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Back to e

Above, we defined $ e$ as the particular real number satisfying

$\displaystyle \lim_{\delta\to 0} \frac{e^\delta-1}{\delta} \isdef 1
$

which gave us $ (a^x)^\prime = a^x$ when $ a=e$ . From this expression, we have, as $ \delta\to 0$ ,

\begin{eqnarray*}
e^\delta - 1 & \rightarrow & \delta \\
\Rightarrow \qquad e^\delta & \rightarrow & 1+\delta \\
\Rightarrow \qquad e & \rightarrow & (1+\delta)^{1/\delta},
\end{eqnarray*}

or

$\displaystyle \zbox {e \isdef \lim_{\delta\to0} (1+\delta)^{1/\delta}.}
$

This is one way to define $ e$ . Another way to arrive at the same definition is to ask what logarithmic base $ e$ gives that the derivative of $ \log_e(x)$ is $ 1/x$ . We denote $ \log_e(x)$ by $ \ln(x)$ .

Numerically, $ e$ is a transcendental number (a type of irrational number3.5), so its decimal expansion never repeats. The initial decimal expansion of $ e$ is given by3.6

$\displaystyle e = 2.7182818284590452353602874713526624977572470937\ldots\,.
$

Any number of digits can be computed from the formula $ (1+\delta)^{1/\delta}$ by making $ \delta$ sufficiently small.


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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-04-21 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA