54 lines
No EOL
15 KiB
HTML
54 lines
No EOL
15 KiB
HTML
<!DOCTYPE html><html><head><meta content="text/html; charset=utf-8" http-equiv="Content-Type" /><meta content="width=device-width, initial-scale=1" name="viewport" /><!--replace-start-0--><!--replace-start-5--><!--replace-start-8--><title>Two’s complement - My Zettelkasten</title><!--replace-end-8--><!--replace-end-5--><!--replace-end-0--><link href="https://cdn.jsdelivr.net/npm/fomantic-ui@2.8.7/dist/semantic.min.css" rel="stylesheet" /><link href="https://fonts.googleapis.com/css?family=Merriweather|Libre+Franklin|Roboto+Mono&display=swap" rel="stylesheet" /><!--replace-start-1--><!--replace-start-4--><!--replace-start-7--><link href="https://raw.githubusercontent.com/srid/neuron/master/assets/neuron.svg" rel="icon" /><meta content="Two’s complement is a method for representing signed numbers (negative integers) in binary." name="description" /><meta content="Two’s complement" property="og:title" /><meta content="My Zettelkasten" property="og:site_name" /><meta content="article" property="og:type" /><meta content="Twos_complement" property="neuron:zettel-id" /><meta content="Twos_complement" property="neuron:zettel-slug" /><meta content="binary" property="neuron:zettel-tag" /><script type="application/ld+json">[]</script><style type="text/css">body{background-color:#eeeeee !important;font-family:"Libre Franklin", serif !important}body .ui.container{font-family:"Libre Franklin", serif !important}body h1, h2, h3, h4, h5, h6, .ui.header, .headerFont{font-family:"Merriweather", sans-serif !important}body code, pre, tt, .monoFont{font-family:"Roboto Mono","SFMono-Regular","Menlo","Monaco","Consolas","Liberation Mono","Courier New", monospace !important}body div.z-index p.info{color:#808080}body div.z-index ul{list-style-type:square;padding-left:1.5em}body div.z-index .uplinks{margin-left:0.29999em}body .zettel-content h1#title-h1{background-color:rgba(33,133,208,0.1)}body nav.bottomPane{background-color:rgba(33,133,208,2.0e-2)}body div#footnotes{border-top-color:#2185d0}body p{line-height:150%}body img{max-width:100%}body .deemphasized{font-size:0.94999em}body .deemphasized:hover{opacity:1}body .deemphasized:not(:hover){opacity:0.69999}body .deemphasized:not(:hover) a{color:#808080 !important}body div.container.universe{padding-top:1em}body div.zettel-view ul{padding-left:1.5em;list-style-type:square}body div.zettel-view .pandoc .highlight{background-color:#ffff00}body div.zettel-view .pandoc .ui.disabled.fitted.checkbox{margin-right:0.29999em;vertical-align:middle}body div.zettel-view .zettel-content .metadata{margin-top:1em}body div.zettel-view .zettel-content .metadata div.date{text-align:center;color:#808080}body div.zettel-view .zettel-content h1{padding-top:0.2em;padding-bottom:0.2em;text-align:center}body div.zettel-view .zettel-content h2{border-bottom:solid 1px #4682b4;margin-bottom:0.5em}body div.zettel-view .zettel-content h3{margin:0px 0px 0.4em 0px}body div.zettel-view .zettel-content h4{opacity:0.8}body div.zettel-view .zettel-content div#footnotes{margin-top:4em;border-top-style:groove;border-top-width:2px;font-size:0.9em}body div.zettel-view .zettel-content div#footnotes ol > li > p:only-of-type{display:inline;margin-right:0.5em}body div.zettel-view .zettel-content aside.footnote-inline{width:30%;padding-left:15px;margin-left:15px;float:right;background-color:#d3d3d3}body div.zettel-view .zettel-content .overflows{overflow:auto}body div.zettel-view .zettel-content code{margin:auto auto auto auto;font-size:100%}body div.zettel-view .zettel-content p code, li code, ol code{padding:0.2em 0.2em 0.2em 0.2em;background-color:#f5f2f0}body div.zettel-view .zettel-content pre{overflow:auto}body div.zettel-view .zettel-content dl dt{font-weight:bold}body div.zettel-view .zettel-content blockquote{background-color:#f9f9f9;border-left:solid 10px #cccccc;margin:1.5em 0px 1.5em 0px;padding:0.5em 10px 0.5em 10px}body div.zettel-view .zettel-content.raw{background-color:#dddddd}body .ui.label.zettel-tag{color:#000000}body .ui.label.zettel-tag a{color:#000000}body nav.bottomPane ul.backlinks > li{padding-bottom:0.4em;list-style-type:disc}body nav.bottomPane ul.context-list > li{list-style-type:lower-roman}body .footer-version img{-webkit-filter:grayscale(100%);-moz-filter:grayscale(100%);-ms-filter:grayscale(100%);-o-filter:grayscale(100%);filter:grayscale(100%)}body .footer-version img:hover{-webkit-filter:grayscale(0%);-moz-filter:grayscale(0%);-ms-filter:grayscale(0%);-o-filter:grayscale(0%);filter:grayscale(0%)}body .footer-version, .footer-version a, .footer-version a:visited{color:#808080}body .footer-version a{font-weight:bold}body .footer-version{margin-top:1em !important;font-size:0.69999em}@media only screen and (max-width: 768px){body div#zettel-container{margin-left:0.4em !important;margin-right:0.4em !important}}body span.zettel-link-container span.zettel-link a{color:#2185d0;font-weight:bold;text-decoration:none}body span.zettel-link-container span.zettel-link a:hover{background-color:rgba(33,133,208,0.1)}body span.zettel-link-container span.extra{color:auto}body span.zettel-link-container.errors{border:solid 1px #ff0000}body span.zettel-link-container.errors span.zettel-link a:hover{text-decoration:none !important;cursor:not-allowed}body [data-tooltip]:after{font-size:0.69999em}body div.tag-tree div.node{font-weight:bold}body div.tag-tree div.node a.inactive{color:#555555}body .tree.flipped{-webkit-transform:rotate(180deg);-moz-transform:rotate(180deg);-ms-transform:rotate(180deg);-o-transform:rotate(180deg);transform:rotate(180deg)}body .tree{overflow:auto}body .tree ul.root{padding-top:0px;margin-top:0px}body .tree ul{position:relative;padding:1em 0px 0px 0px;white-space:nowrap;margin:0px auto 0px auto;text-align:center}body .tree ul::after{content:"";display:table;clear:both}body .tree ul:last-child{padding-bottom:0.1em}body .tree li{display:inline-block;vertical-align:top;text-align:center;list-style-type:none;position:relative;padding:1em 0.5em 0em 0.5em}body .tree li::before{content:"";position:absolute;top:0px;right:50%;border-top:solid 2px #cccccc;width:50%;height:1.19999em}body .tree li::after{content:"";position:absolute;top:0px;right:50%;border-top:solid 2px #cccccc;width:50%;height:1.19999em}body .tree li::after{right:auto;left:50%;border-left:solid 2px #cccccc}body .tree li:only-child{padding-top:0em}body .tree li:only-child::after{display:none}body .tree li:only-child::before{display:none}body .tree li:first-child::before{border-style:none;border-width:0px}body .tree li:first-child::after{border-radius:5px 0px 0px 0px}body .tree li:last-child::after{border-style:none;border-width:0px}body .tree li:last-child::before{border-right:solid 2px #cccccc;border-radius:0px 5px 0px 0px}body .tree ul ul::before{content:"";position:absolute;top:0px;left:50%;border-left:solid 2px #cccccc;width:0px;height:1.19999em}body .tree li div.forest-link{border:solid 2px #cccccc;padding:0.2em 0.29999em 0.2em 0.29999em;text-decoration:none;display:inline-block;border-radius:5px 5px 5px 5px;color:#333333;position:relative;top:2px}body .tree.flipped li div.forest-link{-webkit-transform:rotate(180deg);-moz-transform:rotate(180deg);-ms-transform:rotate(180deg);-o-transform:rotate(180deg);transform:rotate(180deg)}</style><script
|
||
async=""
|
||
id="MathJax-script"
|
||
src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"
|
||
></script>
|
||
<link
|
||
href="https://cdnjs.cloudflare.com/ajax/libs/prism/1.23.0/themes/prism.min.css"
|
||
rel="stylesheet"
|
||
/><link rel="preconnect" href="https://fonts.googleapis.com" /><link
