U2T}_{w}[(u + v) \mod 2^{w}]
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">s</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0905559999999999em;vertical-align:-0.247em;"></span><span class="mord mathdefault">u</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin"><span class="mbin">+</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8435559999999999em;"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02691em;">w</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord text"><span class="mord">U2T</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02691em;">w</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">[</span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:1em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">m</span><span class="mord mathrm">o</span><span class="mord mathrm">d</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02691em;">w</span></span></span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span
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<p>Negative overflow and positive overflows are possible.
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Diagram showing negative overflows becoming positive, and positive overflows becoming negative.</p>
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<h2 id="closer-look-at-signed-addition-overflow">Closer look at signed addition overflow</h2>
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<ul>
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<li>w = 8 -> [-128..127]</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>90</mtext><mn>10</mn></msub><mo>=</mo><msub><mtext>01011010</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{90}_{10} = \text{01011010}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">90</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">01011010</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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<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>45</mtext><mn>10</mn></msub><mo>=</mo><msub><mtext>00101101</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{45}_{10} = \text{00101101}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">45</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">00101101</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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<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>-45</mtext><mn>10</mn></msub><mo>=</mo><msub><mtext>11010011</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{-45}_{10} = \text{11010011}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">-45</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">11010011</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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<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>-90</mtext><mn>10</mn></msub><mo>=</mo><msub><mtext>10100110</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{-90}_{10} = \text{10100110}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">-90</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">10100110</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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<h3 id="example-1-1">Example 1:</h3>
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<p>Decimal: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>90</mtext><mn>10</mn></msub><mo>+</mo><msub><mtext>45</mtext><mn>10</mn></msub><mo>=</mo><msub><mtext>135</mtext><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">\text{90}_{10} + \text{45}_{10} = \text{135}_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">90</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:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">45</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">135</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></span></span></p>
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<p>Binary: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>01011010</mtext><mn>2</mn></msub><mo>+</mo><msub><mtext>00101101</mtext><mn>2</mn></msub><mo>=</mo><msub><mtext>010000111</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{01011010}_{2} + \text{00101101}_{2} = \text{010000111}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">01011010</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">00101101</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span 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">010000111</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></p>
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<p>The binary result is -121</p>
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<h3 id="example-2-1">Example 2:</h3>
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<p>Decimal: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>-90</mtext><mn>10</mn></msub><mo>+</mo><msub><mtext>-45</mtext><mn>10</mn></msub><mo>=</mo><msub><mtext>-135</mtext><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">\text{-90}_{10} + \text{-45}_{10} = \text{-135}_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">-90</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:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">-45</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">-135</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></span></span></p>
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<p>Binary: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>10100110</mtext><mn>2</mn></msub><mo>+</mo><msub><mtext>11010011</mtext><mn>2</mn></msub><mo>=</mo><msub><mtext>101111001</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{10100110}_{2} + \text{11010011}_{2} = \text{101111001}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">10100110</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">11010011</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span 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">101111001</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></p>
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<p>Binary result is 121</p>
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<h3 id="example-3">Example 3:</h3>
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<p>Decimal: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>-90</mtext><mn>10</mn></msub><mo>+</mo><msub><mtext>45</mtext><mn>10</mn></msub><mo>=</mo><msub><mtext>-45</mtext><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">\text{-90}_{10} + \text{45}_{10} = \text{-45}_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">-90</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:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">45</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">-45</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></span></span></p>
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<p>Binary: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>10100110</mtext><mn>2</mn></msub><mo>+</mo><msub><mtext>00101101</mtext><mn>2</mn></msub><mo>=</mo><msub><mtext>011010011</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{10100110}_{2} + \text{00101101}_{2} = \text{011010011}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">10100110</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">00101101</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span 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">011010011</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></p>
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<h3 id="example-4">Example 4:</h3>
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<p>Decimal: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>90</mtext><mn>10</mn></msub><mo>+</mo><msub><mtext>-45</mtext><mn>10</mn></msub><mo>=</mo><msub><mtext>45</mtext><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">\text{90}_{10} + \text{-45}_{10} = \text{45}_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">90</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:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">-45</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">45</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></span></span></p>
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<p>Binary: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>01011010</mtext><mn>2</mn></msub><mo>+</mo><msub><mtext>11010011</mtext><mn>2</mn></msub><mo>=</mo><msub><mtext>100101101</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{01011010}_{2} + \text{11010011}_{2} = \text{100101101}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">01011010</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">11010011</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span 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">100101101</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></p>
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<p>A chart showing the relationship between true sum and actual (overflowed) sum.
