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251 lines
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<title>Index Notation - Powers of 10</title>
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<h1 align="center">Index Notation and Powers of 10</h1>
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<p style="float:left; margin: 0 20px 5px 0;"><img src="algebra/images/exponent-10-2.svg" alt="10 to the Power 2" /></p>
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<p><span class="large"><span class="larger">The exponent (or index or power) of a number says<br />
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<b>how many times</b> to use the number in a <b>multiplication</b>.</span></span></p>
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<p><span class="large">10<sup>2</sup></span> means <b>10</b> × <b>10</b> = 100</p>
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<p>(It says <b>10</b> is used <b>2</b> times in the multiplication)</p>
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<div style="clear:both"></div>
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<div class="example">
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<h3 class="Larger">Example: 10<sup>3</sup> = 10 × 10 × 10 = 1,000 </h3>
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<ul>
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<li>In words: 10<sup>3</sup> could be called "10 to the third power", "10 to the power 3" or
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simply "10 cubed"</li>
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</ul>
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</div>
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<div class="example">
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<h3 class="Larger">Example: 10<sup>4</sup> = 10 × 10 × 10 × 10 = 10,000
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</h3>
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<ul>
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<li>In words: 10<sup>4</sup> could be called "10 to the fourth power", "10 to the power 4" or
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"10 to the 4"</li>
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</ul>
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</div>
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<p class="Larger"></p>
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<p class="Larger">You can multiply <i>any</i> number by itself <i>as many times</i> as you want using this notation (see <a href="exponent.html">Exponents</a>),
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but powers of 10 have a special use ...
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</p>
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<h2>Powers of 10</h2>
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<p>"Powers of 10" is a very useful way of writing down large or small numbers. </p>
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<p>Instead of having lots of zeros, you show how many <b>powers
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of 10</b> will make that many zeros</p>
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<div class="example">
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<h3>Example: 5,000 = 5 × 1,000 = 5 × 10<sup>3</sup> </h3>
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<p class="Larger"> 5 thousand is 5 times a thousand. And a thousand is 10<sup>3</sup>. So 5 times 10<sup>3</sup> = 5,000</p>
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<p class="larger">Can you see that 10<sup>3</sup> is a handy way of making 3 zeros?</p>
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</div>
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<p class="Larger"></p>
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<p>Scientists and Engineers (who often use very big or very small numbers) like to write numbers
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this way.</p>
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<div class="example">
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<h3>Example: The Mass of the Sun</h3>
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<p>The Sun has a Mass of 1.988 × 10<sup>30</sup> kg.</p>
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<p>It is too hard to write 1,988,000,000,000,000,000,000,000,000,000 kg</p>
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<p>(And very easy to make a mistake counting the zeros!)</p>
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</div>
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<div class="example">
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<h3>Example: A Light Year (the distance light travels in one year) </h3>
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<p>It is easier to use <b>9.461 × 10<sup>15</sup></b> meters, rather than <b>9,461,000,000,000,000 </b>meters</p>
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</div>
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<p>It is commonly called <a href="numbers/scientific-notation.html"><b>Scientific Notation</b></a>, or <b>Standard Form.</b></p>
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<h2>Other Way of Writing It</h2>
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<p>Sometimes people use the <b>^</b> symbol (above the 6 on your keyboard), as it is easy to type.</p>
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<div class="example">
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<h3 class="Larger">Example: <b>3 × </b><b>10^4 is the same as 3 × 10<sup>4</sup></b></h3>
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<ul>
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<li><span class="Larger"><b>3 × 10^4 = 3 × 10 × 10 × 10 × 10 = 30,000</b></span></li>
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</ul>
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</div>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="numbers/images/calculator-e-notation.jpg" width="200" height="50" alt="calculator e notation" /></p>
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<p>Calculators often use "E" or "e" like this: </p><div style="clear:both"></div>
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<div class="example">
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<h3 class="Larger">Example: <b>6E+</b><b>5 is the same as 6 × 10<sup>5</sup></b></h3>
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<ul>
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<li><span class="Larger"><b>6E+5 = 6 × 10 × 10 × 10 × 10 × 10 = 600,000</b></span></li>
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</ul>
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</div>
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<div class="example">
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<h3 class="Larger">Example: <b>3.12E4</b><b> is the same as 3.12 × 10<sup>4</sup></b></h3>
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<ul>
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<li><span class="Larger"><b>3.12E4 = 3.12 × 10 × 10 × 10 × 10 = 31,200</b></span></li>
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</ul>
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</div>
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<h2>The Trick</h2>
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<p>While at first it may look hard, there is an easy "trick":</p>
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<div class="center80">
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<p><span class="larger">The index of 10 says ...</span> </p>
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<p align="center">...<span class="larger"> <b>how many places</b> to move the decimal point
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to the right.</span></p>
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</div>
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<p> </p>
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<div class="example">
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<h3>Example: What is 1.35 × 10<sup>4</sup> ?</h3>
