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<!--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">Binary number system</h1><h2 id="decimal-denary-number-system">Decimal (denary) number system</h2><p>Binary is a <strong>positional number system</strong>, just like the decimal number system. This means that the value of an individual digit is conferred by its position relative to other digits. Another way of expressing this is to say that number systems work on the basis of <strong>place value</strong>.</p><p>In the decimal system the columns increase by <strong>powers of 10</strong>. This is because there are ten total integers in the system:</p><p><span class="math inline">\(1, 2, 3, 4, 5, 6, 7, 8, 9\)</span></p><p>When we have completed all the possible intervals between <span class="math inline">\(0\)</span> and <span class="math inline">\(9\)</span>, we start again in a new column.</p><p>The principle of counting in decimal:</p><pre><code class="language-none">0009
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0010
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0011
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0012
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0013
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...
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0019
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0020
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0021</code></pre><p>Thus each column is ten times larger than the previous column:</p><ul><li>Ten (<span class="math inline">\(10^1\)</span>) is ten times larger than one (<span class="math inline">\(10^0\)</span>)</li><li>A hundred (<span class="math inline">\(10^2\)</span>) is ten times larger than ten (<span class="math inline">\(10^1\)</span>)</li></ul><p>We use this knowledge of the exponents of the base of 10 to read integers that contain multiple digits (i.e. any number greater than 9).</p><p>Thus 264 is the sum of</p><p><span class="math display">$$[4 \cdot (10^0)] + [6 \cdot (10^1)] + [2 \cdot 10^2] $$</span></p><h2 id="binary-number-system-1">Binary number system</h2><p>In the binary number system, the columns increase by powers of two. This is because there are only two integers: 0 and 1. As a result, you are required to begin a new column every time you complete an interval from 0 to 1.</p><p>So instead of:</p><p><span class="math display">$$ 10^0, 10^1, 10^2, 10^3 ... (1, 10, 100, 1000) $$</span></p><p>You have:</p><p><span class="math display">$$ 2^0, 2^1, 2^2, 2^3, 2^4... (0, 2, 4, 8, 16) $$</span></p><p>When counting in binary, we put zeros as placeholders in the columns we have not yet filled. This helps to indicate when we need to begin a new column. Thus the counting sequence:</p><p><span class="math display">$$ 1, 2, 3, 4 $$</span></p><p>is equal to:</p><p><span class="math display">$$ 0001, 0010, 0011, 0100 $$</span></p><p>Counting in binary:</p><pre><code class="language-none">000000
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000001
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000010
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000011
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000100
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000101
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000111</code></pre><h2 id="binary-prefix">Binary prefix</h2><p>To distinguish numbers in binary from decimal or <span class="zettel-link-container cf"><span class="zettel-link" title="Zettel: Hexadecimal number system"><a href="Hexadecimal_number_system.html">hexadecimal</a></span></span> numbers, it is common to use the prefix <code>0b</code>. Thus, e.g, <code>0b110</code> for decimal <code>6</code>.</p><h2 id="converting-decimal-to-binary">Converting decimal to binary</h2><p>Let’s convert 6 into binary:</p><p>If we have before us the binary place values (<span class="math inline">\(1, 2, 4, 8\)</span>). We know that 6 is equal to: <strong>1 in the two column and 1 in the 4 column → 110</strong></p><p>More clearly:</p><p><img src="/static/Pasted_image_20220319135558.png" /></p><p>And for comparison:</p><p><img src="/static/Pasted_image_20220319135805.png" /></p><p>Or we can express the binary as:</p><p><span class="math display">$$ (1 _ 2) + (1 _ 4) $$</span></p><p>Or more concisely as:</p><p><span class="math display">$$ 2^1 + 2^2 $$</span></p><h3 id="another-example">Another example</h3><p>Let’s convert 23 into binary:</p><p><img src="/static/Pasted_image_20220319135823.png" /></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="Hexadecimal_number_system.html">Hexadecimal number system</a></span></span><ul class="context-list" style="zoom: 85%;"><li class="item"><div class="pandoc"><p>Hexadecimal is the other main number system used in computing. It works in tandem with the <span class="zettel-link-container cf"><span class="zettel-link" title="Zettel: Binary number system"><a href="Binary_number_system.html">binary number system</a></span></span> and provides an easier and more accessible means of working with long sequences of binary numbers.</p></div></li></ul></li><li><span class="zettel-link-container cf"><span class="zettel-link"><a href="Binary_units_of_measurement.html">Binary units of measurement</a></span></span><ul class="context-list" style="zoom: 85%;"><li class="item"><div class="pandoc"><p>The equivalent entity in the <span class="zettel-link-container cf"><span class="zettel-link" title="Zettel: Binary number system"><a href="Binary_number_system.html">binary number system</a></span></span> is the <strong>bit</strong>. For example the binary number 110 has three bits. A bit can only have one of two values in contrast to a digit which can have one of ten values. These values are 0 and 1.</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><span class="ui basic label zettel-tag" title="Tag">number-systems</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> |