Logarithmic number system: Difference between revisions

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Revision as of 15:54, 17 July 2013

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+{{Portal:Mathematics/Feature article|img=Exp derivative at 0.svg|img-cap= ''e'' is the unique number such that the slope of ''y''=''e<sup>x</sup>'' (blue curve) is exactly 1 when ''x''=0 (illustrated by the red [[Point_of_tangency|tangent line]]).  For comparison, the curves ''y''=2<sup>x</sup> (dotted curve) and ''y''=4<sup>x</sup> (dashed curve) are shown.|img-cred=[[User:Dicklyon|Dick Lyon]]|more=e (mathematical constant)|desc=[[e (mathematical constant)|The mathematical constant '''''e''''']] is occasionally called '''Euler's number''' after the [[Switzerland|Swiss]] [[mathematician]] [[Leonhard Euler]], or '''Napier's constant''' in honor of the [[Scotland|Scottish]] mathematician [[John Napier]] who introduced [[logarithm]]s. It is one of the most important numbers in mathematics, alongside the additive and multiplicative identities [[0 (number)|0]] and [[1 (number)|1]], the [[imaginary unit]] ''i'', and [[pi|&pi;]], the circumference to diameter ratio for any circle. It has a number of equivalent definitions. One is given in the caption of the image to the right, and three more are:
+# The sum of the [[infinite series]]
+#:<math>\begin{align} e & = \sum_{n = 0}^\infty \frac{1}{n!} \\
+& = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots \\ \end{align}</math>
+#:where ''n''! is the [[factorial]] of ''n''.
+#:
+# The [[global maximum]] of the function
+#:<math> f(x) = x^{1 \over x}. </math>
+#:
+# The [[limit (mathematics)|limit]]:
+#:<math>e = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n</math>
+#::
+The number ''e'' is also the base of the [[natural logarithm]]. Since ''e'' is [[transcendental number|transcendental]], and therefore [[irrational number|irrational]], its value can not be given exactly.  The numerical value of ''e'' truncated to 20 [[decimal|decimal places]] is 2.71828 18284 59045 23536. |class={{{class}}}}}
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Logarithmic number system: Difference between revisions

Revision as of 15:54, 17 July 2013

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