2.06:_Even_Numbers

<p class="lt-math-9833">How do we know if a number is even? What does it mean?</p>

<section class="box-definition"><span id="Definition"></span><h2 class="box-legend lt-math-9833"><span class="lt-icon-default">Definition</span></h2>

<p class="lt-math-9833">Some number of dots is <strong>even</strong> if I can divide the dots into pairs, and every dot has a partner.</p>

<p class="lt-math-9833">Some number of dots is <strong>odd</strong> if, when I try to pair up the dots, I always have a single dot left over with no partner.</p>
</section>

<p class="lt-math-9833">The number of dots is either even or odd. It&rsquo;s a property of the quantity and is doesn&rsquo;t change when you represent that quantity in different bases.</p>

<section class="box-example"><span id="Problem_13"></span><h2 class="box-legend lt-math-9833"><span class="lt-icon-default">Problem 13</span></h2>

<p class="lt-math-9833">Which of these numbers represent an even number of dots? Explain how you decide.</p>

<p class="lt-math-9833">\[22_{ten} \quad 319_{ten} \quad 133_{five} \quad 222_{five} \quad 11_{seven} \quad 11_{four} \nonumber \]</p>
</section>

<section class="box-exercise"><span id="Think_.2F_Pair_.2F_Share"></span><h2 class="box-legend lt-math-9833"><span class="lt-icon-default">Think / Pair / Share</span></h2>

<p class="lt-math-9833">Compare your answers to problem 13 with a partner.  Then try these together:</p>

<ol>
    <li class="lt-math-9833">Count by twos to $20_{ten}$.</li>
    <li class="lt-math-9833">Count by twos to $30_{four}$.</li>
    <li class="lt-math-9833">Count by twos to $51_{seven}$.</li>
</ol>
</section>

<p class="lt-math-9833">You know that you can tell if a base ten number is even just by looking at the ones place. But why is that true?  That&rsquo;s not the definition of an even number. There are a few key ideas behind this handy trick:</p>

<ul>
    <li class="lt-math-9833">In base ten, every number looks like $$\text{(some multiple of ten) + (ones digit)}$$ $$\begin{split} 53 &amp;= 50 + 3 \\ 492 &amp;= 490 + 2 \\ 45637289108 &amp;= 45637289100 + 8 \end{split}$$</li>
    <li class="lt-math-9833">Every multiple of ten is an even number, since $$10n = 2(5n),$$and two times a whole number is always even.</li>
    <li class="lt-math-9833">Your whole number looks like this: $$\text{(some multiple of ten) + (ones digit)}$$ $$\text{(even number) + (ones digit),}$$</li>
    <li class="lt-math-9833">Even plus even is even, and even plus odd is odd, so your whole number is even when the ones digit is even, and it&rsquo;s odd when the ones digit is odd.</li>
</ul>

<section class="box-exercise"><span id="Think_.2F_Pair_.2F_Share_2"></span><h2 class="box-legend lt-math-9833"><span class="lt-icon-default">Think / Pair / Share</span></h2>

<ul>
    <li class="lt-math-9833">Make sure you understand the explanation above.  Does each piece make sense to you?</li>
    <li class="lt-math-9833">In particular: Use the definition of even and odd above to explain the last step.  Why is it true that even + even = even and even + odd = odd?</li>
    <li class="lt-math-9833">What about odd + odd?  Is that odd or even?  Justify what you say.</li>
</ul>
</section>

<section class="box-example"><span id="Problem_14"></span><h2 class="box-legend lt-math-9833"><span class="lt-icon-default">Problem 14</span></h2>

<ol>
    <li class="lt-math-9833">Write the numbers zero through fifteen in base seven:</li>
</ol>

