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Revision as of 12:37, 21 January 2003 edit217.158.106.193 (talk) a better continued fraction expansion, with a more elegant pattern← Previous edit Revision as of 01:27, 23 January 2003 edit undoAxelBoldt (talk | contribs)Administrators44,507 edits standard notation for continued fractions with semicolonNext edit →
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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>


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.

Revision as of 01:27, 23 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.