|
||
rel="preconnect"
|
||
href="https://fonts.gstatic.com"
|
||
crossorigin
|
||
/><link
|
||
href="https://fonts.googleapis.com/css2?family=IBM+Plex+Mono:ital,wght@0,100;0,200;0,300;0,400;0,500;0,600;0,700;1,100;1,200;1,300;1,400;1,500;1,600;1,700&family=IBM+Plex+Sans+Condensed:ital,wght@0,100;0,200;0,300;0,400;0,500;0,600;0,700;1,100;1,200;1,300;1,400;1,500;1,600;1,700&family=IBM+Plex+Sans:ital,wght@0,100;0,200;0,300;0,400;0,500;0,600;0,700;1,100;1,200;1,300;1,400;1,500;1,600;1,700&family=IBM+Plex+Serif:ital,wght@0,100;0,200;0,300;0,400;0,500;0,600;0,700;1,100;1,200;1,300;1,400;1,500;1,600;1,700&display=swap"
|
||
rel="stylesheet"
|
||
/>
|
||
<script src="https://cdnjs.cloudflare.com/ajax/libs/prism/1.23.0/components/prism-core.min.js"></script>
|
||
<script src="https://cdnjs.cloudflare.com/ajax/libs/prism/1.23.0/plugins/autoloader/prism-autoloader.min.js"></script>
|
||
<style>
|
||
body .ui.container,
|
||
body ul {
|
||
font-family: "IBM Plex Sans" !important;
|
||
}
|
||
body blockquote {
|
||
border-left-width: 3px !important;
|
||
font-style: italic;
|
||
}
|
||
.headerFont,
|
||
.ui.header,
|
||
body h1,
|
||
h2,
|
||
h3,
|
||
h4,
|
||
h5,
|
||
h6 {
|
||
font-family: "IBM Plex Sans Condensed" !important;
|
||
}
|
||
body p {
|
||
line-height: 1.4;
|
||
}
|
||
.monoFont,
|
||
body code,
|
||
pre,
|
||
tt {
|
||
font-family: "IBM Plex Mono" !important;
|
||
font-size: 12px !important;
|
||
line-height: 1.4 !important;
|
||
}
|
||
</style>
|
||
<!--replace-end-7--><!--replace-end-4--><!--replace-end-1--></head><body><div class="ui fluid container universe"><!--replace-start-2--><!--replace-start-3--><!--replace-start-6--><div class="ui text container" id="zettel-container" style="position: relative"><div class="zettel-view"><article class="ui raised attached segment zettel-content"><div class="pandoc"><h1 id="title-h1">Two’s complement</h1><h2 id="summary">Summary</h2><ul><li><p><em>Two’s complement</em> is a method for representing signed numbers (negative integers) in binary.</p></li><li><p>It is derived by inverting the values of an unsigned binary integer to create its signed equivalent.</p></li><li><p>A benefit is that hardware implementation of the binary arithmetic of signed and unsigned numbers can be handled in the same manner as unsigned numbers, requiring no additional handling. A drawback is that it halves the informational capacity of the given word length for the binary system.</p></li></ul><h2 id="detail">Detail</h2><h3 id="procedural-steps">Procedural steps</h3><p>Two’s complement divides the available word length (see <span class="zettel-link-container cf"><span class="zettel-link" title="Zettel: Binary encoding"><a href="Binary_encoding.html">binary encoding</a></span></span>) into two subsets: one for negative integrs and one for positive integers.</p><p>Take the binary encoding of decimal five (<code>0101</code>). Its complement is <code>1011</code>.</p><p>The procedure for deriving the complement is as follows.</p><p>To derive the complement of an unsigned number:</p><ol><li>Take the unsigned number and invert its digits: <code>0</code> becomes <code>1</code>, <code>1</code> becomes <code>0</code></li><li>Add one</li></ol><p><img src="/static/unsigned-to-signed.png" /></p><p>To derive the unsigned equivalent of a signed number you invert the process but still make the smallest digit <code>1</code>:</p><p><img src="/static/signed-to-unsigned.png" /></p><h3 id="formal-expression">Formal expression</h3><p><span class="math display">$$
|
||
2^n - x
|
||
$$</span></p><ul><li>where <span class="math inline">\(x\)</span> is the negative integer in binary that we wish to derive</li><li>where <span class="math inline">\(n\)</span> is the word length of the binary system in bits.</li></ul><p>Applied to the earlier example we have <span class="math inline">\(2^4 -5\)</span> which is:</p><p><span class="math display">$$