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Actual sum has a possible value between 127 to -128.
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A true sum, however has a range between 255 and -256.
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As your true sum goes further down from -128, its actual sum becomes lower as well. Starting at -129 = 127.
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As you true sum goes futher up from 127, the actual sum also rises, satrting with 128 = -128.</p>
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<h2 id="visualizing-signed-addition-overflow-w--4">Visualizing signed addition overflow (w = 4)</h2>
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<p>A 3D chart which has its x axis go from -8 to +7, its z axis go from -8 to +6, and a y axis which goes from :wq</p>
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<p>Positive Overflow</p>
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<p>For example: 7 (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>0111</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{0111}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">0111</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>) + 1 (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>0001</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{0001}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">0001</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>) = 8 (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>1000</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{1000}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">1000</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>) is the true sum and = -8 (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>1000</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{1000}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">1000</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:0.15
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<h2 id="what-about-subtraction---addition">What about subtraction? -> Addition</h2>
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<p>x + (-x) = 0</p>
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<ul>
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<li>Subtracting a number is equivalent to adding its additive inverse
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<ul>
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<li>Instead of subtracting a positive number, we could add its negative version:</li>
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</ul>
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</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><mn>107</mn><mo>−</mo><mn>118</mn><mo>=</mo><mo>−</mo><mn>11</mn></mrow><annotation encoding="application/x-tex">
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107 - 118 = -11
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1</span><span class="mord">0</span><span class="mord">7</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">1</span><span class="mord">8</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mord">1</span></span></span></span></span></p>
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<p>becomes:</p>
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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><mn>107</mn><mo>+</mo><mo stretchy="false">(</mo><mo>−</mo><mn>118</mn><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>18</mn></mrow><annotation encoding="application/x-tex">
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107 + (-118) = -18
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</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1</span><span class="mord">0</span><span class="mord">7</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mord">1</span><span class="mord">8</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mord">8</span></span></span></span></span></p>
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<ul>
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<li>Let‘s try:</li>
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</ul>
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<p>Decimal:</p>
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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>107</mtext><mn>10</mn></msub><mo>−</mo><msub><mtext>118</mtext><mn>10</mn></msub><mo>=</mo><mo>−</mo><mn>11</mn></mrow><annotation encoding="application/x-tex">\text{107}_{10} - \text{118}_{10} = -11</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">107</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:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">118</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">−</span><span class="mord">11</span></span></span></span></span>
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<p>Binary:</p>
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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>01101011</mtext><mn>2</mn></msub><mo>−</mo><msub><mtext>01110110</mtext><mn>2</mn></msub><mo>=</mo></mrow><annotation encoding="application/x-tex">\text{01101011}_{2} - \text{01110110}_{2} =</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">01101011</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">01110110</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span></span></span></span></span>
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<p>Binary subtraction by addition:</p>
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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>01101011</mtext><mn>2</mn></msub><mo>+</mo><msub><mtext>10001010</mtext><mn>2</mn></msub><mo>=</mo><msub><mtext>11110101</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{01101011}_{2} + \text{10001010}_{2} = \text{11110101}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">01101011</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">10001010</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span 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">11110101</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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<p>Binary subtraction by addition is equal to -11</p>
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<p>Check: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>128</mn><mo>+</mo><mn>64</mn><mo>+</mo><mn>32</mn><mo>+</mo><mn>16</mn><mo>+</mo><mn>4</mn><mo>+</mo><mn>1</mn><mo>=</mo><msub><mtext>-11</mtext><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">-128 + 64 + 32 + 16 + 4 + 1 = \text{-11}_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">−</span><span class="mord">128</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">64</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">32</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">16</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">-11</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></span></span></p>
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<p>T2B(X) conversion:</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><mtext> </mtext><mo stretchy="false">(</mo><mtext>U2B</mtext><mo stretchy="false">(</mo><mi mathvariant="normal">∣</mi><mi>X</mi><mi mathvariant="normal">∣</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">(~(\text{U2B}(|X|)))+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mspace nobreak"> </span><span class="mopen">(</span><span class="mord text"><span class="mord">U2B</span></span><span class="mopen">(</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">∣</span><span class="mclose">)))</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></span>
