6.05:_More_x_-mals

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<p class="lt-math-9860">It should come as no surprise that we can use this reasoning about division in the &ldquo;Dots &amp; Boxes&rdquo; model in other bases as well.</p>

<p class="lt-math-9860">The following picture shows that working in base 5,</p>

<p class="lt-math-9860">\[1432_{five} \div 13_{five} = 110_{five} R2_{five},\; \text{meaning}\; 1432_{five} = 110_{five} \cdot 13_{five} + 2_{five} \ldotp \nonumber \]</p>

<section class="box-exercise"><span id="Think_.2F_Pair_.2F_Share"></span><h2 class="box-legend lt-math-9860"><span class="lt-icon-default">Think / Pair / Share</span></h2>

<p class="lt-math-9860">Carefully explain the connection between the picture and the equation shown above.</p>

<ul>
    <li class="lt-math-9860">Show in the picture where you see $1432_{five}$ from the equation.</li>
    <li class="lt-math-9860">Where do you see $13_{five}$?</li>
    <li class="lt-math-9860">Where do you see $110_{five}$ and $2_{five}$?</li>
</ul>
</section>

<section class="box-example"><span id="Example:_.5C(1432_.7Bfive.7D_.5Cdiv_13_.7Bfive.7D.5C)"></span><h2 class="box-legend lt-math-9860"><span class="lt-icon-default">Example: $1432_{five} \div 13_{five}$</span></h2>

<p class="lt-math-9860">Here&rsquo;s where we left off the division, with a remainder of 2:</p>

<p class="lt-math-9860">Now we can unexplode one of those two remaining dots.  Then we&rsquo;re able to make another group of $13_{five}$.</p>

<p class="lt-math-9860">Once again, there are two dots left over, not in any group.  So let&rsquo;s unexplode one of them.</p>

<p class="lt-math-9860">And we still have two dots left over.  Why not do it again?</p>

<p class="lt-math-9860">It seems like we&rsquo;re going to be doing the same thing forever:</p>

<ul>
    <li class="lt-math-9860">Start with two dots in some box.</li>
    <li class="lt-math-9860">Unexplode one one of the dots, so you have one dot in your original box and five in the box to the right.</li>
    <li class="lt-math-9860">Form a group of $3_{five}$.  That uses the one dot in your original box and three dots in the box to the right.</li>
    <li class="lt-math-9860">So you have two dots left in a box.</li>
    <li class="lt-math-9860">Unexplode one of the dots, so you have one dot in your original box and five in the box to the right.</li>
    <li class="lt-math-9860">This feels familiar&hellip;</li>
</ul>

<p class="lt-math-9860">We conclude:</p>

<p class="lt-math-9860">\[1432_{five} \div 13_{five} = 110.111 \ldots_{five} = 110. \bar{1}_{five} \ldotp \nonumber \]</p>
</section>

<section class="box-exercise"><span id="Think_.2F_Pair_.2F_Share_2"></span><h2 class="box-legend lt-math-9860"><span class="lt-icon-default">Think / Pair / Share</span></h2>

<p class="lt-math-9860">The equation</p>

<p class="lt-math-9860">\[1432_{five} \div 13_{five} = 110. \bar{1}_{five} \ldotp \nonumber \]</p>

<p class="lt-math-9860">is a statement in base five. What is it saying in base ten?</p>

<p class="lt-math-9860">&ldquo;$1432_{five}$&rdquo; is the number</p>

<p class="lt-math-9860">\[1 \cdot 125 + 4 \cdot 25 + 3 \cdot 5 + 2 \cdot 1 = 242_{ten} \ldotp \nonumber \]</p>

<ul>
    <li class="lt-math-9860">What is $13_{five}$ in base 10?  Be sure to explain your answer.</li>
    <li class="lt-math-9860">What is $110. \bar{1}_{five}$ in base 10?  Explain how you got your answer.</li>
    <li class="lt-math-9860">Translate the equation above to a statement in base ten and check that it is correct.</li>
</ul>
</section>

<section class="box-example"><span id="Problem_2"></span><h2 class="box-legend lt-math-9860"><span class="lt-icon-default">Problem 2</span></h2>

