2.03:_Binary_Numbers

<p class="lt-math-9830">Let&rsquo;s go back to the 1&larr;2 rule and examine what&rsquo;s really going on.</p>

<p class="lt-math-9830">Whenever there are two dots in single box, they &ldquo;explode,&rdquo; disappear, and become one dot in the box to the left.</p>
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<p class="lt-math-9830">Two dots in the right-most box is worth one dot in the next box to the left.</p>

<p class="lt-math-9830">If each of the original dots is worth &ldquo;one,&rdquo; then the single dot on the left must be worth two.</p>

<p class="lt-math-9830">But we also have two dots in the box of value 2 is worth one dot in the box just to the left...</p>

<p class="lt-math-9830">So that next box must be worth two 2&rsquo;s, which is four!</p>

<p class="lt-math-9830">And two of these fours make eight.</p>

<p class="lt-math-9830">We said earlier that the 1&larr;2 code for nine dots was 1001. Let&rsquo;s check:</p>

<p class="lt-math-9830">\[8 + 1 = 9 \nonumber \]</p>

<p class="lt-math-9830">so this works!</p>
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<p class="lt-math-9830">You should have found that ten dots has 1&larr;2 code 1010.</p>

<p class="lt-math-9830">\[8 + 2 = 10 \ldotp \nonumber \]</p>
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<section class="box-example"><span id="Problem_7"></span><h5 class="box-legend lt-math-9830"><span class="lt-icon-default">Problem 7</span></h5>

<ol>
    <li class="lt-math-9830">If there were a box to the left of the 8 box, what would the value of that box be?</li>
    <li class="lt-math-9830">What would be the value of a box <i>two</i> spots to the left of the 8 box? Three spots to the left?</li>
    <li class="lt-math-9830">What number has 1&larr;2 code 100101?</li>
    <li class="lt-math-9830">What is the 1&larr;2 code for two hundred dots?</li>
</ol>
</section>

<section class="box-definition"><span id="Definition_and_Notation"></span><h5 class="box-legend lt-math-9830"><span class="lt-icon-default">Definition and Notation</span></h5>

<p class="lt-math-9830">Numbers written in the 1&larr;2 code are called <strong>binary numbers</strong> or <strong>base two numbers</strong>. (The prefix &ldquo;bi&rdquo; means &ldquo;two.&rdquo;)</p>

<p class="lt-math-9830">From now on, when we want to indicate that a number is written in base two, we will write a subscript &ldquo;two&rdquo; on the number.</p>

<p class="lt-math-9830">So $1001_{two}$ means &ldquo;the number of dots that has 1&larr;2 code 1001,&rdquo; which we already saw was nine.</p>
</section>

<p class="lt-math-9830">Important! When we read we say &ldquo;one zero zero one base two.&rdquo; We don&rsquo;t say &ldquo;one thousand and one,&rdquo; because &ldquo;thousand&rdquo; is not a binary number.</p>

<section class="box-exercise"><span id="Think_.2F_Pair_.2F_Share"></span><h5 class="box-legend lt-math-9830"><span class="lt-icon-default">Think / Pair / Share</span></h5>

<ol>
    <li class="lt-math-9830">Your first goal: come up with a <i>general</i> <i>method</i> to find the number of dots represented by any binary number. Clearly describe your method. Test your method out on these numbers, and check your work by actually &ldquo;unexploding&rdquo; the dots. $$1_{two} \qquad 101_{two} \qquad 1011_{two} \qquad 11111_{two}$$</li>
    <li class="lt-math-9830">Explain why binary numbers only contain the digits 0 and 1.</li>
    <li class="lt-math-9830">Here is a new (harder) goal: come up with a <i>general method</i> to find the binary number related to any number of dots <i>without actually going through the &ldquo;exploding dot&rdquo; process</i>. Clearly describe your method. Test your method out on these numbers, and find a way to check your work.</li>
</ol>

<p class="mt-align-center lt-math-9830">two dots = ___<sub>two</sub>               seventeen dots = ___<sub>two</sub></p>

<p class="mt-align-center lt-math-9830">sixty-three dots = ___<sub>two</sub>               one hundred dots = ___<sub>two</sub></p>
</section>

<span id="History"></span><h2 class="lt-math-9830">History</h2>

<p class="lt-math-9830">You probably realize by now that a number is an abstract concept with many representations. The standard decimal representation of a number is only one of these. For computers, numbers are always represented in binary. The basic units are transistors which are either on (1) or off (0).</p>

<p class="mt-align-center lt-math-9830"><em>Electronic circuit board from Samsung Galaxy S III.</em></p>

<p class="lt-math-9830">A transistor<u><sup>[1]</sup></u> is said to store <b><i>one bit</i></b> of information. Eight bits make a byte and a typical home computer&rsquo;s central processing unit performs computations on registries that are each 8 bytes (64-bits).</p>

<p class="lt-math-9830">Using the 1&larr;2 rule we can represent the numbers 0 through 18,446,744,073,709,551,615 with 64 bits.</p>

<hr />
<ol>
    <li class="lt-math-9830">Image of circuit board from <u><a href="http://www.publicdomainpictures.net/" rel="nofollow">http://www.publicdomainpictures.net/</a></u>, licensed under <u><a href="http://creativecommons.org/publicdomain/zero/1.0/" itemprop="license" rel="license noopener" target="_blank">CC0 Public Domain</a></u>. <u>&crarr;</u></li>
</ol>

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