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<h1 id="cmpt-295">CMPT 295</h1>
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<ul>
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<li>Lecture 16 – Midterm 1 Review Session</li>
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</ul>
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<h2 id="go-over-rounding---lecture-6-slide-13">Go over Rounding - Lecture 6 Slide 13:</h2>
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<h3 id="rounding">Rounding</h3>
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<ol>
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<li>Round up</li>
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<li>Round down</li>
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<li>Round half way -> when bits to right of rounding position are <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mtext>100</mtext><mo>…</mo><mtext>0</mtext></mrow><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{100\dots 0}_{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">100</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">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> <!--_-->
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<ul>
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<li>Round to even number: produces 0 as the least significant bit of rounded result.</li>
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</ul>
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</li>
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</ol>
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<p>Example: Round to nearest <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">1/4</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">1/4</span></span></span></span> (2 bits right of binary point):</p>
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<p>The third bit after the binary point is the 24th bit. Imagine the second bit as bit 23 of frac of IEEE.</p>
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<table>
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<thead>
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<tr>
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<th>Value</th>
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<th>Binary</th>
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<th>Rounded</th>
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<th>Action</th>
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<th>Rounded Value</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mfrac><mn>3</mn><mn>32</mn></mfrac></mrow><annotation encoding="application/x-tex">2 \frac{3}{32}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord">2</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">32</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td>
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<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>10.00011</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{10.00011}_{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">10.00011</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></td>
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<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>10.00</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{10.00}_{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">10.00</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></td>
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<td>(<1/2–down)</td>
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<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span></span></span></span></td>
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</tr>
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<tr>
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<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mfrac><mn>3</mn><mn>16</mn></mfrac></mrow><annotation encoding="application/x-tex">2 \frac{3}{16}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord">2</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">16</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td>
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<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>10.00.110</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{10.00.110}_{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">10.00.110</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></td>
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<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>10.01</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{10.01}_{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">10.01</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></td>
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<td>(>1/2–up)</td>
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<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow><annotation encoding="application/x-tex">2 \frac{1}{4}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord">2</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td>
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</tr>
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<tr>
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<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mfrac><mn>7</mn><mn>8</mn></mfrac></mrow><annotation encoding="application/x-tex">2 \frac{7}{8}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord">2</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td>
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<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>10.11100</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{10.11100}_{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">10.11100</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></td>
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<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>11.00</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{11.00}_{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">11.00</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></td>
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<td>(1/2–up to even))</td>
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<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">3</span></span></span></span></td>
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</tr>
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<tr>
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<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mfrac><mn>5</mn><mn>8</mn></mfrac></mrow><annotation encoding="application/x-tex">2 \frac{5}{8}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord">2</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td>
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<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>10.10100</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{10.10100}_{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">10.10100</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></td>
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<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>10.10</mtext><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{10.10}_{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">10.10</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></td>
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<td>(1/2–down to even)</td>
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<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">2 \frac{1}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord">2</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td>
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</tr>
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</tbody>
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</table>
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<h2 id="assignment-3-question-1-a-iii">Assignment 3 Question 1 a. iii</h2>
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<p>Transcriber’s note: the rounding bit will be highlighted by spelling out the number instaed of using the 0/1 characters</p>
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<p>frac: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1100</mn><mtext> </mtext><mn>1100</mn><mtext> </mtext><mn>1100</mn><mtext> </mtext><mn>1100</mn><mtext> </mtext><mn>1100</mn><mtext> </mtext><mn>11</mn><mtext>zero</mtext><mn>0</mn><mtext> </mtext><mover accent="true"><mn>1100</mn><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">1100 \space 1100 \space 1100 \space 1100 \space 1100 \space 11\text{zero}0 \space \overline{1100}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444400000000001em;vertical-align:0em;"></span><span class="mord">1100</span><span class="mspace"> </span><span class="mord">1100</span><span class="mspace"> </span><span class="mord">1100</span><span class="mspace"> </span><span class="mord">1100</span><span class="mspace"> </span><span class="mord">1100</span><span class="mspace"> </span><span class="mord">11</span><span class="mord text"><span class="mord">zero</span></span><span class="mord">0</span><span class="mspace"> </span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8444400000000001em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1100</span></span></span><span style="top:-3.76444em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span></span></p>
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<h2 id="assignment-3-question-1-a-iv">Assignment 3 Question 1 a. iv</h2>
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<p>frac: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0101</mn><mtext> </mtext><mn>0101</mn><mtext> </mtext><mn>0101</mn><mtext> </mtext><mn>0101</mn><mtext> </mtext><mn>0101</mn><mtext> </mtext><mn>01</mn><mtext>zero </mtext><mover accent="true"><mn>101</mn><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">0101 \space 0101 \space 0101 \space 0101 \space 0101 \space 01\text{zero} \space \overline{101}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444400000000001em;vertical-align:0em;"></span><span class="mord">0101</span><span class="mspace"> </span><span class="mord">0101</span><span class="mspace"> </span><span class="mord">0101</span><span class="mspace"> </span><span class="mord">0101</span><span class="mspace"> </span><span class="mord">0101</span><span class="mspace"> </span><span class="mord">01</span><span class="mord text"><span class="mord">zero</span></span><span class="mspace"> </span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8444400000000001em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">101</span></span></span><span style="top:-3.76444em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span></span></p>
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<h2 id="assignment2-question-2-g-h-i-k">Assignment#2 Question 2 g., h., i., k.</h2>
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<!-- AUTO GENERATED FROM CUSTOM PYTHON CODE -->
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<table>
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<thead>
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<tr>
