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

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

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

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

${\displaystyle e=.}$

The number e is relevant because one can show that the exponential function exp(x) can be written as ${\displaystyle e^{x}}$; 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:

${\displaystyle e^{i\pi }+1=0}$

which is regarded by some mathematicians as The most remarkable formula in the world! Two other closely related important identities involving e are:

${\displaystyle \cos(x)={\frac {e^{ix}+e^{-ix}}{2}}}$

and

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