Euler's Number

Euler’s number, denoted as , is a naturally occurring number related to exponential growth and exponential decay. The number appears conceptually as the base of both the exponential function and the natural logarithm function. The approximate value of Euler’s number is shown below.


For the purposes of computing results of formulas and functions, this site considers Euler’s number, as it appears in the form , as shorthand for the exponential function. This choice is intentional and helps explain the formation of some deeper mathematical ideas.

The value is still a very important constant in calculus and is relevant to exponential growth and decay as a naturally occuring value.

Formal Definition

The value of is formally be defined by the value of the exponential function at . Depending on which definition of the exponential function you are using, the value can either be calculated as a limit or a summation as shown below.