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<th colspan="2">
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<th colspan="3">
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Exponent
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<th colspan="2">
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Fraction
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</th>
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<th colspan="3">
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Value
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<tr>
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<th>
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Description
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Bit Representation/th>
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<th>
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exp
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</th>
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<th>
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E
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</th>
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<th>
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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>E</mi></msup></mrow><annotation encoding="application/x-tex">2^{E}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8413309999999999em;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:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em;">E</span></span></span></span></span></span></span></span></span></span></span></span>
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</th>
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<th>
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frac
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</th>
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<th>
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M
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</th>
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<th>
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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><msup><mn>2</mn><mi>E</mi></msup></mrow><annotation encoding="application/x-tex">M 2^{E}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8413309999999999em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em;">E</span></span></span></span></span></span></span></span></span></span></span></span>
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</th>
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<th>
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V
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</th>
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<th>
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Decimal
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</th>
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</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td>
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zero
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</td>
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<td>
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0 000 00
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</td>
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<td>
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0
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</td>
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<td>
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-2
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</td>
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<td>
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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">1/4</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">1</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
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</td>
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<td>
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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">0/4</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">0</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
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</td>
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<td>
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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">0/4</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">0</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
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</td>
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<td>
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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mi mathvariant="normal">/</mi><mn>16</mn></mrow><annotation encoding="application/x-tex">0/16</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">0</span><span class="mord">/</span><span class="mord">1</span><span class="mord">6</span></span></span></span>
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</td>
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<td>
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0
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</td>
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<td>
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0.0
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</td>
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</tr>
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<tr>
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<td>
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Smallest positive denormalized
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</td>
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<td>
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0 000 01
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</td>
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<td>
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0
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</td>
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<td>
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-2
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</td>
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<td>
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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">1/4</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">1</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
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</td>
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<td>
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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">1/4</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">1</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
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</td>
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<td>
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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">1/4</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">1</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
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</td>
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<td>
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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>16</mn></mrow><annotation encoding="application/x-tex">1/16</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">1</span><span class="mord">/</span><span class="mord">1</span><span class="mord">6</span></span></span></span>
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</td>
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<td>
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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>16</mn></mrow><annotation encoding="application/x-tex">1/16</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">1</span><span class="mord">/</span><span class="mord">1</span><span class="mord">6</span></span></span></span>
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</td>
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<td>
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0.0625
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</td>
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</tr>
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<tr>
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<td>
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</td>
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<td>
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0 000 10
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</td>
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<td>
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0
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</td>
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<td>
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-2
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</td>
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<td>
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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">1/4</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">1</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
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</td>
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<td>
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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi mathvariant="normal">/</mi><mn>4</mn><mo>=</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">2/4=1/2</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">2</span><span class="mord">/</span><span class="mord">4</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mord">/</span><span class="mord">2</span></span></span></span>
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</td>
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<td>
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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi mathvariant="normal">/</mi><mn>4</mn><mo>=</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">2/4=1/2</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">2</span><span class="mord">/</span><span class="mord">4</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mord">/</span><span class="mord">2</span></span></span></span>
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</td>
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<td>
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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi mathvariant="normal">/</mi><mn>16</mn></mrow><annotation encoding="application/x-tex">2/16</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">2</span><span class="mord">/</span><span class="mord">1</span><span class="mord">6</span></span></span></span>
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</td>
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<td>
