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<title>Subtraction by Addition (Complements Method)</title>
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<h1 align="center">Subtraction by Addition</h1>
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<p align="center"><i>(also called the Complements Method)</i></p>
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<p>Here we see how to <b>do subtraction using addition!</b> </p>
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<p><i>(I don't recommend this for normal subtraction work, but it is still a valid and interesting way to subtract. And in some cases it may save time.)</i></p>
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<h2>Steps</h2>
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<p>Follow these steps:</p>
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<ul>
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<li>take the "<b>complement</b>" of the number we are subtracting <i>(we will see how soon)</i></li>
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<li><b>add</b> it to to the number we are subtracting from</li>
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<li><b>discard</b> the extra "1" on the left</li>
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</ul>
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<h2>Complement</h2>
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<p>The "complement" is the number <b>to add</b> to make 10 (or 100, 1000, etc, depending on how many digits we have)</p>
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<div class="example">
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<p>Example The complement of 3 is <b>7</b>, because 3+7=10 (we <b>add 7</b> to make 10)</p>
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</div>
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<div class="example">
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<p>Example: the complement of 85 is <b>15</b>, because 85+15=100</p>
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</div>
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<div class="example">
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<p>Example: the complement of 111 is <b>889</b>, because 111+889=1000</p>
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</div>
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<h2>Calculating the Complement</h2>
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<p>The complement is easy to find! </p>
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<p>The basic idea is to find the <b>difference between each digit and 9</b>. That will get us to "999...", so we only need to add 1 to make it "1000..."</p>
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<p>In practice it is easy to follow this method:</p>
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<ul>
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<li>Start at the the "ones" position</li>
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<li>Skip over any zeros </li>
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<li>Then:</li>
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</ul>
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<table border="0" align="center">
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<tr>
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<td width="260" align="right">For the <b>first digit</b> that isn't zero:</td>
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<td width="25"> </td>
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<td width="250" valign="bottom">find what makes it to <b>10</b></td>
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</tr>
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<tr>
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<td width="260" align="right">For <b>all other digits</b>:</td>
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<td width="25"> </td>
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<td width="250" valign="bottom">find what makes it to <b>9</b></td>
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</tr>
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</table>
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<p>Here are two examples:</p>
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<script type="text/javascript">playerMain("images/complements.txt",600,140);</script>
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<p align="center"><i>(You can check it works by adding the number and its complement,<br />
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for example <b>372+628=1000</b>)</i></p>
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<p>With a bit of experience the "what adds to 10" or "what adds to 9" becomes automatic, and taking the complement becomes quick and easy.</p>
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<p>Here is another example where we have to skip over some zeros:<b></b></p>
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<div class="example">
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<p>Example: What is the complement of <b>1700</b>?</p>
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<ul>
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<li>Skip over the two zeros</li>
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<li>The "10" complement of the 7 is <b>3</b>, </li>
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<li>The "9" complement of 1 is <b>8</b>, </li>
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</ul>
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<p>So the answer is:</p>
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<p align="center" class="large">8300</p>
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<p>(Check: 1700+8300 = <b>10000</b>)</p>
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</div>
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<h2>Now Add Them!</h2>
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<p>Now add the two numbers (using <a href="addition-column.html">column addition</a>), but don't forget to discard the extra "1" on the left.</p>
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<p>Here are the 3 steps (<b>complement</b>, <b>add</b>, <b>discard</b>):</p>
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<script type="text/javascript">playerMain("images/subtract-complement-1.txt",600,220);</script>
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<p>And we have found that 653 − 372 = 281 (check it if you want!)</p>
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<h2>What if the number we are subtracting has fewer digits?</h2>
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<p>How can we, for example, do <span class="large">4567 − 56</span> ?</p>
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<p>After taking the complement we just fill any missing spaces with <span class="large">9</span>s. </p>
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<div class="example">
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<p>Example: <span class="large">4567 − 56</span></p>
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<p>Well, the complement of <span class="large">56</span> is <span class="large">44</span> , but we need to "pad it" out to 4 digits, making it <span class="large">9944</span>. Now we add them:</p>
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<p class="center mono"> 4567<br />
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<u>+9944</u><br />
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<u>14511</u><br /></p>
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<p>Then discard the extra "1" on the left, and the answer is:</p>
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<p align="center" class="large">4511</p>
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</div>
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<p>That case would be easier using <a href="subtraction-quick.html">Quick Subtraction</a>, but it does show how this "<b>complement</b>, <b>add</b>, <b>discard</b>" method works.</p>
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<p>Now, you can practice with these <a href="../worksheets/subtraction.html">Subtraction Worksheets</a>.</p>
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<p> </p>
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