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584 lines
20 KiB
HTML
584 lines
20 KiB
HTML
<!doctype html>
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<head>
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<title>Adding and Subtracting Positive and Negative Numbers</title>
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<div id="hdr"></div>
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<div id="cookOK"></div>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">How to Add and Subtract<br>
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Positive and Negative Numbers</h1>
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<h2>Numbers Can be Positive or Negative</h2>
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<p>This is the <a href="number-line.html">Number Line</a>:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td colspan="2" align="center"><img src="numbers/images/number-line.svg" alt="number line -10 to +10" height="47" width="595" ></td>
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</tr>
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<tr>
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<td style="text-align:center;">Negative Numbers (−)</td>
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<td style="text-align:center;">Positive Numbers (+)</td>
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</tr>
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</tbody></table><br>
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<div class="simple">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:center;">
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<td style="width:200px;"><img src="numbers/images/minus.svg" alt="minus" height="47" width="47" ></td>
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<td style="width:200px;"><img src="numbers/images/plus.svg" alt="plus" height="47" width="47" ></td>
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</tr>
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<tr style="text-align:center;">
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<td style="width:200px;">"−" is the negative sign.</td>
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<td style="width:200px;">"+" is the positive sign</td>
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</tr>
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</tbody></table>
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<h2>No Sign Means Positive</h2>
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</div>
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<p>If a number has <b>no sign</b> it usually means that it is a <b>positive</b> number.</p>
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<div class="example">
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<p>Example: <b>5</b> is really <b>+5</b></p>
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</div>
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<h2>Play with it!</h2>
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<p>On the Number Line positive goes to the right and negative to the left.</p>
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<p>Try the sliders below and see what happens:</p>
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<div class="script" style="height: 230px;">
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numbers/images/number-line-add.js?sub=n
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</div>
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<h2>Balloons and Weights</h2>
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<p>Let us think about numbers as balloons (positive) and weights (negative):</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="numbers/images/add-subtract-gen.svg" alt="add subtract general" height="170" width="151" ></p>
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<p>This basket has balloons and weights tied to it:</p>
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<div>
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<ul style="overflow:auto;">
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<li>The balloons pull up (<b>positive</b>)</li>
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<li>And the weights drag down (<b>negative</b>)</li>
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</ul>
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</div>
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<div style="clear:both">
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</div>
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<h2>Adding a Positive Number</h2>
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<p>Adding positive numbers is just simple addition.</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="numbers/images/add-subtract-pp.svg" alt="add subtract ++ goes up" height="179" width="130" ></p>
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<p> </p>
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<p>We can add balloons (we are <b>adding positive</b> value)</p>
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<p class="so">the basket gets pulled upwards (positive)</p>
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<div class="example">
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<h3>Example: 2 + 3 = 5</h3>
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<p>is really saying</p>
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<p class="center">"Positive 2 plus Positive 3 equals Positive 5"</p>
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<p>We could write it as <b>(+2) + (+3) = (+5)</b></p>
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</div>
