6.03:_x-mals

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<p class="lt-math-9858">Just like in base 10, we can add boxes to the right of the decimal point in other bases, like base 5.</p>

<p class="lt-math-9858">However, the prefix &ldquo;dec&rdquo; in &ldquo;decimal point&rdquo; means ten.  So we really shouldn&rsquo;t call it a decimal point anymore.  Maybe a &ldquo;pentimal point&rdquo;?  (In fact, the general term is <strong>radix point</strong>.)</p>

<section class="box-exercise"><span id="Think_.2F_Pair_.2F_Share"></span><h5 class="box-legend lt-math-9858"><span class="lt-icon-default">Think / Pair / Share</span></h5>

<ul>
    <li class="lt-math-9858">Use reasoning like you saw on page 6 for the base ten system to think about other number systems:</li>
</ul>

<ul>
    <li class="lt-math-9858">Figure out the values of <em>x</em> and <em>y</em> in the picture of the base-5 system above. Be sure you can explain your reasoning.</li>
</ul>

<ul>
    <li class="lt-math-9858">Draw a base-4 &ldquo;Dots &amp; Boxes&rdquo; model, including a radix point and some boxes to the right. Label at least three boxes to the left of the ones place and three boxes to the right of the ones place.</li>
</ul>

<ul>
    <li class="lt-math-9858">Draw a base-6 &ldquo;Dots &amp; Boxes&rdquo; model, including a radix point and some boxes to the right. Label at least three boxes to the left of the ones place and three boxes to the right of the ones place</li>
</ul>
</section>

<p class="lt-math-9858">In general, in a base-<em>b</em> system, the boxes to the left of the ones place represent positive powers of the base <em>b</em>.  Boxes to the right of the ones place represent reciprocals of those powers.</p>

<p class="lt-math-9858">Work on the following exercises on your own or with a partner.</p>

<ol>
    <li class="lt-math-9858">Draw a &ldquo;Dots &amp; Boxes&rdquo; picture of each number. $$(a)\; 0.03_{five} \quad (b)\; 0.22_{six} \quad (c)\; 0.103_{four} \quad (d)\; 0.002_{three} \ldotp$$</li>
    <li class="lt-math-9858">Find a familiar (base-10) fraction value for each number. $$(a)\; 0.04_{five} \quad (b)\; 0.3_{six} \quad (c)\; 0.02_{four} \quad (d)\; 0.03_{nine} \ldotp$$</li>
    <li class="lt-math-9858">Find a familiar (base-10) fraction value for each number. (You might want to re-read the example of $0.31$ in base ten from the previous section.) $$(a)\; 0.13_{five} \quad (b)\; 0.25_{six} \quad (c)\; 0.101_{two} \quad (d)\; 0.24_{seven} \quad (e)\; 0.55_{eight} \ldotp$$</li>
</ol>

<section class="box-exercise"><span id="Think_.2F_Pair_.2F_Share_2"></span><h5 class="box-legend lt-math-9858"><span class="lt-icon-default">Think / Pair / Share</span></h5>

<p class="lt-math-9858">Tami and Courtney were working on converting $0.44_{five}$ to a familiar base-10 fraction.  Courtney said this:</p>

<blockquote>
<p class="lt-math-9858">The places in base five to the right of the point are like $\frac{1}{5}$ and then $\frac{1}{25}$.  Since this has two places, the answer should be $\frac{44}{25}$.</p>
</blockquote>

<p class="lt-math-9858">Tami thought about what Courtney said and replied:</p>

<blockquote>
<p class="lt-math-9858">I don&rsquo;t know what the right answer is, but I know that  can&rsquo;t be right.  The number $0.44_{five}$ is less than one, since there are no numbers in the ones place and no explosions that we can do. But the fraction $\frac{44}{25}$ is more than one. It&rsquo;s almost two. So they can&rsquo;t be the same number.</p>
</blockquote>

<ul>
    <li class="lt-math-9858">Who makes the most sense, Courtney or Tami? Why do you think so?</li>
    <li class="lt-math-9858">Find the right answer to the problem Courtney and Tami were working on.</li>
</ul>
</section>

<section class="box-example"><span id="Problem_1"></span><h5 class="box-legend lt-math-9858"><span class="lt-icon-default">Problem 1</span></h5>

<p class="lt-math-9858">Find the &ldquo;decimal&rdquo; representation of $\frac{1}{4}$ in each of the following bases.  Be sure that you can justify your answer. (You might want to review the example of $12 \frac{3}{4}$ in the previous section.) \[\begin{split} base\; 2 \qquad base\; 4&amp; \qquad base\; 6 \\ base\; 8 \qquad base\; 10&amp; \qquad base\; 12 \end{split} \nonumber \]</p>
</section>

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