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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi mathvariant="normal">/</mi><mn>16</mn></mrow><annotation encoding="application/x-tex">2/16</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">2</span><span class="mord">/</span><span class="mord">1</span><span class="mord">6</span></span></span></span>
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</td>
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<td>
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0.125
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</td>
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</tr>
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<tr>
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<td>
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Largest positive denormalized
|
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</td>
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<td>
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0 000 11
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</td>
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<td>
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0
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</td>
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<td>
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-2
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</td>
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<td>
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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">1/4</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">1</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
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</td>
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<td>
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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">3/4</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">3</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
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</td>
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<td>
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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">3/4</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">3</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
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</td>
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<td>
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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><mi mathvariant="normal">/</mi><mn>16</mn></mrow><annotation encoding="application/x-tex">3/16</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">3</span><span class="mord">/</span><span class="mord">1</span><span class="mord">6</span></span></span></span>
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</td>
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<td>
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<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><mi mathvariant="normal">/</mi><mn>16</mn></mrow><annotation encoding="application/x-tex">3/16</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">3</span><span class="mord">/</span><span class="mord">1</span><span class="mord">6</span></span></span></span>
|
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</td>
|
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<td>
|
|
|
0.1875
|
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</td>
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</tr>
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<tr>
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<td>
|
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Smallest positive normalized
|
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</td>
|
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<td>
|
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0 001 00
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</td>
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<td>
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1
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</td>
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<td>
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-2
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</td>
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<td>
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|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">1/4</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">1</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">0/4</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">0</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mi mathvariant="normal">/</mi><mn>4</mn><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">4/4=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="mord">4</span><span class="mord">/</span><span class="mord">4</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mi mathvariant="normal">/</mi><mn>16</mn></mrow><annotation encoding="application/x-tex">4/16</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">4</span><span class="mord">/</span><span class="mord">1</span><span class="mord">6</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mi mathvariant="normal">/</mi><mn>16</mn></mrow><annotation encoding="application/x-tex">4/16</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">4</span><span class="mord">/</span><span class="mord">1</span><span class="mord">6</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
0.25
|
|
|
</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>
|
|
|
</td>
|
|
|
<td>
|
|
|
0 001 01
|
|
|
</td>
|
|
|
<td>
|
|
|
1
|
|
|
</td>
|
|
|
<td>
|
|
|
-2
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">1/4</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">1</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">1/4</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">1</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>5</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">5/4</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">5</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>5</mn><mi mathvariant="normal">/</mi><mn>16</mn></mrow><annotation encoding="application/x-tex">5/16</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">5</span><span class="mord">/</span><span class="mord">1</span><span class="mord">6</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>5</mn><mi mathvariant="normal">/</mi><mn>16</mn></mrow><annotation encoding="application/x-tex">5/16</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">5</span><span class="mord">/</span><span class="mord">1</span><span class="mord">6</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
0.3125
|
|
|
</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>
|
|
|
</td>
|
|
|
<td>
|
|
|
0 001 10
|
|
|
</td>
|
|
|
<td>
|
|
|
1
|
|
|
</td>
|
|
|
<td>
|
|
|
-2
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">1/4</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">1</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi mathvariant="normal">/</mi><mn>4</mn><mo>=</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">2/4=1/2</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">2</span><span class="mord">/</span><span class="mord">4</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mord">/</span><span class="mord">2</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>6</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">6/4</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">6</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>6</mn><mi mathvariant="normal">/</mi><mn>16</mn></mrow><annotation encoding="application/x-tex">6/16</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">6</span><span class="mord">/</span><span class="mord">1</span><span class="mord">6</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>6</mn><mi mathvariant="normal">/</mi><mn>16</mn></mrow><annotation encoding="application/x-tex">6/16</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">6</span><span class="mord">/</span><span class="mord">1</span><span class="mord">6</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
0.375
|
|
|
</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>
|
|
|
</td>
|
|
|
<td>
|
|
|
0 001 11
|
|
|
</td>
|
|
|
<td>
|
|
|
1
|
|
|
</td>
|
|
|
<td>
|
|
|
-2
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">1/4</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">1</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">3/4</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">3</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>7</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">7/4</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">7</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>7</mn><mi mathvariant="normal">/</mi><mn>16</mn></mrow><annotation encoding="application/x-tex">7/16</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">7</span><span class="mord">/</span><span class="mord">1</span><span class="mord">6</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>7</mn><mi mathvariant="normal">/</mi><mn>16</mn></mrow><annotation encoding="application/x-tex">7/16</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">7</span><span class="mord">/</span><span class="mord">1</span><span class="mord">6</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
0.4375
|
|
|
</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>
|
|
|
</td>
|
|
|
<td>
|
|
|
0 010 00
|
|
|
</td>
|
|
|
<td>
|
|
|
2
|
|
|
</td>
|
|
|
<td>
|
|
|
-1
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">1/2</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">1</span><span class="mord">/</span><span class="mord">2</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">0/4</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">0</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mi mathvariant="normal">/</mi><mn>4</mn><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">4/4=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="mord">4</span><span class="mord">/</span><span class="mord">4</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mi mathvariant="normal">/</mi><mn>8</mn></mrow><annotation encoding="application/x-tex">4/8</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">4</span><span class="mord">/</span><span class="mord">8</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mi mathvariant="normal">/</mi><mn>8</mn></mrow><annotation encoding="application/x-tex">4/8</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">4</span><span class="mord">/</span><span class="mord">8</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
0.5
|
|
|
</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>
|
|
|
</td>
|
|
|
<td>
|
|
|
0 010 01
|
|
|
</td>
|
|
|
<td>
|
|
|
2
|
|
|
</td>
|
|
|
<td>
|
|
|
-1