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<h2>Subtracting A Positive Number</h2>
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<p>Subtracting positive numbers is just simple subtraction.</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="numbers/images/add-subtract-np.svg" alt="add subtract -+ goes down" height="164" width="134" ></p>
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<p><br></p>
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<p>We can take away balloons (we are <b>subtracting positive</b> value)</p>
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<p class="so">the basket gets pulled downwards (negative)</p>
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<div style="clear:both">
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</div>
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<div class="example">
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<h3>Example: 6 − 3 = 3</h3>
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<p>is really saying</p>
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<p class="center">"Positive 6 minus Positive 3 equals Positive 3"</p>
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<p>We could write it as <b>(+6) − (+3) = (+3)</b></p>
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</div>
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<h2>Adding A Negative Number</h2>
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<p>Now let's see what adding and subtracting <b>negative</b> numbers looks like:</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="numbers/images/add-subtract-pn.svg" alt="add subtract +- goes down" height="180" width="130" ></p>
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<p><br></p>
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<p>We can add weights (we are <b>adding negative</b> values)</p>
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<p class="so">the basket gets pulled downwards (negative)</p>
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<div class="example">
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<h3>Example: 6 + (−3) = 3</h3>
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<p>is really saying</p>
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<p class="center">"Positive 6 plus Negative 3 equals Positive 3"</p>
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<p class="center"> </p>
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<p>We could write it as <b>(+6) + (−3) = (+3)</b></p>
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</div>
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<p>The last two examples showed us that taking away balloons (subtracting a positive) or adding weights (adding a negative) both make the basket go down.</p>
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<p>So these have the <b>same result</b>:</p>
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<ul>
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<li><b>(+6) − (+3) = (+3)</b></li>
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<li><b>(+6) + (−3) = (+3)</b></li>
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</ul>
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<p>In other words <b>subtracting a positive</b> is the same as <b>adding a negative</b>.</p>
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<h2>Subtracting A Negative Number</h2>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="numbers/images/add-subtract-nn.svg" alt="add subtract -- goes up" height="171" width="129" ></p>
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<p><br></p>
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<p>Lastly, we can take away weights (we are <b>subtracting negative</b> values)</p>
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<p class="so">the basket gets pulled upwards (positive)</p>
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<div class="example">
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<h3>Example: What is 6 − (−3) ?</h3>
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<p class="center large">6−(−3) = 6 <b>+</b> 3 = 9</p>
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</div>
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<p>Yes indeed! Subtracting a Negative is the same as adding!</p>
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<p class="center large">Two Negatives make a Positive</p>
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<h2>What Did We Find?</h2>
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<h3>Adding a positive number is simple addition ...</h3>
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<div class="flexx">
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<div><img src="numbers/images/plus-plus-plus.svg" alt="plus plus gives plus" height="47" width="203" ></div>
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<div style="text-align: center; width:20rem;">Adding a Positive is <b>Addition</b></div>
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</div>
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<p> </p>
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<h3>Positive and Negative Together ...</h3>
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<div class="flexx">
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<div><img src="numbers/images/plus-minus-minus.svg" alt="plus minus gives minus" height="103" width="203" ></div>
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<div style="text-align: center; width:20rem;">Subtracting a Positive<br>
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<i>or<br>