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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><mo stretchy="false">(</mo><mtext> </mtext><mo stretchy="false">(</mo><mtext>U2B</mtext><mo stretchy="false">(</mo><mi mathvariant="normal">∣</mi><mo>−</mo><mn>118</mn><mi mathvariant="normal">∣</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">(~(\text{U2B}(|-118|)))+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mspace nobreak"> </span><span class="mopen">(</span><span class="mord text"><span class="mord">U2B</span></span><span class="mopen">(</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">118∣</span><span class="mclose">)))</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></span>
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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><mo stretchy="false">(</mo><mtext> </mtext><mo stretchy="false">(</mo><mtext>U2B</mtext><mo stretchy="false">(</mo><mn>118</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">(~(\text{U2B}(118)))+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mspace nobreak"> </span><span class="mopen">(</span><span class="mord text"><span class="mord">U2B</span></span><span class="mopen">(</span><span class="mord">118</span><span class="mclose">)))</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></span>
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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><mo stretchy="false">(</mo><mo>∼</mo><mo stretchy="false">(</mo><mn>0111011</mn><msub><mn>0</mn><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">(\sim(01110110_{2}))+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0111011</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></span>
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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><mo stretchy="false">(</mo><msub><mtext>10001001</mtext><mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">(\text{10001001}_{2})+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord text"><span class="mord">10001001</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></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>10001010</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{10001010}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">10001010</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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</ol>
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<h2 id="multiplication-u_w-t_w-and-overflow">Multiplication (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∗</mo><mi>w</mi><mi>u</mi></msubsup><mo separator="true">,</mo><msubsup><mo>∗</mo><mi>w</mi><mi>t</mi></msubsup></mrow><annotation encoding="application/x-tex">*^{u}_{w}, *^{t}_{w}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.040556em;vertical-align:-0.247em;"></span><span class="mord"><span class="mbin">∗</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mbin">∗</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7935559999999999em;"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>) and overflow</h2>
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<p>True product would be the result of integer multiplication with unlimited space: expected product
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Actual product is the result of multiplication with limited space.</p>
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<ul>
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<li>Operands: w bits</li>
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<li>True Product: 2w bits <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mo>×</mo><mi>v</mi></mrow><annotation encoding="application/x-tex">u \times v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span></span></li>
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<li>
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<p>Discard: w bits</p>
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</li>
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<li>Discarding high order w bits has same effect as applying modular arithmetic</li>
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</ul>
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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>p</mi><mo>=</mo><mi>u</mi><msubsup><mo>∗</mo><mi>w</mi><mi>u</mi></msubsup><mi>v</mi><mo>=</mo><mo stretchy="false">(</mo><mi>u</mi><mo>×</mo><mi>v</mi><mo stretchy="false">)</mo><mspace></mspace><mspace width="1em"/><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><msup><mn>2</mn><mi>w</mi></msup></mrow><annotation encoding="application/x-tex">p = u *^{u}_{w}v = (u \times v) \mod 2^{w}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.9613919999999999em;vertical-align:-0.247em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin"><span class="mbin">∗</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:1em;"></span></span><span class="base"><span class="strut" style="height:0.7143919999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span></span></span></span></span></span></span></span></span></span>
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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>p</mi><mo>=</mo><mi>u</mi><msubsup><mo>∗</mo><mi>w</mi><mi>t</mi></msubsup><mi>v</mi><mo>=</mo><msub><mtext>U2T</mtext><mi>w</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>u</mi><mo>×</mo><mi>v</mi><mo stretchy="false">)</mo><mspace></mspace><mspace width="1em"/><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><msup><mn>2</mn><mi>w</mi></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">p = u *^{t}_{w}v = \text{U2T}_{w}[(u \times v) \mod 2^{w}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0905559999999999em;vertical-align:-0.247em;"></span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin"><span class="mbin">∗</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8435559999999999em;"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord text"><span class="mord">U2T</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">[(</span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:1em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><s
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<ul>
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<li>Example: w = 4</li>
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</ul>
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<p>Decimal:</p>
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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><mn>5</mn><mn>10</mn></msub><mo>×</mo><msub><mn>5</mn><mn>10</mn></msub><mo>=</mo><msub><mtext>25</mtext><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">5_{10} \times 5_{10} = \text{25}_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">25</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></span></span></span>
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<p>Binary:</p>
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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>0101</mtext><mn>2</mn></msub><mo>×</mo><msub><mtext>0101</mtext><mn>2</mn></msub><mo>=</mo><menclose notation="horizontalstrike"><mn>001</mn></menclose><msub><mtext>1001</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{0101}_{2} \times \text{0101}_{2} = \sout{001}\text{1001}_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">0101</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">0101</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.64444em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">001</span></span></span><span style="top:-3.2155em;"><span class="pstrut" style="height:3em;"></span><span class="stretchy sout"></span></span></span></span></span></span><span class="mord"><span class="mord text"><span class="mord">1001</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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<h2 id="multiplication-with-power-of-2-versus-shifting">Multiplication with power-of-2 versus shifting</h2>
