E (mathematical constant): Difference between revisions

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	The ] ''e'' is relevant because one can show that the ] exp(''x'') can be written as <math>e^x</math>; the exponential function is important because it is, up to multiplication by a scalar, the unique function which is its own ] and is hence commonly used to model growth or decay processes.		The ] ''e'' is relevant because one can show that the ] exp(''x'') can be written as <math>e^x</math>; the exponential function is important because it is, up to multiplication by a scalar, the unique function which is its own ] and is hence commonly used to model growth or decay processes.

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

	: <math>e^{i\pi}+1=0</math>		: <math>e^{i\pi}+1=0</math>

Revision as of 12:11, 14 September 2003

The constant $e$ (occasionally called 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

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

and can also be written as the infinite series

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 $n!$ stands for the factorial of $n$ .

The number e is relevant because one can show that the exponential function exp(x) can be written as $e^{x}$ ; 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:

e^{i\pi }+1=0

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

The infinite continued fraction expansion of $e$ contains an interesting pattern as follows:

e=

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

e=

Extenal link

Wikisource - E to 10,000 Places