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</i> Adding a Negative<br>
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<i>is</i><br>
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<b>Subtraction</b></div>
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</div>
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||
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<div class="example">
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||
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<h3>Example: What is 6 − (+3) ?</h3>
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<p class="center large">6−(+3) = 6 <b>−</b> 3 = 3</p>
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</div>
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<div class="example">
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||
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<h3>Example: What is 5 + (−7) ?</h3>
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<p class="center large">5+(−7) = 5 <span class="large"><b>−</b></span> 7 = −2</p>
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</div>
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<p> </p>
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<h3>Subtracting a negative ...</h3>
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<div class="flexx">
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<div><img src="numbers/images/minus-minus-plus.svg" alt="minus minus gives plus" height="47" width="203" ></div>
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<div style="text-align: center; width:20rem;">Subtracting a Negative is the same as <b>Adding</b></div>
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</div>
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<div class="example">
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||
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<h3>Example: What is 14 − (−4) ?</h3>
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<p class="center large">14−(−4) = 14<span class="large"> + </span>4 = 18</p>
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</div>
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<h2>The Rules:</h2>
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||
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<p>It can all be put into <b>two rules</b>:</p>
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||
<table style="border: 0; margin:auto;">
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||
<tbody>
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||
<tr>
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||
<td style="text-align:center; width:100px;"> </td>
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||
<th>Rule</th>
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||
<td style="width:10px;"> </td>
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||
<td style="width:40px;"> </td>
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||
<th width="20" nowrap="nowrap"> </th>
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<th nowrap="nowrap">Example</th>
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||
</tr>
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||
<tr>
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<th class="large" align="center" width="100">+(+) </th>
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<td rowspan="2">Two like signs become a <b>positive sign</b></td>
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||
<td rowspan="2"> </td>
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||
<td rowspan="2" width="40"><img src="numbers/images/plus.svg" alt="plus" height="47" width="47" ></td>
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||
<td align="center" nowrap="nowrap"> </td>
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<td align="center" nowrap="nowrap">3+(+2) = 3<b> +</b> 2 = 5</td>
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||
</tr>
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<tr>
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<th class="large" align="center" width="100">−(−)</th>
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<td align="center" nowrap="nowrap"> </td>
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<td align="center" nowrap="nowrap">6−(−3) = 6<b> +</b> 3 = 9</td>
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||
</tr>
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||
<tr>
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<td style="text-align:center; width:100px;"> </td>
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||
<td> </td>
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||
<td> </td>
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||
<td style="width:40px;"> </td>
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||
<td align="center" nowrap="nowrap"> </td>
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||
<td align="center" nowrap="nowrap"> </td>
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||
</tr>
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||
<tr>
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<th align="center" width="100"><span class="large">+(−) </span></th>
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<td rowspan="2">Two unlike signs become a <b>negative sign</b></td>
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<td rowspan="2"> </td>
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<td rowspan="2" width="40"><img src="numbers/images/minus.svg" alt="minus" height="47" width="47" ></td>
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<td align="center" nowrap="nowrap"> </td>