<table class="mt-responsive-table">
    <tbody>
        <tr>
            <th class="mt-align-center">base ten</th>
            <th class="mt-align-center">base seven</th>
        </tr>
        <tr>
            <td class="mt-align-center lt-math-9833" data-th="base ten">0</td>
            <td class="mt-align-center lt-math-9833" data-th="base seven">$0_{seven}$</td>
        </tr>
        <tr>
            <td class="mt-align-center lt-math-9833" data-th="base ten">1</td>
            <td class="mt-align-center lt-math-9833" data-th="base seven">$1_{seven}$</td>
        </tr>
        <tr>
            <td class="mt-align-center lt-math-9833" data-th="base ten">2</td>
            <td class="mt-align-center lt-math-9833" data-th="base seven">&nbsp;</td>
        </tr>
        <tr>
            <td class="mt-align-center lt-math-9833" data-th="base ten">3</td>
            <td class="mt-align-center lt-math-9833" data-th="base seven">&nbsp;</td>
        </tr>
        <tr>
            <td class="mt-align-center lt-math-9833" data-th="base ten">4</td>
            <td class="mt-align-center lt-math-9833" data-th="base seven">&nbsp;</td>
        </tr>
        <tr>
            <td class="mt-align-center lt-math-9833" data-th="base ten">5</td>
            <td class="mt-align-center lt-math-9833" data-th="base seven">&nbsp;</td>
        </tr>
        <tr>
            <td class="mt-align-center lt-math-9833" data-th="base ten">6</td>
            <td class="mt-align-center lt-math-9833" data-th="base seven">&nbsp;</td>
        </tr>
        <tr>
            <td class="mt-align-center lt-math-9833" data-th="base ten">7</td>
            <td class="mt-align-center lt-math-9833" data-th="base seven">&nbsp;</td>
        </tr>
        <tr>
            <td class="mt-align-center lt-math-9833" data-th="base ten">8</td>
            <td class="mt-align-center lt-math-9833" data-th="base seven">&nbsp;</td>
        </tr>
        <tr>
            <td class="mt-align-center lt-math-9833" data-th="base ten">9</td>
            <td class="mt-align-center lt-math-9833" data-th="base seven">&nbsp;</td>
        </tr>
        <tr>
            <td class="mt-align-center lt-math-9833" data-th="base ten">10</td>
            <td class="mt-align-center lt-math-9833" data-th="base seven">&nbsp;</td>
        </tr>
        <tr>
            <td class="mt-align-center lt-math-9833" data-th="base ten">11</td>
            <td class="mt-align-center lt-math-9833" data-th="base seven">&nbsp;</td>
        </tr>
        <tr>
            <td class="mt-align-center lt-math-9833" data-th="base ten">12</td>
            <td class="mt-align-center lt-math-9833" data-th="base seven">&nbsp;</td>
        </tr>
        <tr>
            <td class="mt-align-center lt-math-9833" data-th="base ten">13</td>
            <td class="mt-align-center lt-math-9833" data-th="base seven">&nbsp;</td>
        </tr>
        <tr>
            <td class="mt-align-center lt-math-9833" data-th="base ten">14</td>
            <td class="mt-align-center lt-math-9833" data-th="base seven">&nbsp;</td>
        </tr>
        <tr>
            <td class="mt-align-center lt-math-9833" data-th="base ten">15</td>
            <td class="mt-align-center lt-math-9833" data-th="base seven">&nbsp;</td>
        </tr>
    </tbody>
</table>

<ol start="2">
    <li class="lt-math-9833">Circle all of the even numbers in your list.  How do you know they are even?</li>
    <li class="lt-math-9833">Find a rule: how can you tell if a number is even when it&rsquo;s written in base seven?</li>
</ol>
</section>

<section class="box-example"><span id="Problem_15"></span><h2 class="box-legend lt-math-9833"><span class="lt-icon-default">Problem 15</span></h2>

<ol>
    <li class="lt-math-9833">Write the numbers zero through fifteen in base four:</li>
</ol>

<section class="box-exercise"><span id="Think_.2F_Pair_.2F_Share_3"></span><h2 class="box-legend lt-math-9833"><span class="lt-icon-default">Think / Pair / Share</span></h2>

<ul>
    <li class="lt-math-9833">Why are the rules for recognizing even numbers different in different bases?</li>
    <li class="lt-math-9833">For either your base four rule or your base seven rule, can you explain <i>why</i> it works that way?</li>
</ul>
</section>

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