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<title>Scientific Notation</title>
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Scientific Notation</h1>
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<p><b>Scientific Notation </b>(also called <a href="../algebra/standard-form.html">Standard Form</a> in Britain) is a special way of writing numbers:</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:right;">Like this:</td>
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<td style="width:30px;"> </td>
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<td style="text-align:center;"><img src="images/scientific-notation-a.svg" alt="scientific notation 700 = 7x10^2"></td>
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</tr>
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<tr>
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<td style="text-align:right;"> </td>
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<td style="width:30px;"> </td>
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<td style="text-align:center;"> </td>
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</tr>
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<tr>
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<td style="text-align:right;">Or this:</td>
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<td style="width:30px;"> </td>
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<td style="text-align:center;"><img src="images/scientific-notation-b.svg" alt="scientific notation 4,900,000,000 = 4.9x10^9"></td>
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</tr>
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</tbody></table>
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<p>It makes it easy to use big and small values.</p>
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<h2>OK, How Does it Work?</h2>
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<div class="example">
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<h3>Example: 700</h3>
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<p>Why is 700 written as <span class="center"><b>7 × 10<sup>2</sup></b> in Scientific Notation ?</span></p>
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<div class="so"> 700 = 7 × 100 </div>
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<div class="so">and 100 = 10<sup>2</sup> <i>(see <a href="../index-notation-powers.html">powers of 10</a>)</i></div>
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<div class="so">so 700 = <b>7 × 10<sup>2</sup> </b></div>
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<p class="center">Both <b>700</b> and <b>7 × 10<sup>2</sup></b> have the same value, just shown in different ways.</p>
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</div>
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<div class="example">
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<h3>Example: 4,900,000,000</h3>
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<p> 1,000,000,000 = 10<sup>9</sup> ,<br>
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so 4,900,000,000 = <b>4.9 × 10<sup>9</sup></b> in Scientific Notation</p>
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</div>
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<p class="larger">The number is written in <b>two parts</b>:</p>
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<div class="bigul">
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<ul>
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<li>Just the <b>digits</b>, with the decimal point placed after the first digit, followed by</li>
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<li><b>× 10 to a <a href="../index-notation-powers.html">power</a></b> that puts the decimal point where it should be<br>
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(i.e. it shows how many places to move the decimal point).</li>
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</ul>
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</div>
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<p class="center"><img src="images/scientific-notation.svg" alt="scientific notation 5326.6 = 5.3266x10^3"><br>
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In this example, 5326.6 is written as <b>5.3266 × 10<sup>3</sup></b>,<br>
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because 5326.6 = 5.3266 × 1000 = 5.3266 × 10<sup>3</sup></p>
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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>scinotationMain();</script>
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<p>Now try to use Scientific Notation yourself:</p>
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<script>scinotationnumMain();</script>
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<h2>Other Ways of Writing It</h2>
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<p style="float:right; margin: 0 0 5px 30px;"><b style="font: 36px Arial; color: blue;">3.1 × 10^8</b></p>
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<p>We can use the <b>^</b> symbol (above the 6 on a keyboard), as it is easy to type.</p>
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<div class="example">
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<p class="larger">Example: <b>3 × </b><b>10^4 is the same as 3 × 10<sup>4</sup></b></p>
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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="images/calculator-e-notation.jpg" alt="calculator E notation" height="50" width="200"></p>
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<p>Calculators often use "E" or "e" like this:</p>
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<div style="clear:both"></div>
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<div class="example">
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<p class="larger">Example: <b>6E+</b><b>5 is the same as 6 × 10<sup>5</sup></b></p>
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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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<p class="larger">Example: <b>3.12E4</b><b> is the same as 3.12 × 10<sup>4</sup></b></p>
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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>How to Do it</h2>
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<p class="center"><b>To figure out the power of 10, think "how many places do I move the decimal point?"</b></p>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td><img src="../images/style/left-arrow.gif" alt="left arrow" height="46" width="46"></td>
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<td>When the number is 10 or greater, the decimal point has to move <b>to the left</b>, and the power of 10 is <b>positive</b>.</td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td><img src="../images/style/right-arrow.gif" alt="right arrow" height="46" width="46"></td>
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<td>
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<p>When the number is smaller than 1, the decimal point has to move <b>to the right</b>, so the power of 10 is <b>negative</b>.</p></td>
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</tr>
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</tbody></table>
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<div class="example">
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<h3>Example: 0.0055 is written <b>5.5 × 10<sup>-3</sup></b></h3>
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<p><br>
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Because 0.0055 = 5.5 × 0.001 = 5.5 × 10<sup>-3</sup></p>
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</div>
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<div class="example">
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<h3>Example: 3.2 is written <b>3.2 × 10<sup>0</sup></b></h3>
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<p><br>
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We didn't have to move the decimal point at all, so the power is <b>10<sup>0</sup></b></p>
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<p>But it is now in Scientific Notation</p>
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</div>
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<h2>Check!</h2>