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||
<td align="center" nowrap="nowrap">7+(−2) = 7 <b>−</b> 2 = 5</td>
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||
</tr>
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||
<tr>
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<th class="large" align="center" width="100">−(+)</th>
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<td align="center" nowrap="nowrap"> </td>
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||
<td align="center" nowrap="nowrap">8−(+2) = 8 <b>−</b> 2 = 6</td>
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||
</tr>
|
||
<tr>
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||
<td style="text-align:center; width:100px;"> </td>
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||
<td> </td>
|
||
<td> </td>
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||
<td style="width:40px;"> </td>
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||
<td nowrap="nowrap"> </td>
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||
<td nowrap="nowrap"> </td>
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||
</tr>
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||
</tbody></table>
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<p class="center">They are "like signs" when they are like each other (in other words: the same).</p>
|
||
<p class="center"> </p>
|
||
<p>So, all you have to remember is:</p>
|
||
<div class="center80">
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<p class="larger">Two <b>like</b> signs become a <b>positive sign</b></p>
|
||
<p class="larger">Two <b>unlike</b> signs become a <b>negative sign</b></p>
|
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</div>
|
||
<p><br></p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: What is 5+(−2) ?</h3>
|
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<p>+(−) are <b>unlike</b> signs (they are not the same), so they become a <b>negative sign</b>.</p>
|
||
<p class="center large">5+(−2) = 5 <span class="large"><b>−</b></span> 2 = 3</p>
|
||
</div>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: What is 25−(−4) ?</h3>
|
||
<p>−(−) are <b>like</b> signs, so they become a <b>positive sign</b>.</p>
|
||
<p class="center large">25−(−4) = 25+4 = 29</p>
|
||
</div>
|
||
|
||
|
||
<h2>Starting Negative</h2>
|
||
|
||
<p>What if we <b>start</b> with a negative number?</p>
|
||
<p><a href="numbers/number-line-using.html">Using The Number Line</a> can help:</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: What is −3+(+2) ?</h3>
|
||
<p>+(+) are <b>like</b> signs, so they become a <b>positive sign</b>.</p>
|
||
<p class="center large">−3+(+2) = −3 <span class="large"><b>+</b></span> 2</p>
|
||
<p class="center"><img src="numbers/images/number-line-m3p2.gif" alt="number line -3 + 2 = -1" height="99" width="172" ><br>
|
||
Start at −3 on the number line,<br>
|
||
move forward 2 and you end up at −1</p>
|
||
<p class="center large">−3+(+2) = −3 <span class="large"><b>+</b></span> 2 = −1</p>
|
||
</div>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: What is −3+(−2) ?</h3>
|
||
<p>+(−) are <b>unlike</b> signs, so they become a <b>negative sign</b>.</p>
|
||
<p class="center large">−3+(−2) = −3 <span class="large"><b>−</b></span> 2</p>
|
||
<p class="center"><img src="numbers/images/number-line-m3m2.gif" alt="number line -3 - 2 = -5" height="99" width="192" ><br>
|
||
Start at −3 on the number line,<br>
|
||
move back 2 and you end up at −5</p>
|
||
<p class="center large">−3+(−2) = −3 <span class="large"><b>−</b></span> 2 = −5</p>
|
||
</div>
|
||
|
||
|
||
<h2>Now Play With It!</h2>
|
||
|
||
<table align="center" width="80%" border="0">
|
||
<tbody>
|
||
<tr>
|
||
<td><img src="images/pat-athlete.gif" alt="pat athlete" height="86" width="51" ></td>
|
||
<td> </td>
|
||
<td>Try playing <a href="numbers/casey-runner.html">Casey Runner</a>, you need to know the rules of positive and negative to succeed!</td>
|
||
</tr>
|
||
</tbody></table>
|
||
|
||
|
||
<h2>A Common Sense Explanation</h2>
|
||
|
||
<p>And there is a "common sense" explanation:</p>
|
||
<div class="fun"><img src="images/art/sage-brocolli.gif" alt="dog brocolli" style="float:left; margin: 10px;" height="127" width="250" >
|
||
<p>If I say "Eat!" I am encouraging you to eat (positive)</p>
|
||
<p>If I say "Do not eat!" I am saying the opposite (negative).</p>
|
||
<p>Now if I say "Do <b>NOT</b> not eat!", I am saying I don't
|
||
want you to starve, so I am back to saying "Eat!" (positive).</p>
|
||
</div>
|
||
<p>So, two negatives make a positive, and if that satisfies you, then
|
||
you are done!</p>
|
||
<p> </p>
|
||
|
||
|
||
<h2>Another Common Sense Explanation</h2>
|
||
|
||
<p>A friend is <span class="largest">+</span>, an enemy is <span class="largest">−</span></p>
|
||
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td class="largest">+ + ⇒ +</td>
|
||
<td style="width:30px;"> </td>
|
||
<td>a friend of a friend is my friend </td>
|
||
</tr>
|
||
<tr>
|
||
<td class="largest">+ − ⇒ −</td>
|
||
<td> </td>
|
||
<td>a friend of an enemy is my enemy</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="largest">− + ⇒ −</td>
|
||
<td> </td>
|
||
<td>an enemy of a friend is my enemy</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="largest">− − ⇒ +</td>
|
||
<td> </td>
|
||
<td>an enemy of an enemy is my friend</td>
|
||
</tr>
|
||
</tbody></table>
|
||
|
||
|
||
<h2>A Bank Example</h2>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: Last year the bank subtracted $10 from your account by mistake, and they want to fix it.</h3>
|
||
<p>So the bank must <b>take away a negative $10</b>.</p>
|
||
