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      <text>1.2.6.1 Scientific notation</text>
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      <text><![CDATA[<h3>Scientific notation
</h3>
<p>It is often necessary to use very large or very small numbers such as \(78000000\) and \(0.00000034\). <strong><br></strong></p><p><strong>Scientific notation</strong> can be used to express such numbers in a more concise form. Each number is written in the form <br></p><p>\( a\times 10^n\) where \(a\) is a number between \(1\) and \(10\). <br></p><p>We can make use of the following facts:</p>
<ul><li>\(10=10^1, \quad 100=10^2, \quad 1000=10^3\) and so on,</li>
<li>\( 0.1=10^{-1},\quad 0.01=10^{-2},\quad 0.001=10^{-3}\) and so on. </li>
</ul>
<p>For example,</p>
<ul> 
<li>the number \(5000\) can be written \(5\times 1000=5\times 10^3\),</li>
 <li>the number \(403\) can be written \(4.03\times 100=4.03\times 10^2\),</li>
 <li>the number \(0.009\) can be written \(9\times 0.001=9\times 10^{-3}\).</li>
</ul>
<p>Furthermore, to multiply a number by \(10^n\) the decimal point is moved \(n\) places to the right if \(n\) is a positive integer, and \(n\) places to the left if \(n\) is a negative integer. (If necessary additional zeros are inserted to make up the required number of digits before the decimal point.)</p>

<hr>
<h4 class="HELM_example">Example</h4>
<p>Write the following numbers  in scientific notation.</p>
<ol class="HELM_parts_inline">
<li>\(0.00678\)  </li>
<li> \(123456.7\)</li>
</ol>
<h4 class="HELM_solution">Solution</h4>
<ol class="HELM_parts">
<li>\(0.00678=6.78\times 10^{-3}\)</li>
<li>\(123456.7=1.234567\times 10^5\)</li>
</ol>
<p>&nbsp;</p>
<hr>

<h3>Engineering constants</h3>
<p>Many constants appearing in engineering calculations are expressed in scientific notation. For example the charge on an electron equals \(1.6\times 10^{-19}\) coulomb and the speed of light is \(3\times 10^8\) m s\(^{-1}\). Avogadro’s constant is equal to \(6.023\times 10^{26}\) and is the number of atoms in one kilomole of an element. Clearly the use of scientific notation avoids writing lengthy strings of zeros.</p>
 <p>Your scientific calculator will be able to accept numbers in scientific notation. Often the \(E\) button is used and a number like \(4.2\times 10^7\) will be entered as \(4.2 E7\). Note that \(10E4\) means \(10\times 10^4\), that is \(10^5\). To enter the number \(10^3\) say, you would key in \(1E3\). Entering powers of 10 incorrectly is a common cause of error. You must check how your particular calculator accepts numbers in scientific notation.</p>
<p> The following example is designed to check that you can enter numbers given in scientific notation into your calculator.</p>

<hr>
<h4 class="HELM_example">Example</h4>
<p>Use your calculator to find \(4.2\times 10^{-3}\times 3.6\times 10^{-4}\).</p>
<h4 class="HELM_solution">Solution</h4>
<p>\(1.512 \times 10^{−6}\).</p>
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