<ol>
    <li class="lt-math-9860">Draw pictures to compute $8 \div 3$ in a base ten system, and show the answer is $2. \bar{6}$.</li>
    <li class="lt-math-9860">Draw the pictures to compute $8_{nine} \div 3_{nine}$ in a base 9 system, and write the answer as a decimal.  (Or is it a &ldquo;nonimal&rdquo;?)</li>
</ol>
</section>

<section class="box-example"><span id="Problem_3"></span><h2 class="box-legend lt-math-9860"><span class="lt-icon-default">Problem 3</span></h2>

<ol>
    <li class="lt-math-9860">Draw the pictures to compute $1 \div 11$ in a base ten system, and show the answer is $0. \overline{09}$.</li>
    <li class="lt-math-9860">Draw the base 3 pictures to compute $1_{three} \div 11_{three}$, and write the answer as a decimal (&ldquo;trimal&rdquo;?) number.</li>
    <li class="lt-math-9860">Draw the base four pictures to compute $1_{four} \div 11_{four}$, and write the answer as a decimal (&ldquo;quadimal&rdquo;?) number.</li>
    <li class="lt-math-9860">Draw the base six pictures to compute $1_{six} \div 11_{six}$, and write the answer as a decimal (&ldquo;heximal&rdquo;?) number.</li>
    <li class="lt-math-9860">Describe any patterns you notice in the computations above.  Do you have a conjecture of a general rule?  Can you prove your general rule is true?</li>
</ol>
</section>

<section class="box-example"><span id="Problem_4"></span><h2 class="box-legend lt-math-9860"><span class="lt-icon-default">Problem 4</span></h2>

<p class="lt-math-9860">Remember that the fraction $\frac{2}{5}$ represents the division problem $2 \div 5$.  (This is all written in base ten.)</p>

<ol>
    <li class="lt-math-9860">What is the decimal expansion (in base ten) of the fraction $\frac{2}{5}$?</li>
    <li class="lt-math-9860">Rewrite the base-ten fraction $\frac{2}{5}$ as a base four division problem.  Then find the decimal expansion for that fraction in base four.</li>
    <li class="lt-math-9860">Rewrite the base-ten fraction $\frac{2}{5}$ as a base five division problem.  Then find the decimal expansion for that fraction in base five.</li>
    <li class="lt-math-9860">Rewrite the base-ten fraction $\frac{2}{5}$ as a base seven division problem.  Then find the decimal expansion for that fraction in base seven.</li>
    <li class="lt-math-9860">Barry said that in base fifteen, the division problem looks like $$2_{fifteen} \div 5_{fifteen},$$and the decimal representation would be $0.6_{fifteen}$<span style="text-align:justify;">. Check Barry&rsquo;s answer. Is he right?</span></li>
</ol>
</section>

<section class="box-example"><span id="Problem_5"></span><h2 class="box-legend lt-math-9860"><span class="lt-icon-default">Problem 5</span></h2>

<p class="lt-math-9860">Expand each of the following as a &ldquo;decimal&rdquo; number in the base given.  (The fraction is given in base ten.)</p>

<p class="lt-math-9860">\[\begin{split} (a)\; \frac{1}{9}\; \text{in base 10} \quad \qquad &amp;(b)\; \frac{1}{2}\; \text{in base 3} \\ (c)\; \frac{1}{3}\; \text{in base 4} \quad \qquad &amp;(d)\; \frac{1}{4}\; \text{in base 5} \\ (e)\; \frac{1}{5}\; \text{in base 6} \quad \qquad &amp;(f)\; \frac{1}{6}\; \text{in base 7} \\ (g)\; \frac{1}{7}\; \text{in base 8} \quad \qquad &amp;(h)\; \frac{1}{8}\; \text{in base 9} \end{split} \nonumber \]</p>

<p class="lt-math-9860">Do you notice any patterns? Any conjectures?</p>
</section>

<section class="box-example"><span id="Problem_6_(Challenge)"></span><h2 class="box-legend lt-math-9860"><span class="lt-icon-default">Problem 6 (Challenge)</span></h2>

<p class="lt-math-9860">What fraction has decimal expansion $0. \bar{3}_{seven}$?  How do you know you are right?</p>
</section>

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