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<!-- #BeginEditable "Body" -->
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<h1 class="center">Inverse</h1>
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<p class="center">Inverse means the <b>opposite in effect</b>. The reverse of.</p>
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<p>It is a <b>general idea</b> in mathematics and has many meanings. Here are a few.</p>
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<h2>The Inverse of Adding is Subtracting</h2>
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<p>Adding moves us one way, subtracting moves us the opposite way.</p>
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<div class="example">
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<p>Example: <b>20 <span class="hilite">+ 9</span> = 29</b> can be reversed by<b> 29 <span class="hilite">− 9</span> = 20</b> (back to where we started)</p>
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</div>
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<p>And the other way around:</p>
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<div class="example">
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<p>Example: <b>15 <span class="hilite">− 3</span> = 12</b> can be reversed by<b> 12 <span class="hilite">+ 3</span> = 15</b> (back to where we started)</p>
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</div>
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<h4>Additive Inverse</h4>
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<p>The <b>additive inverse</b> is what we add to a number to <b>get zero</b>.</p>
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<div class="example">
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<h3>Example: The additive inverse of −5 is +5, because −5 + 5 = 0.</h3>
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<p>Another example: the additive inverse of +7 is −7.</p>
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</div>
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<h2>The Inverse of Multiplying is Dividing</h2>
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<p>Multiplying can be "undone" by dividing.</p>
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<div class="example">
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<p>Example: <b>5 <span class="hilite">× 9</span> = 45</b> can be reversed by <b>45 <span class="hilite">/ 9</span> = 5</b></p>
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</div>
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<p>It works the other way around too, dividing can be undone by multiplying.</p>
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<div class="example">
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<p>Example: <b>10 <span class="hilite">/ 2</span> = 5 </b> can be reversed by <b>5 <span class="hilite">× 2</span> = 10</b></p>
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</div>
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<h4>Multiplicative Inverse</h4>
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<p>The <b>multiplicative inverse</b> is what we multiply a number by to get 1.</p>
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<p>It is the <a href="../reciprocal.html">reciprocal</a> of a number.</p>
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<div class="example">
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<div>
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<h3>Example: The multiplicative inverse of 5 is <span class="intbl"><em>1</em><strong>5</strong></span>, because 5 × <span class="intbl"><em>1</em><strong>5</strong></span> = 1</h3>
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</div>
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</div>
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<h4>But Not With 0</h4>
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<p>We <a href="dividing-by-zero.html">can't divide by 0</a>, so don't try!</p>
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<div class="example">
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<h3>Example: <b>5 × 0 = 0</b> cannot be reversed by <b>0/0 = ???</b></h3>
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</div>
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<h2>Inverse of a Function</h2>
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<p>Doing a function and then its inverse will give us back the original value:</p>
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<p class="center"><img src="../sets/images/function-apple-banana.svg" alt="banana f() to apple f()-inverse back to banana"><br>
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When the function <span class="large">f</span> turns the apple into a banana,<br>
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Then the <b>inverse</b> function <span class="large">f<sup>-1</sup></span> turns the banana back to the apple</p>
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<p class="center"> </p>
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<div class="example">
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<p>Here we have the function <b>f(x) = 2x+3</b>, written as a flow diagram:</p>
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<p class="center"><img src="../sets/images/flow-2xp3.svg" alt="2x+3"></p>
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<p>The <b>Inverse Function</b> goes the other way:</p>
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<p class="center"><img src="../sets/images/flow-2xp3-inv.svg" alt="Inverse"></p>
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<p class="center">So the inverse of: 2x+3 is: (y−3)/2</p>
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</div>
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<p>Read <a href="../sets/function-inverse.html">Inverse of a Function</a> to find out more.</p>
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<h2>Inverse Sine, Cosine and Tangent</h2>
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<div class="example">
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<h3>Example: the sine function</h3>
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<p class="center"><img src="../algebra/images/sin-sin-1.svg" alt="Inverse"></p>
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<p class="center">The <b>sine</b> function <span class="largest">sin</span> takes angle θ and gives the ratio <span class="intbl"> <em>opposite</em> <strong>hypotenuse </strong></span></p>
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<p class="center">The <b>inverse sine</b> function <span class="largest">sin<sup>-1</sup></span> takes the ratio <span class="intbl"> <em>opposite</em><strong>hypotenuse </strong></span> and gives angle θ</p>
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</div>
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<p>Read <a href="../algebra/trig-inverse-sin-cos-tan.html">Inverse Sine, Cosine, Tangent</a> to find out more.</p>
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<h2>The Inverse of an Exponent is a Logarithm</h2>
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<p>Read <a href="../algebra/logarithms.html">logarithms</a>to find out more, but basically:</p>
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<p class="center"><img src="../algebra/images/exponent-to-logarithm.gif" alt="2^3=8 becomes log_2(8)=3" height="125" width="133"></p>
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<p class="center">The logarithm tells us what the exponent is!</p>
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<p> </p>
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<a href="index.html">Numbers Index</a>
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