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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 ]: ''e<sup>i&pi</sup>''+1=0. The number ''e'' is known to be ] and even ]. It features (along with a few other fundamental constants) in ]: ''e<sup>i&pi</sup>''+1=0, which is regarded by some mathematicians as '''The most remarkable formula in the world'''!

Two other important identities involving ''e'' are:

<math>cos(x)=\frac{e^{ix}+e^{-ix}}{2}</math>

and

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

Revision as of 06:35, 19 February 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 is known to be irrational and even transcendental. It features (along with a few other fundamental constants) in Euler's identity: e+1=0, which is regarded by some mathematicians as The most remarkable formula in the world!

Two other important identities involving e are:

c o s ( x ) = e i x + e − i x 2 {\displaystyle cos(x)={\frac {e^{ix}+e^{-ix}}{2}}}

and

s i n ( x ) = e i x − e − i x 2 i . {\displaystyle sin(x)={\frac {e^{ix}-e^{-ix}}{2i}}.}