<p>Let's say your current balance is $80, so you will then have:</p>
|
||
<p class="center large">$80−(−$10) = $80<b> + $10</b> = $90</p>
|
||
<p>So you get <b>$10 more</b> in your account.</p>
|
||
</div>
|
||
|
||
|
||
<h2>A Long Example You Might Like</h2>
|
||
|
||
<h3>Ally's Points</h3>
|
||
<p style="float:left; margin: 0 30px 5px 0;"><i><img src="images/alexa.jpg" alt="alexa" height="128" width="118" ></i></p>
|
||
<p>Ally can be naughty or nice. So Ally's parents have said</p>
|
||
<p class="indent50px"><i>"If you are nice we will add 3 points (+3).<br>
|
||
If you are naughty, we take away 3 points (−3).<br>
|
||
When you reach 30 Points you get a toy."</i></p>
|
||
<p> </p>
|
||
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td style="text-align:right;">Ally starts the day with 9 Points:</td>
|
||
<td align="center" nowrap="nowrap"> </td>
|
||
<td align="center" nowrap="nowrap"><b>9</b></td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:right;">Ally's Mom discovers spilt milk:</td>
|
||
<td align="center" nowrap="nowrap"> </td>
|
||
<td align="center" nowrap="nowrap"><b>9 − 3 = 6</b></td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:center;">
|
||
<p>Then Dad confesses he spilt the milk and writes "undo".</p>
|
||
<p>How do we "undo" a minus 3?<br>
|
||
We <b>add</b> 3 back again!</p></td>
|
||
<td align="center" nowrap="nowrap"> </td>
|
||
<td align="center" nowrap="nowrap"> </td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:right;">So Mom calculates:</td>
|
||
<td align="center" nowrap="nowrap"> </td>
|
||
<td align="center" nowrap="nowrap"><b>6 − (−3) = 6 + 3 = 9</b></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p class="center"><img src="images/allys-points.gif" alt="alexas points" height="99" width="177" ></p>
|
||
<p class="center large"><span class="larger">So when we subtract a negative, we gain points<br>
|
||
(i.e. the same as adding points).</span></p><br>
|
||
<div class="flexa">
|
||
<div class="flex1"><img src="numbers/images/minus-minus-plus.svg" alt="minus minus gives plus" height="47" width="203" ></div>
|
||
<div class="flex2 larger">So Subtracting a Negative is the same as <b>Adding</b></div>
|
||
</div>
|
||
<p class="center large"> </p>
|
||
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td style="text-align:right;">A few days later. Ally has 12 points.</td>
|
||
<td align="center" nowrap="nowrap"> </td>
|
||
<td align="center" nowrap="nowrap"> </td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:right;"><br>
|
||
</td>
|
||
<td nowrap="nowrap"><br>
|
||
</td>
|
||
<td nowrap="nowrap"><br>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:right;">Mom adds 3 points because Ally's room is clean.</td>
|
||
<td align="center" nowrap="nowrap"> </td>
|
||
<td align="center" nowrap="nowrap"><b>12 + 3 = 15</b></td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:right;"><br>
|
||
</td>
|
||
<td nowrap="nowrap"><br>
|
||
</td>
|
||
<td nowrap="nowrap"><br>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:right;">Dad says "I cleaned that room" and writes "undo" on the chart. Mom calculates:</td>
|
||
<td align="center" nowrap="nowrap"> </td>
|
||
<td align="center" nowrap="nowrap"><b>15 − (+3) = 12</b></td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:right;"><br>
|
||
</td>
|
||
<td nowrap="nowrap"><br>
|
||
</td>
|
||
<td nowrap="nowrap"><br>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:right;"> Dad sees Ally brushing the dog. Writes "+3" on the chart. Mom calculates:</td>
|
||
<td align="center" nowrap="nowrap"> </td>
|
||
<td align="center" nowrap="nowrap"><b>12 + (+3) = 15</b></td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:right;"><br>
|
||
</td>
|
||
<td nowrap="nowrap"><br>
|
||
</td>
|
||
<td nowrap="nowrap"><br>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:right;"> Ally throws a stone against the window. Dad writes "−3" on the chart. Mom calculates:</td>
|
||
<td align="center" nowrap="nowrap"> </td>
|
||
<td align="center" nowrap="nowrap"><b>15 + (−3) = 12</b></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p class="center"><img src="images/allys-points2.gif" alt="alexas points2" height="192" width="169" ></p><br>
|
||
<p class="center large">See: both "<b>15 − (+3)</b>" and "<b>15 + (−3)</b>" result in 12.</p>
|
||
<p>So:</p>
|
||
<p class="center large">It doesn't matter if you subtract positive points<br>
|
||
or add negative points,<br>
|
||
you still end up losing points.</p>
|
||
<div class="flexa">
|
||
<div class="flex1"><img src="numbers/images/plus-minus-minus.svg" alt="plus minus gives minus" height="103" width="203" ></div>
|
||
<div class="flex2 larger"><span class="larger">So Subtracting a Positive<br>
|
||
<i>or<br>
|
||
</i> Adding a Negative</span><br>
|
||
<span class="larger"><i>is</i></span><br>
|
||
<span class="larger"><b>Subtraction</b></span></div>
|
||
</div>
|
||
<p> </p>
|
||
|
||
|
||
<h2>Try These Exercises ...</h2>
|
||
|
||
<p class="center">Now try <a href="worksheets/ws-opsb74f.html?ops=as&nops=1&n=12&ansType=step&amin=-12&amax=12&seed=9424">This Worksheet</a>, and see how you go.</p>
|
||
<p class="center">And also try these questions:</p>
|
||
<div class="questions">11715, 11716, 11717, 11718, 11719, 11720, 11721, 3445, 3446</div>
|
||
|
||
<div class="related">
|
||
<a href="number-line.html">Number Line</a>
|
||
<a href="numbers/number-line-using.html">Using The Number Line</a>
|
||
<a href="whole-numbers.html">Whole Numbers and Integers</a>
|
||
<a href="multiplying-negatives.html">Multiplying Negatives</a>
|
||
<a href="numbers/index.html">Number Index</a>
|
||
<a href="algebra/index.html">Algebra Index</a>
|
||
</div>
|
||
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