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">1/2</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">1</span><span class="mord">/</span><span class="mord">2</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">1/4</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">1</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>5</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">5/4</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">5</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>5</mn><mi mathvariant="normal">/</mi><mn>8</mn></mrow><annotation encoding="application/x-tex">5/8</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">5</span><span class="mord">/</span><span class="mord">8</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>5</mn><mi mathvariant="normal">/</mi><mn>8</mn></mrow><annotation encoding="application/x-tex">5/8</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">5</span><span class="mord">/</span><span class="mord">8</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
0.625
|
|
|
</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>
|
|
|
</td>
|
|
|
<td>
|
|
|
0 010 11
|
|
|
</td>
|
|
|
<td>
|
|
|
2
|
|
|
</td>
|
|
|
<td>
|
|
|
-1
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">1/2</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">1</span><span class="mord">/</span><span class="mord">2</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">3/4</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">3</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>7</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">7/4</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">7</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>7</mn><mi mathvariant="normal">/</mi><mn>8</mn></mrow><annotation encoding="application/x-tex">7/8</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">7</span><span class="mord">/</span><span class="mord">8</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>7</mn><mi mathvariant="normal">/</mi><mn>8</mn></mrow><annotation encoding="application/x-tex">7/8</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">7</span><span class="mord">/</span><span class="mord">8</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
0.875
|
|
|
</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>
|
|
|
One
|
|
|
</td>
|
|
|
<td>
|
|
|
0 011 00
|
|
|
</td>
|
|
|
<td>
|
|
|
3
|
|
|
</td>
|
|
|
<td>
|
|
|
0
|
|
|
</td>
|
|
|
<td>
|
|
|
1
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">0/4</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">0</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mi mathvariant="normal">/</mi><mn>4</mn><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">4/4=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="mord">4</span><span class="mord">/</span><span class="mord">4</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">4/4</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">4</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">4/4</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">4</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
1.0
|
|
|
</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>
|
|
|
</td>
|
|
|
<td>
|
|
|
0 011 01
|
|
|
</td>
|
|
|
<td>
|
|
|
3
|
|
|
</td>
|
|
|
<td>
|
|
|
0
|
|
|
</td>
|
|
|
<td>
|
|
|
1
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">1/4</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">1</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>5</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">5/4</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">5</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>5</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">5/4</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">5</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>5</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">5/4</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">5</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
1.25
|
|
|
</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>
|
|
|
</td>
|
|
|
<td>
|
|
|
0 011 10
|
|
|
</td>
|
|
|
<td>
|
|
|
3
|
|
|
</td>
|
|
|
<td>
|
|
|
0
|
|
|
</td>
|
|
|
<td>
|
|
|
1
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi mathvariant="normal">/</mi><mn>4</mn><mo>=</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">2/4=1/2</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">2</span><span class="mord">/</span><span class="mord">4</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mord">/</span><span class="mord">2</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>6</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">6/4</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">6</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>6</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">6/4</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">6</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>6</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">6/4</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">6</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
1.5
|
|
|
</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>
|
|
|
</td>
|
|
|
<td>
|
|
|
0 011 11
|
|
|
</td>
|
|
|
<td>
|
|
|
3
|
|
|
</td>
|
|
|
<td>
|
|
|
0
|
|
|
</td>
|
|
|
<td>
|
|
|
1
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">3/4</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">3</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>7</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">7/4</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">7</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>7</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">7/4</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">7</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>7</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">7/4</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">7</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
1.75
|
|
|
</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>
|
|
|
</td>
|
|
|
<td>
|
|
|
0 100 00
|
|
|
</td>
|
|
|
<td>
|
|
|
4
|
|
|
</td>
|
|
|
<td>
|
|
|
1
|
|
|
</td>
|
|
|
<td>
|
|
|
2
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">0/4</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">0</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mi mathvariant="normal">/</mi><mn>4</mn><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">4/4=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="mord">4</span><span class="mord">/</span><span class="mord">4</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>8</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">8/4</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">8</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>8</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">8/4</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">8</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
2
|
|
|
</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>
|
|
|
</td>
|
|
|
<td>
|
|
|
0 100 01
|
|
|
</td>
|
|
|
<td>
|
|
|
4
|
|
|
</td>
|
|
|
<td>
|
|
|
1
|
|
|
</td>
|
|
|
<td>
|
|
|
2
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">1/4</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">1</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>5</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">5/4</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">5</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">10/4</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">1</span><span class="mord">0</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">10/4</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">1</span><span class="mord">0</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
2.5
|
|
|
</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>
|
|
|
</td>
|
|
|
<td>
|
|
|
0 100 10
|
|
|
</td>
|
|
|
<td>
|
|
|
4
|
|
|
</td>
|
|
|
<td>
|
|
|
1
|
|
|
</td>
|
|
|
<td>
|
|
|
2
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi mathvariant="normal">/</mi><mn>4</mn><mo>=</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">2/4=1/2</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">2</span><span class="mord">/</span><span class="mord">4</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mord">/</span><span class="mord">2</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>6</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">6/4</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">6</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>12</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">12/4</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">1</span><span class="mord">2</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>12</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">12/4</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">1</span><span class="mord">2</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
3
|
|
|
</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>
|
|
|
</td>
|
|
|
<td>
|
|
|
0 100 11
|
|
|
</td>
|
|
|
<td>
|
|
|
4
|
|
|
</td>
|
|
|
<td>
|
|
|
1
|
|
|
</td>
|
|
|
<td>
|
|
|
2
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">3/4</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">3</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>7</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">7/4</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">7</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>14</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">14/4</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">1</span><span class="mord">4</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>14</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">14/4</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">1</span><span class="mord">4</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
3.5
|
|
|
</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>
|
|
|
</td>
|
|
|
<td>
|
|
|
0 101 00
|
|
|
</td>
|
|
|
<td>
|
|
|
5
|
|
|
</td>
|
|
|
<td>
|
|
|
2
|
|
|
</td>
|
|
|
<td>
|
|
|
4
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">0/4</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">0</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mi mathvariant="normal">/</mi><mn>4</mn><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">4/4=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="mord">4</span><span class="mord">/</span><span class="mord">4</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>16</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">16/4</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">1</span><span class="mord">6</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>16</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">16/4</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">1</span><span class="mord">6</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
4
|
|
|
</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>
|
|
|
</td>
|
|
|
<td>
|
|
|
0 101 01
|
|
|
</td>
|
|
|
<td>
|
|
|
5
|
|
|
</td>
|
|
|
<td>
|
|
|
2
|
|
|
</td>
|
|
|
<td>
|
|
|
4
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">1/4</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">1</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>5</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">5/4</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">5</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>20</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">20/4</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">2</span><span class="mord">0</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>20</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">20/4</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">2</span><span class="mord">0</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
