Definition and Understanding of e Number

<div style="text-align: justify;"> <table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody> <tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFJmFVpq1_0j8XdN4xxBUOSYdeYKxJpGnLftSrbKD1npj6Pj3LGy16cI8R1SCGYd26m6N1opGobqE_ZpsWAf9KQlNn9Kb53aiVzA9kL2iz36yrXR1ljJx_EHg2z_yrKINDmR5VZEUfF9o/s1600/Definition+and+Understanding+of+E+numbers.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="216" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFJmFVpq1_0j8XdN4xxBUOSYdeYKxJpGnLftSrbKD1npj6Pj3LGy16cI8R1SCGYd26m6N1opGobqE_ZpsWAf9KQlNn9Kb53aiVzA9kL2iz36yrXR1ljJx_EHg2z_yrKINDmR5VZEUfF9o/s320/Definition+and+Understanding+of+E+numbers.gif" width="320" /></a></td></tr> <tr><td class="tr-caption" style="text-align: center;">Image From http://www.gogeometry.com/</td></tr> </tbody></table> <b><i>e</i></b> number is a number that is the base of natural logarithms. There are several ways of defining it. Probably the most satisfactory is this. First, define natural logarithms (ln) as in approach <b>2</b> to the logarithmic function. Then define exponential (exp) as the inverse function of ln in exponential function. Then define <b><i>e</i></b> as equal to exp 1. This amounts to saying that <b><i>e</i></b> is the number that makes<br /> <br /></div> <div style="text-align: justify;"> <div style="text-align: center;"> <a href="http://www.codecogs.com/eqnedit.php?latex=\int_{1}^{e}\frac{1}{t}dt" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\int_{1}^{e}\frac{1}{t}dt" title="\int_{1}^{e}\frac{1}{t}dt" /></a></div> <div style="text-align: center;"> <br /></div> <div style="text-align: justify;"> It is necessary to go on to show that <a href="http://www.codecogs.com/eqnedit.php?latex=e^{x}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?e^{x}" title="e^{x}" /></a> and exp <i>x</i> are equal and so are identical as functions, and also that ln and <a href="http://www.codecogs.com/eqnedit.php?latex=log_{e}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?log_{e}" title="log_{e}" /></a> are identical functions.<br /> <br /> The number <i style="font-weight: bold;">e</i> has important properties derived from some of the properties of ln and exp. For example,<br /> <br /></div> </div> <div style="text-align: center;"> <a href="http://www.codecogs.com/eqnedit.php?latex=e=\lim_{h\rightarrow&space;0}(1+h)^\frac{1}{h}=\lim_{n\rightarrow&space;\mathbb{\sim&space;}}\left&space;(&space;1+\frac{1}{n}&space;\right&space;)^n" target="_blank"><img src="http://latex.codecogs.com/gif.latex?e=\lim_{h\rightarrow&space;0}(1+h)^\frac{1}{h}=\lim_{n\rightarrow&space;\mathbb{\sim&space;}}\left&space;(&space;1+\frac{1}{n}&space;\right&space;)^n" title="e=\lim_{h\rightarrow 0}(1+h)^\frac{1}{h}=\lim_{n\rightarrow \mathbb{\sim }}\left ( 1+\frac{1}{n} \right )^n" /></a></div> <div style="text-align: center;"> <br /></div> <div style="text-align: justify;"> Also, <i style="font-weight: bold;">e</i> is the sum of the series</div> <div style="text-align: justify;"> <br /></div> <div style="text-align: center;"> <a href="http://www.codecogs.com/eqnedit.php?latex=1+\frac{1}{1!}+\frac{1}{2!}+...+\frac{1}{n!}+..." target="_blank"><img src="http://latex.codecogs.com/gif.latex?1+\frac{1}{1!}+\frac{1}{2!}+...+\frac{1}{n!}+..." title="1+\frac{1}{1!}+\frac{1}{2!}+...+\frac{1}{n!}+..." /></a></div> <div style="text-align: center;"> <br /></div> <div style="text-align: justify;"> Another approach, but not a recommended one, is to make of one these properties the definition of <i style="font-weight: bold;">e</i>. Then exp <i>x</i> would be defined as <a href="http://www.codecogs.com/eqnedit.php?latex=e^{x}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?e^{x}" title="e^{x}" /></a> and exp <i>x, </i>ln <i>x</i> would be defined as its inverse function, and the properties of these functions would have to be proved.<br /> <br /> The value of <i style="font-weight: bold;">e</i> is 2.718 281 83 (to 8 decimal places). The proof that <i style="font-weight: bold;">e</i> is irrational is comparatively easy. In 1873, Hermite proved that <i style="font-weight: bold;">e</i> is transcendental, and his proof was subsequently simplified by Hilbert.</div>

Definition and Understanding of e Number

Image From http://www.gogeometry.com/ e  number is a number that is the base of natural logarithms. There are several ways of defining ...

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