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146 lines
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<h1 align="center">Dividing by Zero</h1>
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<table border="0" align="center">
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<tr>
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<td align="center"><p>Don't divide by <a href="zero.html">zero</a> or this could happen!</p>
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<p> </p>
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<p>Just kidding.</p></td>
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<td> </td>
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<td><img src="images/divide-by-zero.jpg" alt="Divide by Zero" width="300" height="159" /></td>
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<td><p> </p></td>
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</tr>
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</table>
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<p>In truth:</p>
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<p align="center" class="larger">Dividing by Zero is <b>undefined</b>.</p>
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<h2>Dividing</h2>
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<p>To see why, let us look at what is meant by "division":</p>
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<div class="def">
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<p>Division is splitting into equal parts or groups. </p>
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<p>It is the result of "fair sharing".</p>
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</div>
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<br />
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<div class="example">
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<p>Example: there are 12 chocolates, and 3 friends want to share them, how do they divide the chocolates?</p>
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<div class="simple">
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<table border="0" align="center">
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<tr align="center">
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<td><img src="images/chocolates-12.svg" alt="12 chocolates" style="max-width:100%" /></td>
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<td> </td>
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<td><img src="images/chocolates-12-div-3.svg" alt="12 chocolates divided into 3" style="max-width:100%" /></td>
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</tr>
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<tr align="center">
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<th>12 Chocolates</th>
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<th> </th>
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<th>12 Chocolates Divided by 3</th>
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</tr>
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</table>
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</div>
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<p align="center">So they get 4 each: 12/3 = 4</p>
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<h2>Dividing by Zero</h2>
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<p>Now, let us try dividing the 12 chocolates among <b>zero</b> people, how much does each person get?</p>
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<p><b>Does that question even make sense?</b> No, of course it doesn't.</p>
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<p>We can't share among zero people, and we can't divide by 0.</p>
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<h2>Another Good Reason</h2>
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<p>After dividing, can we multiply to get back again?</p>
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<p align="center"><img src="images/divide-by-zero-12.svg" alt="multiply by zero" /></p>
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<p>But multiplying by 0 gives 0, so that won't work.</p>
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<p>Once again, dividing by zero gives us difficulties!</p>
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<h2>Imagine We Could Divide by Zero</h2>
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<p>Okay, let us <b>imagine </b>we<b> can</b> divide by zero, and see what happens.</p>
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<p>That means that things like <span class="frac-large"><sup>1</sup>/<sub>0</sub></span> and <span class="frac-large"><sup>0</sup>/<sub>0</sub></span> would behave like normal numbers.</p>
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<h3>Try Multiplying By Zero</h3>
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<p>So let us try using our new "numbers".</p>
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<p>For example, we know that zero times any number is zero:</p>
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<div class="example">
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<p> Example: 0×1 = 0, 0×2 = 0, etc</p>
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</div>
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<p>So that should also be true for <span class="frac-large"><sup>1</sup>/<sub>0</sub></span>:</p>
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<p align="center" class="large">0 × (<sup>1</sup>/<sub>0</sub>) = <b>0</b></p>
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<p>But we could also rearrange it a little like this:</p>
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<p align="center" class="large">0 × (<span class="frac"><sup>1</sup>/<sub>0</sub></span>) = (<span class="frac"><sup>0</sup>/<sub>0</sub></span>) × 1 = <b>1</b></p>
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<p>(Careful! I am <b>not</b> saying this is correct! We are <b>assuming</b> that we <b>can</b> divide by zero, so 0/0 should work the same as 5/5, which is 1).</p>
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<p>Arrggh! If we multiply <span class="frac-large"><sup>1</sup>/<sub>0</sub></span> by zero we could get <span class="large">0 <i>or</i> 1</span>. </p>
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<p>In fact we can't have both possibilites, so we <b>cannot</b> define <span class="frac-large"><sup>1</sup>/<sub>0</sub></span> to be a number.</p>
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<p>So it is <b>undefined</b>.</p>
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<h2>So what is 0/0 ?</h2>
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<p> <span class="larger">0/0</span> is like asking "how many 0s in 0?"</p>
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<p> Are there no zeros in zero at all? Or perhaps there is exactly one zero in zero? Or many zeros?</p>
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<p>So 0/0 is <b>indeterminate</b> (it could be any value).</p>
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<h2>In conclusion:</h2>
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<p align="center" class="large">When we try to divide by zero, things stop making sense</p>
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<p>That is all.</p>
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<h2>But Wait ...</h2>
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<p>There is a special method where we get <b>closer</b> and <b>closer</b> to zero ... just read <a href="../calculus/limits.html">Limits (An Introduction)</a> to find out more.</p>
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<p> </p>
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<div class="related"> <a href="division.html">Division</a> <a href="index.html">Numbers Index</a></div>
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