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<p>You can calculate it as: 1.35 x (10 × 10 × 10 × 10) = 1.35 x 10,000 = 13,500</p>
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<p>But it is easier to think "move the decimal point 4 places to the right" like this:</p>
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<div class="clear">
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<table border="0" cellpadding="2" align="center">
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<tr>
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<td><font color="blue">1<b>.</b>35</font></td>
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<td><img src="images/arrow.gif" width="30" height="30" alt="arrow" /></td>
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<td><font color="blue">13<b>.</b>5</font></td>
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<td><img src="images/arrow.gif" width="30" height="30" alt="arrow" /></td>
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<td><font color="blue">135<b>.</b></font></td>
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<td><img src="images/arrow.gif" width="30" height="30" alt="arrow" /></td>
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<td><font color="blue">1350<b>.</b></font></td>
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<td><img src="images/arrow.gif" width="30" height="30" alt="arrow" /></td>
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<td><font color="blue">13500<b>.</b></font></td>
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</tr>
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</table>
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</div>
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</div>
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<h2>Negative Powers of 10</h2>
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<p>Negative? What could be the opposite of multiplying? <b>Dividing! </b></p>
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<p>A negative power means <b>how many times to
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divide</b> by the number. </p>
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<div class="example">
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<h3>Example: 5 × 10<sup>-3</sup> = 5 ÷ 10 ÷ 10 ÷ 10 = 0.005</h3>
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</div>
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<p>Just remember for negative powers of 10:</p>
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<p align="center" class="Larger"><font color="red"><b>For negative powers of 10, move the decimal point to the left.</b></font></p>
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<p>So Negatives just go the other way.</p>
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<div class="example">
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<h3>Example: What is 7.1 × 10<sup>-3</sup> ?</h3>
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<p>Well, it is really 7.1 x (<span class="frac"><sup>1</sup>/<sub>10</sub></span> × <span class="frac"><sup>1</sup>/<sub>10</sub></span> × <span class="frac"><sup>1</sup>/<sub>10</sub></span>) = 7.1 × 0.001 = 0.0071</p>
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<p>But it is easier to think "move the decimal point 3 places to the <b>left</b>" like this:</p>
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<div class="clear">
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<table border="0" cellpadding="2" align="center">
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<tr>
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<td><font color="red">7<b>.</b>1</font></td>
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<td><img src="images/arrow.gif" width="30" height="30" alt="arrow" /></td>
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<td><font color="red">0<b>.</b>71</font></td>
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<td><img src="images/arrow.gif" width="30" height="30" alt="arrow" /></td>
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<td><font color="red">0<b>.</b>071</font></td>
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<td><img src="images/arrow.gif" width="30" height="30" alt="arrow" /></td>
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<td><font color="red">0<b>.</b>0071</font></td>
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</tr>
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</table>
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</div>
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</div>
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<h2>Try It Yourself</h2>
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<p>Enter a number and see it in Scientific Notation:</p>
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<script type="text/javascript">scinotationMain();</script>
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<p>Now try to use Scientific Notation yourself:</p>
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<script type="text/javascript">scinotationnumMain();</script>
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<h2>Summary</h2>
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<div class="simple">
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<p>The index of 10 says how many places to move the decimal point. Positive means move it to the right, negative means to the left. Example:</p>
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<table width="500" border="0" align="center">
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<tr>
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<td><div align="center"></div></td>
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<td align="center"><b>Number</b></td>
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<td align="center"><div align="center"><b>In Scientific<br />
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Notation</b></div></td>
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<td align="center"><b>In Words</b></td>
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</tr>
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<tr>
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<td><b>Positive Powers</b></td>
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<td align="center">5,000</td>
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<td align="center">5 × 10<sup>3</sup></td>
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<td align="center">5 Thousand</td>
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</tr>
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<tr>
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<td><b>Negative Powers</b></td>
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<td align="center">0.005</td>
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<td align="center">5 × 10<sup>-3</sup></td>
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<td align="center">5 Thousand<i>ths</i></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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<div class="questions">
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<script type="text/javascript">getQ(968, 1354, 969, 1355, 970, 1356, 971, 1357, 3472, 3473);</script>
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</div>
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<div class="related"> <a href="numbers/scientific-notation.html">Scientific Notation</a> <a href="decimals.html">Decimals</a>
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<a href="numbers/number-line-zoom.html">Zoomable Number Line</a>
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<a href="ordering_decimals.html">Ordering Decimals</a> <a href="metric-numbers.html">Metric Numbers</a> <a href="exponent.html">Exponents</a> <a href="rounding-numbers.html">Rounding Numbers</a> <a href="converting-decimals-fractions.html">Convert Decimals to Fractions</a> <a href="converting-fractions-decimals.html">Convert Fractions to Decimals</a> </div>
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Copyright © 2017 MathsIsFun.com
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