4.10:_Fractions_involving_zero

<span id="Zero_in_the_Numerator"></span><h2 class="lt-math-10587">Zero in the Numerator</h2>

<section class="box-exercise"><span id="Think_.2F_Pair_Share"></span><h5 class="box-legend lt-math-10587"><span class="lt-icon-default">Think / Pair Share</span></h5>

<p class="lt-math-10587">Does the fraction \(\frac{0}{11}\) make sense?</p>

<ul>
    <li class="lt-math-10587">Write a &ldquo;pies per child&rdquo; story for the fraction \(\frac{0}{11}\). Does it make sense? How much pie does each individual child receive in your story?</li>
    <li class="lt-math-10587">Think of \(\frac{0}{11}\) as the answer to a division problem. What is that division problem?  Can you solve it?</li>
</ul>
</section>

<p class="lt-math-10587">It seems pretty clear that zero pies among eleven kids gives zero pies per child:</p>

<p class="lt-math-10587">\[\frac{0}{11} = 0 \ldotp \nonumber \]</p>

<p class="lt-math-10587">The same reasoning would lead us to say:</p>

<p class="lt-math-10587">\[\frac{0}{b} = 0\; \text{for any positive number}\; b \ldotp \nonumber \]</p>

<p class="lt-math-10587">The &ldquo;Pies Per Child Model&rdquo; offers one explanation: If there are no pies for us to share, no one gets any pie. It does not matter how many children there are. No pie is no pie is no pie.</p>

<p class="lt-math-10587">We can also justify this claim by thinking about a missing factor multiplication problem:</p>

<p class="lt-math-10587">\[\frac{0}{b}\; \text{is asking us to fill in the blank} :\; \_\_ \cdot \; b = 0 \ldotp \nonumber \]</p>

<p class="lt-math-10587">The only way to fill that in and make a true statement is with 0,  so  \(\frac{0}{b} = 0\).</p>

<span id="Zero_in_the_Denominator"></span><h2 class="lt-math-10587">Zero in the Denominator</h2>

<p class="lt-math-10587">What happens if things are flipped the other way round?</p>

<section class="box-exercise"><span id="Think_.2F_Pair_.2F_Share"></span><h5 class="box-legend lt-math-10587"><span class="lt-icon-default">Think / Pair / Share</span></h5>

<p class="lt-math-10587">Does the fraction \(\frac{11}{0}\) make sense?</p>

<ul>
    <li class="lt-math-10587">Write a &ldquo;pies per child&rdquo; story for the fraction \(\frac{11}{0}\). Does it make sense? How much pie does each individual child receive in your story?</li>
    <li class="lt-math-10587">Think of \(\frac{11}{0}\) as the answer to a division problem. What is that division problem?  Can you solve it?</li>
</ul>
</section>

<p class="lt-math-10587">Students often learn in school that &ldquo;dividing by 0 is undefined.&rdquo; But they learn this as a rule, rather than thinking about why it makes sense or how it connects to other ideas in mathematics. In this case, the most natural connection is to a multiplication fact, the zero property for multiplication:</p>

<p class="lt-math-10587">\[\text{any number} \cdot 0 = 0 \ldotp \nonumber \]</p>

<p class="lt-math-10587">That says we can never find solutions to problems like</p>

<p class="lt-math-10587">\[\_\_ \cdot 0 = 5, \qquad \_\_ \cdot 0 = 17, \qquad \_\_ \cdot 0 = 1 \ldotp \nonumber \]</p>

<p class="lt-math-10587">Using the connection between fractions and division, and the connection between division and multiplication, that means there is no number \(\frac{5}{0}\). There is no number  \(\frac{17}{0}\). And there is no number \(\frac{1}{0}\). They are all &ldquo;undefined&rdquo; because they are not equal to any number at all.</p>

<section class="box-exercise"><span id="Think_.2F_Pair_.2F_Share_2"></span><h5 class="box-legend lt-math-10587"><span class="lt-icon-default">Think / Pair / Share</span></h5>

<p class="lt-math-10587">Can we give meaning to \(\frac{0}{0}\) at least? After all, a zero would appear on both sides of that equation!</p>

<ul>
    <li class="lt-math-10587">Cyril says that \(\frac{0}{0} = 2\) since \(0 \cdot 2 = 0\).</li>
    <li class="lt-math-10587">Ethel says that \(\frac{0}{0} = 17\) since  \(0 \cdot 17 = 0\).</li>
    <li class="lt-math-10587">Wonhi says that \(\frac{0}{0} = 887231243\) since  \(0 \cdot 887231243 = 0\).</li>
</ul>

<p class="lt-math-10587">Who is right? Can they all be correct? What do you think?</p>
</section>

<p class="lt-math-10587">Cyril says that \(\frac{0}{0} = 2\), and he believes he is correct because it passes the check: \(0 \cdot 2 = 0\).</p>

<p class="lt-math-10587">But 17 also passes the check, and so does 887231243.  In fact, I can choose any number for x, and \(0 \cdot x = 0\) will pass the check!</p>

<p class="lt-math-10587">The trouble with the expression \(\frac{a}{0}\) (with <img alt="a" height="8" title="Rendered by QuickLaTeX.com" width="9" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/16537/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.svg?revision=1" /> not zero) is that there is <i>no meaningful value</i> to assign to it. The trouble with \(\frac{0}{0}\) is different:  There are <i>too many possible values</i> to give it!</p>

<p class="lt-math-10587">Dividing by zero is simply too problematic to be done! It is best to avoid doing so and never will we allow zero as the denominator of a fraction. (But all is fine with 0 as a numerator.)</p>

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