Misplaced Pages

E (mathematical constant): Difference between revisions

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
Browse history interactively← Previous editNext edit →Content deleted Content addedVisualWikitext
Revision as of 13:49, 17 September 2002 editZundark (talk | contribs)Extended confirmed users, File movers, Pending changes reviewers29,661 editsm occasionly -> occasionally← Previous edit Revision as of 12:22, 21 January 2003 edit undoAp (talk | contribs)2,137 edits converted to <math> modeNext edit →
Line 1: Line 1:
] ]

The constant ''e'' (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 (1 + 1/''n'')<sup>''n''</sup> as ''n'' goes to infinity and can 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 ]
: <math>e = {1 \over 0!} + {1 \over 1!}
also be written as the ]
: ''e'' = 1 / 0! + 1 / 1! + 1 / 2! + 1 / 3! + 1 / 4! + ... + {1 \over 2!} + {1 \over 3!}
+ {1 \over 4!} + \cdots</math>
Here ''n''! stands for the ] of ''n''. Here <math>n!</math> stands for the ] of <math>n</math>.


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


The number ''e'' is relevant because one can show that the ] exp(''x'') can be written as ''e''<sup>''x''</sup>; 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 ]. The number <math>e</math> is known to be ] and even ]. It features (along with a few other fundamental constants) in ].
It features (along with a few other fundamental constants) in ].

Revision as of 12:22, 21 January 2003


The constant e {\displaystyle 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 of ( 1 + 1 / n ) n {\displaystyle (1+1/n)^{n}} as n {\displaystyle n} goes to infinity and can also be written as the infinite series

e = 1 0 ! + 1 1 ! + 1 2 ! + 1 3 ! + 1 4 ! + ⋯ {\displaystyle e={1 \over 0!}+{1 \over 1!}+{1 \over 2!}+{1 \over 3!}+{1 \over 4!}+\cdots }

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

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

e = [ 2 ; 1 , 2 , 1 , 1 , 4 , 1 , 1 , 6 , 1 , 1 , 8 , 1 , 1 , 10 , … ] . {\displaystyle e=.}

The number e is relevant because one can show that the exponential function exp(x) can be written as e x {\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 {\displaystyle e} is known to be irrational and even transcendental. It features (along with a few other fundamental constants) in Euler's identity.