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<ul>
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<li>If <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>×</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x \times y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><msup><mn>2</mn><mi>k</mi></msup></mrow><annotation encoding="application/x-tex">y = 2^{k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.849108em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span></span> then <code class="language-plaintext highlighter-rouge">x << k</code>
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<ul>
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<li>For both signed and unsigned</li>
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</ul>
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</li>
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<li>Example:
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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><mi>x</mi><mo>×</mo><mn>8</mn><mo>=</mo><mi>x</mi><mo>×</mo><msup><mn>2</mn><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">x \times 8 = x \times 2^{3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">8</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span></span> = <code class="language-plaintext highlighter-rouge">x << 3</code></li>
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<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>×</mo><mn>24</mn><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo>×</mo><mn>25</mn><mo stretchy="false">)</mo><mtext>–</mtext><mo stretchy="false">(</mo><mi>x</mi><mo>×</mo><mn>23</mn><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo>×</mo><mn>32</mn><mo stretchy="false">)</mo><mtext>–</mtext><mo stretchy="false">(</mo><mi>x</mi><mo>×</mo><mn>8</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \times 24 = (x \times 25) – (x \times 23) = (x \times 32) – (x \times 8)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">24</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">25</span><span class="mclose">)</span><span class="mord">–</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">23</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">32</span><span class="mclose">)</span><span class="mord">–</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">8</span><span class="mclose">)</span></span></span></span> = (x « 5) – (x « 3) (decompose 24 in powers of 2 => 32 – 8)</li>
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</ul>
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</li>
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<li>Most machines shift and add faster than multiply
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<ul>
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<li>We’ll soon see that compiler generates this code automatically
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17</li>
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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>Demo of size and sign conversion in C: code and results posted!</li>
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<li>Addition:
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<ul>
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<li>Unsigned/signed:
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<ul>
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<li>Behave the same way at the bit level</li>
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<li>Interpretation of resulting bit vector (sum) may differ</li>
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</ul>
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</li>
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<li>Unsigned addition -> may overflow, i.e., (w+1)th bit is set
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<ul>
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<li>If so, then actual sum obtained => <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo><mspace></mspace><mspace width="0.6666666666666666em"/><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><msup><mn>2</mn><mi>w</mi></msup></mrow><annotation encoding="application/x-tex">(x + y) \mod 2^{w}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.6666666666666666em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span></span></span></span></span></span></span></span></span></li>
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</ul>
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</li>
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<li>Signed addition -> may overflow, i.e., (w+1)th bit is set
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<ul>
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<li>If so, then true sum may be too +ve -> positive overflow OR too -ve -> negative overflow</li>
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<li>Then actual sum obtained => <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>U2T</mtext><mi>w</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo><mspace></mspace><mspace width="0.6666666666666666em"/><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><msup><mn>2</mn><mi>w</mi></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\text{U2T}_{w}[(x + y) \mod 2^{w}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord text"><span class="mord">U2T</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">[(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.6666666666666666em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></li>
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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>Subtraction
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<ul>
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<li>Becomes an addition where negative operands are transformed into their additive inverse (in two’s complement)</li>
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</ul>
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</li>
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|
<li>Multiplication:
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<ul>
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<li>Unsigned: actual product obtained -> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>×</mo><mi>y</mi><mo stretchy="false">)</mo><mspace></mspace><mspace width="0.6666666666666666em"/><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><msup><mn>2</mn><mi>w</mi></msup></mrow><annotation encoding="application/x-tex">(x \times y) \mod 2^{w}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.6666666666666666em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span></span></span></span></span></span></span></span></span></li>
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|
<li>Signed: actual product obtained -> $$\text{U2T}_{w}[(x \times y) \mod 2^{w}]</li>
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|
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|
|
<li>Can be replaced by additions and shifts</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>
|
|
|
|
|
|
<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>
|
|
|
|
|
|
<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>
|
|
|
|
|
|
<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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<p>We’ll illustrate what we covered today by having a demo!</p>
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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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