|
||
16 - 5 = 11
|
||
$$</span></p><p>When we convert the decimal <code>11</code> to binary we get <code>1011</code> which is identical to the signed version of the unsigned integer.</p><p>We can confirm the correctness of the derviation by summing the signed and unsigned binary values. If this results in zeros (ignoring the overflow bit), the derivation is correct as the two values effectively cancel each other out:</p><p>$$</p><p>1011 + 0101 = 0000 $$</p><h3 id="advantages">Advantages</h3><ul><li>The circuit implementation of arithmetic involving positive and negative integers is the same as the implementation of positive integers. There is no need for additional harware or special handling of the values.</li><li>This can be contrasted with the alternative approaches to signing numbers such as <strong>signed magnitude representation</strong> which uses certain bits as designators of negative/positive status.</li></ul><h3 id="limitations">Limitations</h3><ul><li><p>Two’s complement reduces the overall informational capacity of the given binary word length, effectively halving the total number of unique values.</p></li><li><p>In a 4-bit system instead of 16 total unique encodings of integers you have 8 encodings for positive integers and 8 encodings for the their signed equivalent. For integers larger than denary 8 you would need to increase the bit length of the system</p></li><li><p>Consequently two’s complement can necessitate larger overall word lengths.</p></li></ul><h2 id="related-notes">Related notes</h2><p><span class="zettel-link-container cf"><span class="zettel-link" title="Zettel: Signed and unsigned numbers"><a href="Signed_and_unsigned_numbers.html">signed_and_unsigned_numbers</a></span></span>, <span class="zettel-link-container cf"><span class="zettel-link" title="Zettel: Binary addition"><a href="Binary_addition.html">binary addition</a></span></span>, <span class="zettel-link-container cf"><span class="zettel-link"><a href="Signed_magnitude_representation.html">Signed magnitude representation</a></span></span></p></div></article><nav class="ui attached segment deemphasized backlinksPane" id="neuron-backlinks-pane"><h3 class="ui header">Backlinks</h3><ul class="backlinks"><li><span class="zettel-link-container cf"><span class="zettel-link"><a href="Signed_and_unsigned_numbers.html">Signed and unsigned numbers</a></span></span><ul class="context-list" style="zoom: 85%;"><li class="item"><div class="pandoc"><p><span class="zettel-link-container cf"><span class="zettel-link"><a href="Twos_complement.html">Two’s complement</a></span></span>, <span class="zettel-link-container cf"><span class="zettel-link"><a href="Binary_encoding.html">Binary encoding</a></span></span>, <span class="zettel-link-container cf"><span class="zettel-link"><a href="Signed_magnitude_representation.html">Signed magnitude representation</a></span></span></p></div></li></ul></li></ul></nav><nav class="ui attached segment deemphasized bottomPane" id="neuron-tags-pane"><div><span class="ui basic label zettel-tag" title="Tag">binary</span></div></nav><nav class="ui bottom attached icon compact inverted menu blue" id="neuron-nav-bar"><!--replace-start-9--><!--replace-end-9--><a class="right item" href="impulse.html" title="Open Impulse"><i class="wave square icon"></i></a></nav></div></div><!--replace-end-6--><!--replace-end-3--><!--replace-end-2--><div class="ui center aligned container footer-version"><div class="ui tiny image"><a href="https://neuron.zettel.page"><img alt="logo" src="https://raw.githubusercontent.com/srid/neuron/master/assets/neuron.svg" title="Generated by Neuron 1.9.35.3" /></a></div></div></div></body></html> |