5
|
|
|
</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>
|
|
|
</td>
|
|
|
<td>
|
|
|
0 101 10
|
|
|
</td>
|
|
|
<td>
|
|
|
5
|
|
|
</td>
|
|
|
<td>
|
|
|
2
|
|
|
</td>
|
|
|
<td>
|
|
|
4
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi mathvariant="normal">/</mi><mn>4</mn><mo>=</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">2/4=1/2</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">2</span><span class="mord">/</span><span class="mord">4</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mord">/</span><span class="mord">2</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>6</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">6/4</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">6</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>24</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">24/4</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">2</span><span class="mord">4</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>24</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">24/4</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">2</span><span class="mord">4</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
6
|
|
|
</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>
|
|
|
</td>
|
|
|
<td>
|
|
|
0 101 11
|
|
|
</td>
|
|
|
<td>
|
|
|
5
|
|
|
</td>
|
|
|
<td>
|
|
|
2
|
|
|
</td>
|
|
|
<td>
|
|
|
4
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">3/4</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">3</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>7</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">7/4</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">7</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>28</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">28/4</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">2</span><span class="mord">8</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>28</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">28/4</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">2</span><span class="mord">8</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
7
|
|
|
</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>
|
|
|
</td>
|
|
|
<td>
|
|
|
0 110 00
|
|
|
</td>
|
|
|
<td>
|
|
|
6
|
|
|
</td>
|
|
|
<td>
|
|
|
3
|
|
|
</td>
|
|
|
<td>
|
|
|
8
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">0/4</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">0</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mi mathvariant="normal">/</mi><mn>4</mn><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">4/4=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="mord">4</span><span class="mord">/</span><span class="mord">4</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>32</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">32/4</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">3</span><span class="mord">2</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>32</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">32/4</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">3</span><span class="mord">2</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
8
|
|
|
</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>
|
|
|
</td>
|
|
|
<td>
|
|
|
0 110 01
|
|
|
</td>
|
|
|
<td>
|
|
|
6
|
|
|
</td>
|
|
|
<td>
|
|
|
3
|
|
|
</td>
|
|
|
<td>
|
|
|
8
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">1/4</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">1</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>5</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">5/4</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">5</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>40</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">40/4</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">4</span><span class="mord">0</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>40</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">40/4</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">4</span><span class="mord">0</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
10
|
|
|
</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>
|
|
|
</td>
|
|
|
<td>
|
|
|
0 110 10
|
|
|
</td>
|
|
|
<td>
|
|
|
6
|
|
|
</td>
|
|
|
<td>
|
|
|
3
|
|
|
</td>
|
|
|
<td>
|
|
|
8
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi mathvariant="normal">/</mi><mn>4</mn><mo>=</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">2/4=1/2</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">2</span><span class="mord">/</span><span class="mord">4</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mord">/</span><span class="mord">2</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>6</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">6/4</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">6</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>48</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">48/4</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">4</span><span class="mord">8</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>48</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">48/4</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">4</span><span class="mord">8</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
12
|
|
|
</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>
|
|
|
Largest positive normalized
|
|
|
</td>
|
|
|
<td>
|
|
|
0 110 11
|
|
|
</td>
|
|
|
<td>
|
|
|
6
|
|
|
</td>
|
|
|
<td>
|
|
|
3
|
|
|
</td>
|
|
|
<td>
|
|
|
8
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">3/4</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">3</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>7</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">7/4</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">7</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>56</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">56/4</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">5</span><span class="mord">6</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>56</mn><mi mathvariant="normal">/</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">56/4</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">5</span><span class="mord">6</span><span class="mord">/</span><span class="mord">4</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
14
|
|
|
</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>
|
|
|
+ infinity
|
|
|
</td>
|
|
|
<td>
|
|
|
0 111 0
|
|
|
</td>
|
|
|
<td>
|
|
|
-
|
|
|
</td>
|
|
|
<td>
|
|
|
-
|
|
|
</td>
|
|
|
<td>
|
|
|
-
|
|
|
</td>
|
|
|
<td>
|
|
|
-
|
|
|
</td>
|
|
|
<td>
|
|
|
-
|
|
|
</td>
|
|
|
<td>
|
|
|
-
|
|
|
</td>
|
|
|
<td>
|
|
|
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">\infty</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">∞</span></span></span></span>
|
|
|
</td>
|
|
|
<td>
|
|
|
-
|
|
|
</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>
|
|
|
NaN
|
|
|
</td>
|
|
|
<td>
|
|
|
</td>
|
|
|
<td>
|
|
|
-
|
|
|
</td>
|
|
|
<td>
|
|
|
-
|
|
|
</td>
|
|
|
<td>
|
|
|
-
|
|
|
</td>
|
|
|
<td>
|
|
|
-
|
|
|
</td>
|
|
|
<td>
|
|
|
-
|
|
|
</td>
|
|
|
<td>
|
|
|
-
|
|
|
</td>
|
|
|
<td>
|
|
|
NaN
|
|
|
</td>
|
|
|
<td>
|
|
|
-
|
|
|
</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
<!-- END OF GENERATED CODE -->
|
|
|
|
|
|
<p>g. What is the “range” (not contiguous) of fractional decimal numbers that can be represented using this 6-bit floating-point representation?</p>
|
|
|
|
|
|
<p>“range” of real numbers <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mo>−</mo><mn>14.0</mn><mo>…</mo><mn>14.0</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-14.0\dots 14.0]</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><span class="mord">14.0</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">14.0</span><span class="mclose">]</span></span></span></span> not considering <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>±</mo><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">\pm\infty</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">±</span><span class="mord">∞</span></span></span></span> and NaN (since it’s not a contiguous range)</p>
|
|
|
|
|
|
<p>h. What is the range of the normalized exponent E (E found in the equation <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi><mo>=</mo><msup><mtext>(-1)</mtext><mi>s</mi></msup><mi>M</mi><msup><mn>2</mn><mi>E</mi></msup></mrow><annotation encoding="application/x-tex">v=\text{(-1)}^{s} M 2^{E}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;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:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord text"><span class="mord">(-1)</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.804292em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">s</span></span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span></span></span></span></span></span></span></span></span>) which can be represented by this 6-bit floating representation?</p>
|
|
|
|
|
|
<p>Range of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo>→</mo><mo stretchy="false">[</mo><mo>−</mo><mn>2</mn><mo>…</mo><mn>5</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">E \rightarrow [-2\dots 5]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">−</span><span class="mord">2</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">5</span><span class="mclose">]</span></span></span></span> is the same as <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>exp</mtext><mo>→</mo><mo stretchy="false">[</mo><mn>001</mn><mo>…</mo><mn>110</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\text{exp} \rightarrow [001\dots 110]</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 text"><span class="mord">exp</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:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">001</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">110</span><span class="mclose">]</span></span></span></span></p>
|
|
|
|
|
|
<p>Denormalized exponent E:</p>
|
|
|
|
|
|
<table>
|
|
|
<thead>
|
|
|
<tr>
|
|
|
<th>exp</th>
|
|
|
<th>E</th>
|
|
|
<th>type</th>
|
|
|
</tr>
|
|
|
</thead>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td>000</td>
|
|
|
<td>-2</td>
|
|
|
<td>denormalized</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>001</td>
|
|
|
<td>-2</td>
|
|
|
<td>normalized</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>010</td>
|
|
|
<td>-1</td>
|
|
|
<td>normalized</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>011</td>
|
|
|
<td>0</td>
|
|
|
<td>normalized</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>100</td>
|
|
|
<td>1</td>
|
|
|
<td>normlaized</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>101</td>
|
|
|
<td>2</td>
|
|
|
<td>normalized</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>110</td>
|
|
|
<td>3</td>
|
|
|
<td>normalized</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>111</td>
