E (mathematical constant): Difference between revisions

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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 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=.

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	The ] <math>e</math> (occasionally called ''Napier's constant'' in honor of the Scottish ] ] who introduced ]) is the base of the ]. It is approximately equal to		The ] <math>e</math> (occasionally called ''Napier's constant'' in honor of the Scottish ] ] who introduced ]) is the base of the ]. It is approximately equal to

	: ''e'' = 2.71828 18284 59045 23536 02874 .....		: ''e'' = 2.71828 18284 59045 23536 02874 .....

	It is equal to exp(1) where exp is the ] and therefore it is the ]		It is equal to exp(1) where exp is the ] and therefore it is the ]

	: <math>e = \lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n</math>		: <math>e = \lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n</math>

	and can also be written as the ]		and can also be written as the ]

	: <math>e = \sum_{n=0}^\infty {1 \over n!} = {1 \over 0!} + {1 \over 1!}		: <math>e = \sum_{n=0}^\infty {1 \over n!} = {1 \over 0!} + {1 \over 1!}
	+ {1 \over 2!} + {1 \over 3!}		+ {1 \over 2!} + {1 \over 3!}
	+ {1 \over 4!} + \cdots</math>		+ {1 \over 4!} + \cdots</math>

	Here <math>n!</math> stands for the ] of <math>n</math>.		Here <math>n!</math> stands for the ] of <math>n</math>.

	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 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>

	which was described by ] as "The most remarkable formula in ]"!		which was described by ] as "The most remarkable formula in ]"!

	The infinite ] expansion of <math>e</math> contains an interesting pattern as follows:		The infinite ] expansion of <math>e</math> contains an interesting pattern as follows:
	: <math>e = . </math>		: <math>e = . </math>