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190 lines
7.0 KiB
HTML
190 lines
7.0 KiB
HTML
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<!-- Mirrored from www.mathsisfun.com/algebra/reciprocal.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:48:14 GMT -->
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<title>Reciprocal In Algebra</title>
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Reciprocal In Algebra</h1>
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<p class="center">Turn it upside down!</p>
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<h2>Reciprocal of a Number</h2>
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<p>To get the <a href="../reciprocal.html">reciprocal</a> of a <b>number</b>, we divide 1 by the number:</p>
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<p class="center"><img src="../numbers/images/reciprocal.svg" alt="reciprocal of 8 is 1/8" /></p>
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<h3>Examples:</h3>
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<div class="simple">
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<table width="400" border="0" align="center">
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<tr align="center">
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<th>Number</th>
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<th>Reciprocal</th>
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<th>As a Decimal</th>
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</tr>
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<tr align="center">
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<td>2</td>
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<td><sup>1</sup>/<sub>2</sub></td>
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<td>= 0.5</td>
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</tr>
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<tr align="center">
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<td>8</td>
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<td><sup>1</sup>/<sub>8</sub></td>
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<td>= 0.125</td>
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</tr>
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<tr align="center">
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<td>1,000</td>
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<td><sup>1</sup>/<sub>1,000</sub></td>
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<td>= 0.001</td>
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</tr>
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</table>
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</div>
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<h2>Reciprocal of a Variable</h2>
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<p>Likewise, the reciprocal of a variable "x" is "1/x".</p>
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<p>And the reciprocal of something more complicated like "x/y" is "y/x".</p>
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<p class="center">In other words <b>turn it upside down</b>.<br>
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<img src="../images/art/sage-upside-down.gif" width="251" height="114" alt="dog upside down" /></p>
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<div class="example">
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<h3>Example: What is the Reciprocal of x/(x−1) ?</h3>
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<p class="larger">Answer: take <span class="intbl"><em>x</em><strong>(x−1)</strong></span> and flip it upside down: <span class="intbl"><em>(x−1)</em><strong>x</strong></span></p>
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</div>
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<script>reciprocalMain('x','(x−1)');</script>
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<h3>More Examples:</h3>
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<div class="simple">
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<table width="400" border="0" align="center">
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<tr align="center">
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<th>Expression</th>
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<th>Reciprocal</th>
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</tr>
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<tr align="center">
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<td>2x</td>
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<td><span class="intbl"><em>1</em><strong>2x</strong></span></td>
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</tr>
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<tr align="center">
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<td><span class="intbl"><em>3</em><strong>x</strong></span></td>
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<td><span class="intbl"><em>x</em><strong>3</strong></span></td>
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</tr>
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<tr align="center">
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<td><span class="intbl"><em>(2x−3)</em><strong>(x+5)</strong></span></td>
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<td><span class="intbl"><em>(x+5)</em><strong>(2x−3)</strong></span></td>
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</tr>
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</table>
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</div>
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<p> </p>
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<h2>Flipping a Flip</h2>
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<p>When we take the <b>reciprocal of a reciprocal</b> we end up back where we started!</p>
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<p class="center"><img src="images/reciprocal-reciprocal.svg" alt="Reciprocals" /></p>
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<div class="example">
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<h3>Example:</h3>
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<p class="center ">The reciprocal of <span class="intbl large"><em>ax</em><strong>y</strong></span> is <span class="intbl large"><em>y</em><strong>ax</strong></span></p>
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<p class="center">The reciprocal of <span class="intbl large"><em>y</em><strong>ax</strong></span> is <span class="intbl large"><em>ax</em><strong>y</strong></span> (back again)</p>
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</div>
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<div class="example">
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<h3>Example: What is:</h3>
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<p class="center large"><span class="intbl"><em>1</em><strong><sup>1</sup>/<sub>w</sub></strong></span></p>
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<p>Answer: <span class="large">w</span></p>
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<p>Why? because the reciprocal of <sup>1</sup>/<sub>w</sub> is <sup>w</sup>/<sub>1</sub>
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</span> which is just <b>w</b></p>
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<p>Or with numbers: What is <span class="intbl">
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<em>1</em><strong>½</strong>
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</span> ? Divide 1 into halves, and the answer is <b>2</b></p>
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</div>
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<h2>Notation</h2>
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<p>The reciprocal of "x" can be shown as:</p>
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<table style="border: 0; margin:auto;">
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<tr>
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<td width="100" align="center"><p class="center"><span class="intbl"><em>1</em><strong>x</strong></span></p></td>
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<td width="30" align="center"> <i><b>or</b></i> </td>
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<td width="200" align="center"><span class="large">x<sup>−1</sup></span> (see <a href="../exponent.html">exponents</a>)</td>
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</tr>
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</table>
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<p> </p>
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<div class="questions">
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<script>getQ(311, 1621, 308, 1622, 2313, 3153, 2311, 2312, 3151, 3152);</script>
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</div>
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<div class="related">
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<a href="../reciprocal.html">Reciprocal</a>
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<a href="../reciprocal-fraction.html">Reciprocal of a Fraction</a>
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</div>
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Copyright © 2020 MathsIsFun.com
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