|
|
|
<td><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>±</mo><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">\pm\infty</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">±</span><span class="mord">∞</span></span></span></span>, NaN</td>
|
|
|
<td>denormlaized</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
|
|
|
<p>i. Give an example of a fractional decimal number that can be represented using this 6-bit floating-point representaetion, but is within the “range” of representable values.</p>
|
|
|
|
|
|
<p>11.0 cannot be represented but it is within the range.</p>
|
|
|
|
|
|
<h2 id="from-lecture-6-slide-15">From lecture 6 Slide 15</h2>
|
|
|
|
|
|
<p>What does Epsilon mean? A small positive quantity.</p>
|
|
|
<ol>
|
|
|
<li>the 5th letter of the Greek alphabet — see Alphabet Table. 2: an arbitrarily small positive quantity in mathematical analysis.</li>
|
|
|
</ol>
|
|
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<ul>
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<li><em>exp</em> and <em>frac</em>: interpreted as <em>unsigned</em> values.</li>
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<li>if <code class="language-plaintext highlighter-rouge">frac</code> = 000…0 -> M = 1.0</li>
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<li>if <code class="language-plaintext highlighter-rouge">frac</code> = 111…1 -> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mo>=</mo><mn>2.0</mn><mo>−</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">M = 2.0 - \epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">2.0</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">ϵ</span></span></span></span> (where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</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">ϵ</span></span></span></span> means a very small value)</li>
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</ul>
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<p>k. How close is the value of the <code class="language-plaintext highlighter-rouge">frac</code> of the largest normalized number to 1? In other words, how close is M to 2, i.e.What is E (epsilon) in this equation: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>M</mi><mo><</mo><mn>2</mn><mo>−</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">1\leq M \lt 2 - \epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.78041em;vertical-align:-0.13597em;"></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:0.72243em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">ϵ</span></span></span></span>? Express <code class="language-plaintext highlighter-rouge">E</code> as a fractional decimal number.</p>
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<p>First, let’s fix the above equation <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>M</mi><mo><</mo><mn>2</mn><mo>−</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">1 \leq M < 2 - \epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.78041em;vertical-align:-0.13597em;"></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:0.72243em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">ϵ</span></span></span></span>. It should be <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>M</mi><mo><</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">1 \leq M < 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.78041em;vertical-align:-0.13597em;"></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:0.72243em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span></span></span></span>.</p>
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<p>Remember:</p>
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<ul>
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<li>
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<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>1.0</mn><mo>≤</mo><mi>M</mi><mo><</mo><mn>2.0</mn></mrow><annotation encoding="application/x-tex">1.0 \leq M < 2.0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.78041em;vertical-align:-0.13597em;"></span><span class="mord">1.0</span><span class="mspace" style="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:0.72243em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2.0</span></span></span></span></span>
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</li>
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<li>
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<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>1.0</mn><mo>≤</mo><mi>M</mi><mo>≤</mo><mn>2.0</mn><mo>−</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">1.0 \leq M \leq 2.0 - \epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.78041em;vertical-align:-0.13597em;"></span><span class="mord">1.0</span><span class="mspace" style="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:0.8193em;vertical-align:-0.13597em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">2.0</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">ϵ</span></span></span></span></span>
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</li>
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<li>
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<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>1.0</mn><mo>≤</mo><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mtext>frac</mtext><mo stretchy="false">)</mo><mo>≤</mo><mn>2.0</mn><mo>−</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">1.0 \leq (1 + \text{frac}) \leq 2.0 - \epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.78041em;vertical-align:-0.13597em;"></span><span class="mord">1.0</span><span class="mspace" style="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">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">frac</span></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">2.0</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">ϵ</span></span></span></span></span>
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</li>
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<li>
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<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>0.0</mn><mo>≤</mo><mtext>frac</mtext><mo>≤</mo><mn>1.0</mn><mo>−</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">0.0 \leq \text{frac} \leq 1.0 - \epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.78041em;vertical-align:-0.13597em;"></span><span class="mord">0.0</span><span class="mspace" style="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:0.83041em;vertical-align:-0.13597em;"></span><span class="mord text"><span class="mord">frac</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1.0</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">ϵ</span></span></span></span></span>
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</li>
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</ul>
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<p>answer:</p>
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<p>The value of “frac” of the largest denormalized number is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>.11</mtext><mo>=</mo><mn>3</mn><mi mathvariant="normal">/</mi><mn>4</mn><mo>=</mo><msub><mtext>0.75</mtext><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">\text{.11} = 3/4 = \text{0.75}_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">.11</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:1em;vertical-align:-0.25em;"></span><span class="mord">3/4</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">0.75</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>How close is the value of the “frac” of the largest normalized number to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>=</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>4</mn><mo>=</mo><msub><mtext>0.25</mtext><mn>10</mn></msub></mrow><annotation encoding="application/x-tex">1 = 1/4 = \text{0.25}_{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1/4</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">0.25</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> <!--_--></p>
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<h2 id="assignment3-question-1">Assignment#3 Question 1</h2>
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<p>1) [10 points] Memory addressing modes – Marked by Aditi</p>
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<p>Assume the following values are stored at the indicated memory addresses and registers.</p>
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<table>
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<thead>
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<tr>
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<th>Memory Address</th>
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<th>Value</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td>0x230</td>
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<td>0x23</td>
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</tr>
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<tr>
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<td>0x234</td>
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<td>0x00</td>
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</tr>
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<tr>
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<td>0x235</td>
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<td>0x01</td>
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</tr>
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<tr>
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<td>0x23A</td>
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<td>0xED</td>
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</tr>
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<tr>
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<td>0x240</td>
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<td>0xFF</td>
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</tr>
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</tbody>
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</table>
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<table>
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<thead>
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<tr>
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<th>Register</th>
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<th>Value</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td>%rdi</td>
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<td>0x230</td>
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</tr>
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<tr>
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<td>%rsi</td>
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<td>0x234</td>
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</tr>
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<tr>
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<td>%rcx</td>
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<td>0x4</td>
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</tr>
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<tr>
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<td>%rax</td>
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<td>0x1</td>
