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The constant ${\displaystyle e}$ (occasionally called Euler's number or Napier's constant in honor of the Scottish mathematician John Napier who introduced logarithms) is the base of the natural logarithm. It is approximately equal to

e = 2.71828 18284 59045 23536 02874 ...

It is equal to exp(1) where exp is the exponential function and therefore it is the limit

${\displaystyle e=\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}}$

and can also be written as the infinite series

${\displaystyle e=\sum _{n=0}^{\infty }{1 \over n!}={1 \over 0!}+{1 \over 1!}+{1 \over 2!}+{1 \over 3!}+{1 \over 4!}+\cdots }$

Here ${\displaystyle n!}$ stands for the factorial of ${\displaystyle n}$.

The number e is relevant because one can show that the exponential function exp(x) can be written as ${\displaystyle e^{x}}$; the exponential function is important because it is, up to multiplication by a scalar, the unique function which is its own derivative and is hence commonly used to model growth or decay processes.

The number e is known to be irrational and even transcendental. It was the first number to be proved transcendental without having been specifically constructed; the proof was given by Charles Hermite in 1873. It is conjectured to be normal. It features (along with a few other fundamental constants) in Euler's identity:

${\displaystyle e^{i\pi }+1=0}$

which was described by Richard Feynman as "The most remarkable formula in mathematics"!

The infinite continued fraction expansion of ${\displaystyle e}$ contains an interesting pattern as follows:

${\displaystyle e=}$

The apparent iregularity at the beginning disappears if one rewrites the fraction as:

${\displaystyle e=}$

Extenal link

• Wikisource - E to 10,000 Places