E (mathematical constant): Difference between revisions

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	The constant <math>e</math> (occasionally called ''Napier's constant'' in honor of the Scottish mathematician ] who introduced logarithms) is the base of the ]. It is approximately equal to		The constant <math>e</math> (occasionally called ''Napier's constant'' in honor of the Scottish mathematician ] who introduced logarithms) 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 ] of <math>(1 + 1/n)^n</math> ~~as <math>n</math> goes to infinity~~ and can also be written as 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>
			and can also be written as the ]
	: <math>e = {1 \over 0!} + {1 \over 1!}		: <math>e = {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 infinite ] expansion of <math>e</math> contains an interesting pattern as follows:
⚫	: <math>e = . </math>

	The number ''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 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 features (along with a few other fundamental constants) in ]:		The number ''e'' is known to be ] and even ]. 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 is regarded by some mathematicians as ]! ~~Two other closely related important identities involving ''e'' are:~~		which is regarded by some mathematicians as "The most remarkable formula in the world"!

		⚫	The infinite ] expansion of <math>e</math> contains an interesting pattern as follows:
	: <math>\cos(x)=\frac{e^{ix}+e^{-ix}}{2}</math>
		⚫	: <math>e = . </math>

	and

	: <math>\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}.</math>

Revision as of 23:54, 31 March 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={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 is conjectured to be normal. It features (along with a few other fundamental constants) in Euler's identity:

e^{i\pi }+1=0

which is regarded by some mathematicians as "The most remarkable formula in the world"!

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

e=.