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</tr>
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</tbody>
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</table>
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<p>Imagine the operands in the table below are the Src (source) operands for some unspecififed assembly instructions (any instruction except lea*), fill in the following table with the appropriate answers.</p>
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<p>Note: We do not need to know what the assembly instructions are in order to fill the table.</p>
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<table>
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<thead>
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<tr>
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<th>Operand</th>
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<th>Operand Value (expressed in hexidecimal)</th>
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<th>Operand Form (Choices are: immediate, register or one of the 9 memory addressing modes)</th>
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</tr>
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</thead>
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<tbody>
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<tr>
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<td>%rsi</td>
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<td>0x234</td>
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<td>Register</td>
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</tr>
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<tr>
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<td>(%rdi)</td>
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<td>0x23</td>
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<td>Indurect memory address mode</td>
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</tr>
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<tr>
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<td><code class="language-plaintext highlighter-rouge">$0x23A</code></td>
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<td>0x23A</td>
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<td>immedaite value</td>
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</tr>
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<tr>
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<td>0x240</td>
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<td>0xff</td>
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<td>Absolute memory addressing mode (this answer is preferable to “Imm” as it is more specific than “Imm” and highlights the fact that it does not require a “S” – see first row of tables below)</td>
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</tr>
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<tr>
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<td><code class="language-plaintext highlighter-rouge">10(%rdi)</code></td>
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<td><code class="language-plaintext highlighter-rouge">0xED</code></td>
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<td>“Base + displacement” memory addressing mode</td>
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</tr>
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<tr>
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<td><code class="language-plaintext highlighter-rouge">560(%rcx,%rax)</code></td>
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<td><code class="language-plaintext highlighter-rouge">0x01</code></td>
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<td>indexed memory addressing mode</td>
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</tr>
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<tr>
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<td><code class="language-plaintext highlighter-rouge">-550(,%rdi,2)</code></td>
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<td><code class="language-plaintext highlighter-rouge">0xED</code></td>
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<td>Scaled indexed memory addressing mode</td>
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</tr>
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<tr>
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<td><code class="language-plaintext highlighter-rouge">0x6(%rdi,%rax,4)</code></td>
|
|
|
<td><code class="language-plaintext highlighter-rouge">0xED</code></td>
|
|
|
<td>Scaled indexed addressing mode</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
|
|
|
<p>Still using the first table listed above displaying the value stored at various memory addresses and registers, fill in the following tables with three different Src (source) operands for some unspecififed assembly instructions (any instructions except lea*). For each row, this operand must result in the operand <strong>Value</strong> listed and must satified the <strong>Operand Form</strong> listed.</p>
|
|
|
|
|
|
<table>
|
|
|
<thead>
|
|
|
<tr>
|
|
|
<th>Operand</th>
|
|
|
<th>Value</th>
|
|
|
<th>Operand Form (Choices are: immediate, registers or one of the 9 memory addressing modes)</th>
|
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|
</tr>
|
|
|
</thead>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td><code class="language-plaintext highlighter-rouge">0x234</code></td>
|
|
|
<td><code class="language-plaintext highlighter-rouge">0x00</code></td>
|
|
|
<td>Absolute memory addressing mode</td>
|
|
|
</tr>
|
|
|
<tr>
|
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|
<td><code class="language-plaintext highlighter-rouge">(%rdi,%rax,4)</code></td>
|
|
|
<td><code class="language-plaintext highlighter-rouge">0x00</code></td>
|
|
|
<td>Scaled indexed memory address mode</td>
|
|
|
</tr>
|
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|
<tr>
|
|
|
<td><code class="language-plaintext highlighter-rouge">(%rdi,%rcx)</code></td>
|
|
|
<td><code class="language-plaintext highlighter-rouge">0x00</code></td>
|
|
|
<td>Indexed memory addressing mode</td>
|
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|
</tr>
|
|
|
</tbody>
|
|
|
</table>
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|
|
|
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|
<p>Other answers are possible!</p>
|
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|
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|
<h2 id="assignment3-question-2">Assignment#3 Question 2</h2>
|
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|
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|
<p>2) [2 marks] Machine level instructions and their memory locations – Marked by Aditi</p>
|
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|
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|
<p>Consider a function called <code class="language-plaintext highlighter-rouge">arith</code>, defined in a file called <code class="language-plaintext highlighter-rouge">arith.c</code> and called from the main function found in the file called <code class="language-plaintext highlighter-rouge">main.c</code>.</p>
|
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|
|
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|
<p>This function <code class="language-plaintext highlighter-rouge">arith</code>, performs some arithmatic manipulation on three parameters.</p>
|
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|
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|
|
<p>Compiling <code class="language-plaintext highlighter-rouge">main.c</code> and <code class="language-plaintext highlighter-rouge">arith.c</code> files, we create an executable called <code class="language-plaintext highlighter-rouge">ar</code>, then we execute the command:</p>
|
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|
|
|
|
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>objdump -d ar > arith.objdump
|
|
|
</code></pre></div></div>
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|
<p>We display the partial content of <code class="language-plaintext highlighter-rouge">arith.objdump</code> below. The file <code class="language-plaintext highlighter-rouge">arith.objdump</code> is the disassembled version of the executable file <code class="language-plaintext highlighter-rouge">ar</code>.</p>
|
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|
|
|
|
<p>Your task is to fill in its missing parts, which have been underlined:</p>
|
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|
|
|
|
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>0000000000400527 <arith>:
|
|
|
400527: 48 8d 04 37 lea (%rdi,%rsi,1)
|
|
|
40052b: 48 01 d0 add %rdx,%rax
|
|
|
40052e: 48 8d 0c 76 lea (%rsi,%rsi,2),%rcx
|
|
|
400532: 48 c1 e1 04 shl $0x4,%rcx
|
|
|
400536: 48 8d 54 0f 04 lea 0x4(%rdi,%rcx,1),%rdx
|
|
|
40053b: 48 0f af c2 imul %rdx,%rax
|
|
|
40053f: c3 retq
|
|
|
</code></pre></div></div>
|
|
|
|
|
|
<h2 id="assignment4-question-2">Assignment#4 Question 2</h2>
|
|
|
|
|
|
<p>In the assembly code, there are a lot more steps than in the
|
|
|
C code, so how to match them and create the C code.</p>
|
|
|
|
|
|
<p>Consier the following assembly code:</p>
|
|
|
|
|
|
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code># long func(long x,int n)
|
|
|
# x in %rdi, n in %esi, result in %rax
|
|
|
|
|
|
func:
|
|
|
movl %esi,%ecx
|
|
|
movl $1,%edx
|
|
|
movl $0,%eax
|
|
|
jmp cond
|
|
|
loop:
|
|
|
movq %rdi,%r8
|
|
|
addq %rdx,%r8
|
|
|
orq %r8,%rax
|
|
|
salq %c1,%rdx # shift left %rdx by content of %c1*
|
|
|
cond:
|
|
|
testq %rdx,%rdx # %rdx <- %rdx & %rdx
|
|
|
jne loop # jump if not zero (when %rdx & %rdx != 0)
|
|
|
ret # faill thru to ret (when %rdx & %rdx == 0)
|
|
|
</code></pre></div></div>
|
|
|
|
|
|
<p>Hand tracing code!</p>
|
|
|
|
|
|
<h2 id="from-our-lectures-14-and-15">From our Lectures 14 and 15</h2>
|
|
|
|
|
|
<p>Example pt.1 in C</p>
|
|
|
|
|
|
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>// multstore is caller
|
|
|
// x is %rdi
|
|
|
// y is %rsi
|
|
|
// dest is %rdx
|
|
|
void multstore(long x, long y, long *dest) {
|
|
|
long t = mult2(x, y);
|
|
|
*dest = t;
|
|
|
return;
|
|
|
}
|
|
|
</code></pre></div></div>
|
|
|
|
|
|
<p>Example pt.2 in C:</p>
|
|
|
|
|
|
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>// mult2 callee
|
|
|
// a is %rdi
|
|
|
// b is %rsi
|
|
|
long mult2(long a, long b) {
|
|
|
long s = a * b;
|
|
|
return s;
|
|
|
}
|
|
|
</code></pre></div></div>
|
|
|
|
|
|
<p>Example pt.1 in assembly:</p>
|
|
|
|
|
|
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>0000000000400540 <multstore>:
|
|
|
400540: push %rbx # Save %rbx (1)
|
|
|
400541: mov %rdx,%rbx # Save dest (2)
|
|
|
400544: callq 400550 <mult2> # mult2(x,y) (3)
|
|
|
400549: mov %rax,(%rbx) # Save at Dest (7)
|
|
|
40054c: pop %rbx # Restore %rbx (8)
|
|
|
40054d: retq # Return (9)
|
|
|
</code></pre></div></div>
|
|
|
|
|
|
<p>Example pt.2 in assembly:</p>
|
|
|
|
|
|
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>0000000000400550 <mult2>:
|
|
|
400550: mov %rdi,%rax # a (4)
|
|
|
400553: imul %rsi,%rax # a*b (5)
|
|
|
400557: retq # Return (6)
|
|
|
</code></pre></div></div>
|
|
|
|
|
|
<!-- NOTE: Do slide 16+. There are 4.-->
|
|
|
|
|
|
