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      <text>4.1.12 (The number 'e')</text>
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      <text><![CDATA[<h1>The importance of the number \(e\)</h1>

<p>We have seen that the most simple exponential functions have the form \( y =a^x\), where \(a > 0, a \not=1\). There are an infinite number of choices of the base number. However the most important and widely used exponential function has the particular base \(e= 2.7182818\dots\). This is called the <b>natural exponential</b>. There are numerous ways of&nbsp; calculating the value of \(e\). For example, it can be shown that the value of \(e\) is the end-point of the sequence of numbers:
\[ 
\left(\dfrac{2}{1}\right)^1,\quad \left(\dfrac{3}{2}\right)^2,\quad \left(\dfrac{4}{3}\right)^3 ,\quad\dots ,\qquad \left(\dfrac{17}{16}\right)^{16},\quad\dots ,\quad \left(\dfrac{65}{64}\right)^{64}
\]
which, in decimal form (each to 6 d.p.) are 
\[ 
2.000000,\quad 2.250000,\quad 2.370370,\quad\dots ,\quad 2.637929,\quad \dots , 2.697345,\dots
\]</p>

<p>We will examine sequences and series more closely in week 8 and limits in week 10  however, for now let us just note that this leads to a precise definition for the value of \(e\): 
\[
e=\lim_{n\rightarrow\infty}\left(\dfrac{n+1}{n}\right)^n.
\]</p>
<p> An quicker way of calculating \(e\) is to use the (infinite) series: 
\[ 
\quad e=1+\dfrac{1}{1!}+\dfrac{1}{2!}+\dfrac{1}{3!}+\dfrac{1}{4!}+\dots +\dfrac{1}{n!}+\dots
\]
where, 
\[ 
n!=n\times (n-1)\times (n-2)\times \dots (3)\times (2)\times (1).
\]</p>

</p><p style="text-align: center;"><img src="@@PLUGINFILE@@/6-2-Fig2.svg" alt="Exponential function for the base e." class="img-responsive atto_image_button_text-bottom" width="257" height="177"><br></p>

<p style="text-align: center;"><b>&nbsp;\({\bf y = e^x}\).</b></p>

<p>We shall see that the number \(e\) is the number that governs all exponential growth or decay. The function \(y = e^x\) has the property that its slope is the value of the function. Also the area under the curve, up to some value \(x\) is also the value of the function (i.e. \(e^x\)).</p>

<p>Exponential functions (and&nbsp; variants) appear in various areas of mathematics and engineering. For example, the shape of a hanging chain or rope, under the effect of gravity, is well described by a combination of the exponential curves \(e^{kx}\), \(e^{-kx}\). The function \(e^{-x^2}\) plays a major role in statistics; it being fundamental in the important normal distribution which describes the variability in many naturally occurring phenomena. The exponential function \(e^{-kx}\) appears directly, again in the area of statistics, in the Poisson distribution which (amongst other things) is used to predict the number of events (which occur randomly) in a given time interval. <br></p>]]></text>
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