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<p>After putting the number in Scientific Notation, just check that:</p>
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<ul>
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<li>The "digits" part is between 1 and 10 (it can be 1, but never 10)</li>
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<li>The "power" part shows exactly how many places to move the decimal point</li>
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</ul>
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<h2>Why Use It?</h2>
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<p>Because it makes it easier when dealing with very big or very small numbers, which are common in Scientific and Engineering work.</p>
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<div class="example">
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<p>Example: it is easier to write (and read) 1.3 × 10<sup>-9</sup> than 0.0000000013</p>
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</div>
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<p>It can also make calculations easier, as in this example:</p>
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<div class="example">
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<p><b>Example: a tiny space inside a computer chip has been measured to be
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0.00000256m wide, 0.00000014m long and 0.000275m high. </b></p>
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<p><b>What is its volume?</b></p>
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<p>Let's first convert the three lengths into scientific notation:</p>
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<ul>
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<li>width:
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0.000 002 56m = 2.56×10<sup>-6</sup></li>
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<li>length: 0.000 000 14m = 1.4×10<sup>-7</sup></li>
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<li>height: 0.000 275m = 2.75×10<sup>-4</sup></li>
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</ul>
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<p>Then multiply the digits together (ignoring the ×10s):</p>
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<p>2.56 × 1.4 × 2.75 = 9.856</p>
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<p>Last, multiply the ×10s:</p>
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<p>10<sup>-6</sup> × 10<sup>-7</sup> × 10<sup>-4</sup> = 10<sup>-17</sup> (easier than it looks, just <b>add −6, −4 and −7</b> together)</p>
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<p class="larger">The result is <b>9.856×10<sup>-17</sup> m<sup>3</sup></b></p>
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</div>
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<p>It is used a lot in Science:</p>
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<div class="example">
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<h3>Example: Suns, Moons and Planets</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>Easier than writing 1,988,000,000,000,000,000,000,000,000,000 kg<br>
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(and that number gives a false sense of many digits of accuracy.)</p>
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</div>
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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>
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<td>
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<h3>Play With It!</h3>
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<p>Use Scientific Notation<br>
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in <a href="gravity-freeplay.html">Gravity Freeplay</a></p></td>
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<td><a href="gravity-freeplay.html"><img src="images/gravity-freeplay-thumb.gif" alt="gravity freeplay" height="103" width="245" border="0"></a></td>
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</tr>
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</tbody></table>
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</div>
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<p> </p>
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<p>It can also save space! Here is what happens when you double on each square of a chess board:</p>
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<p class="center"><img src="images/chess-board-double.svg" alt="Chess board doubling"><br>
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Values are rounded off, so
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53,6870,912 is shown as just 5×10<sup>8</sup></p>
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<p>That last value, shown as 9×10<sup>18</sup> is actually
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9,223,372,036,854,775,808</p>
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<h2>Engineering Notation</h2>
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<p>Engineering Notation is like Scientific Notation, except that we only use powers of ten that are multiples of 3 (such as 10<sup>3</sup>, 10<sup>-3</sup>, 10<sup>12</sup> etc).</p>
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<div class="example">
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<h3>Examples:</h3>
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<ul>
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<li>2,700 is written <b>2.7 × 10<sup>3</sup></b></li>
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<li>27,000 is written <b>27 × 10<sup>3</sup></b></li>
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<li>270,000 is written <b>270 × 10<sup>3</sup></b></li>
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<li>2,700,000 is written <b>2.7 × 10<sup>6</sup></b></li>
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</ul>
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</div>
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<div class="example">
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<h3>Example: 0.00012 is written <b>120 × 10<sup>-6</sup></b></h3>
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</div>
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<p>Notice that the "digits" part can now be between 1 and 1,000 (it can be 1, but never 1,000).</p>
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<p>The advantage is that we can replace the <b>×10</b>s with <a href="../metric-numbers.html">Metric Numbers</a>. So we can use standard words (such as thousand or million), prefixes (such as kilo, mega) or the symbol (k, M, etc)</p>
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<div class="example">
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<p>Example: 19,300 meters is written <b>19.3 × 10<sup>3</sup> m, or 19.3 km</b></p>
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</div>
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<div class="example">
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<p>Example: 0.00012 seconds is written <b>120 × 10<sup>-6</sup> s, or 120 microseconds</b></p>
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</div>
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<p> </p>
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<div class="questions">
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<script>getQ(1340, 1341, 1342, 1344, 1345, 5077, 5078, 5079, 1343, 1346, 1347, 5080);</script>
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</div>
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<div class="related">
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<a href="../index-notation-powers.html">Powers of 10</a>
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<a href="../decimals.html">Decimals</a>
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<a href="../metric-numbers.html">Metric Numbers</a>
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<a href="../measure/lengths-small-large.html">Lengths From Very Small to Very Large</a>
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<a href="../exponent.html">Exponents</a>
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</div>
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