<h2 id="example--steps-1-and-2">Example – Steps 1 and 2</h2>
|
|
|
|
|
|
<table>
|
|
|
<thead>
|
|
|
<tr>
|
|
|
<th>Register/Address</th>
|
|
|
<th>M[] Stack Value</th>
|
|
|
<th>Note</th>
|
|
|
</tr>
|
|
|
</thead>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td> </td>
|
|
|
<td>ret address</td>
|
|
|
<td>return address of caller of multstore</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>(deleted) <code class="language-plaintext highlighter-rouge">%rsp</code></td>
|
|
|
<td>(new) <code class="language-plaintext highlighter-rouge">%rbx</code></td>
|
|
|
<td>(deleted) top</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>(new) <code class="language-plaintext highlighter-rouge">%rsp</code></td>
|
|
|
<td> </td>
|
|
|
<td>(new) top</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
|
|
|
<p>Registers:</p>
|
|
|
|
|
|
<table>
|
|
|
<thead>
|
|
|
<tr>
|
|
|
<th>Register</th>
|
|
|
<th>Value</th>
|
|
|
</tr>
|
|
|
</thead>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td><code class="language-plaintext highlighter-rouge">%rbx</code></td>
|
|
|
<td>dest (5)</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td><code class="language-plaintext highlighter-rouge">%rsp</code></td>
|
|
|
<td>(deleted) 0x120 (new) 118</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td><code class="language-plaintext highlighter-rouge">%rax</code></td>
|
|
|
<td> </td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td><code class="language-plaintext highlighter-rouge">%rip</code> (PC)</td>
|
|
|
<td>0x400 (deleted) 540 (deleted) 541 (1) (new) 544 (4)</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td><code class="language-plaintext highlighter-rouge">%rdi</code></td>
|
|
|
<td>x</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td><code class="language-plaintext highlighter-rouge">%rsi</code></td>
|
|
|
<td>y</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td><code class="language-plaintext highlighter-rouge">%rdx</code></td>
|
|
|
<td>dest (mem. address)</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
|
|
|
<h2 id="example--steps-3-and-4">Example – Steps 3 and 4</h2>
|
|
|
|
|
|
<table>
|
|
|
<thead>
|
|
|
<tr>
|
|
|
<th>Register/Address</th>
|
|
|
<th>M[] Stack Value</th>
|
|
|
<th>Note</th>
|
|
|
</tr>
|
|
|
</thead>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td> </td>
|
|
|
<td>ret address</td>
|
|
|
<td> </td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>(deleted) <code class="language-plaintext highlighter-rouge">%rsp</code></td>
|
|
|
<td>(deleted) <code class="language-plaintext highlighter-rouge">%rbx</code> (new) ret. address 0x400549 (3)</td>
|
|
|
<td> </td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>(2) <code class="language-plaintext highlighter-rouge">%rsp</code></td>
|
|
|
<td> </td>
|
|
|
<td>top</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
|
|
|
<p>Registers:</p>
|
|
|
|
|
|
<table>
|
|
|
<thead>
|
|
|
<tr>
|
|
|
<th>Register</th>
|
|
|
<th>Value</th>
|
|
|
</tr>
|
|
|
</thead>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td><code class="language-plaintext highlighter-rouge">%rbx</code></td>
|
|
|
<td>dest</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td><code class="language-plaintext highlighter-rouge">%rsp</code></td>
|
|
|
<td>(deleted) 0x118 (new) 110</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td><code class="language-plaintext highlighter-rouge">%rax</code></td>
|
|
|
<td>a</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td><code class="language-plaintext highlighter-rouge">%rip</code> (PC)</td>
|
|
|
<td>0x400 (deleted)544 (deleted)549(1) (deleted)550(4) (new)553</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td><code class="language-plaintext highlighter-rouge">%rdi</code></td>
|
|
|
<td>x -> a</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td><code class="language-plaintext highlighter-rouge">%rsi</code></td>
|
|
|
<td>y -> b</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td><code class="language-plaintext highlighter-rouge">%rdx</code></td>
|
|
|
<td>dest</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
|
|
|
<h2 id="example--steps-5-and-6">Example – Steps 5 and 6</h2>
|
|
|
|
|
|
<table>
|
|
|
<thead>
|
|
|
<tr>
|
|
|
<th>Register/Address</th>
|
|
|
<th>M[] Stack Value</th>
|
|
|
<th>Note</th>
|
|
|
</tr>
|
|
|
</thead>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td> </td>
|
|
|
<td>ret address</td>
|
|
|
<td> </td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>(5) (new) <code class="language-plaintext highlighter-rouge">%rsp</code></td>
|
|
|
<td><code class="language-plaintext highlighter-rouge">%rbx</code></td>
|
|
|
<td> </td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td>(deleted) <code class="language-plaintext highlighter-rouge">%rsp</code></td>
|
|
|
<td>(new) 0x400549</td>
|
|
|
<td>top</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
|
|
|
<p>Registers:</p>
|
|
|
|
|
|
<table>
|
|
|
<thead>
|
|
|
<tr>
|
|
|
<th>Register</th>
|
|
|
<th>Value</th>
|
|
|
</tr>
|
|
|
</thead>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td><code class="language-plaintext highlighter-rouge">%rbx</code></td>
|
|
|
<td>dest</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td><code class="language-plaintext highlighter-rouge">%rsp</code></td>
|
|
|
<td>(deleted) 0x110 (new) 118</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td><code class="language-plaintext highlighter-rouge">%rax</code></td>
|
|
|
<td>a*b</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td><code class="language-plaintext highlighter-rouge">%rip</code> (PC)</td>
|
|
|
<td>0x400549</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td><code class="language-plaintext highlighter-rouge">%rdi</code></td>
|
|
|
<td>x -> a</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td><code class="language-plaintext highlighter-rouge">%rsi</code></td>
|
|
|
<td>y -> b</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td><code class="language-plaintext highlighter-rouge">%rdx</code></td>
|
|
|
<td>dest</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
|
|
|
<h2 id="example--steps-7-8-and-9">Example – Steps 7, 8 and 9</h2>
|
|
|
|
|
|
<table>
|
|
|
<thead>
|
|
|
<tr>
|
|
|
<th>Register/Address</th>
|
|
|
<th>M[] Stack Value</th>
|
|
|
<th>Note</th>
|
|
|
</tr>
|
|
|
</thead>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td> </td>
|
|
|
<td>ret address</td>
|
|
|
<td> </td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td><code class="language-plaintext highlighter-rouge">%rsp</code></td>
|
|
|
<td><code class="language-plaintext highlighter-rouge">%rbx</code></td>
|
|
|
<td>Top</td>
|
|
|
</tr>
|
|
|
</tbody>
|
|
|
</table>
|
|
|
|
|
|
<p>Registers:</p>
|
|
|
|
|
|
<table>
|
|
|
<thead>
|
|
|
<tr>
|
|
|
<th>Register</th>
|
|
|
<th>Value</th>
|
|
|
</tr>
|
|
|
</thead>
|
|
|
<tbody>
|
|
|
<tr>
|
|
|
<td><code class="language-plaintext highlighter-rouge">%rbx</code></td>
|
|
|
<td>dest</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td><code class="language-plaintext highlighter-rouge">%rsp</code></td>
|
|
|
<td>0x118</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td><code class="language-plaintext highlighter-rouge">%rax</code></td>
|
|
|
<td>a*b</td>
|
|
|
</tr>
|
|
|
<tr>
|
|
|
<td><code class="language-plaintext highlighter-rouge">%rip</code></td>
|
|
|
<td>0x400549</td>
|
|
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</tr>
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<tr>
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<td><code class="language-plaintext highlighter-rouge">%rdi</code></td>
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<td>x (a)</td>
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</tr>
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<tr>
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<td><code class="language-plaintext highlighter-rouge">%rsi</code></td>
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<td>y (b)</td>
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</tr>
|
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<tr>
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<td><code class="language-plaintext highlighter-rouge">%rdx</code></td>
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|
<td>dest</td>
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</tr>
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</tbody>
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</table>
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|
<h2 id="next-next-lecture">Next next lecture</h2>
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|
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<ul>
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<li>Introduction
|
|
|
<ul>
|
|
|
<li>C program -> assembly code -> machine level code</li>
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|
</ul>
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</li>
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<li>Assembly language basics: data, move operation
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<ul>
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<li>Memory addressing modes</li>
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</ul>
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</li>
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<li>Operation leaq and Arithmetic & logical operations</li>
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<li>Conditional Statement – Condition Code + cmovX</li>
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<li>Loops</li>
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|
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<li>(highlighted) Function call – Stack – Recursion
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|
|
<ul>
|
|
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<li>Overview of Function Call</li>
|
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|
<li>Memory Layout and Stack - x86-64 instructions and registers</li>
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|
|
<li>Passing control</li>
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|
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<li>(highlighted) Passing data – Calling Conventions</li>
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<li>Managing local data</li>
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<li>Recursion</li>
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</ul>
|
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</li>
|
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<li>Array</li>
|
|
|
<li>Buffer Overflow</li>
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|
|
<li>Floating-point operations</li>
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</ul>
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|
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<footer>
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</footer>
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|
</div>
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</main>
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</body>
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</html>
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