# E

Euler's number (not to be confused with Euler number), or *e*, is an extremely important and one of the most fundamental numbers in mathematics, approximately equal to 2.72, and is almost as famous as pi. It appears very often in mathematics and nature, it is the base of natural logarithm, its digits after the decimal point go on forever without showing a simple pattern (just as those of pi), and it has many more interesting properties.

It can be defined in several ways:

- Number *e* is such number for which a function *f(x) = e^x* (so called exponential function) equals its own derivative, i.e. *f(x) = f'(x)*.
- Number *e* is a limit of the infinite series 1/0! + 1/1! + 1/2! + 1/3! + ... (! signifies factorial). I.e. adding all these infinitely many numbers gives exactly *e*.
- Number *e* is a number greater than 1 for which integral of function 1/x from 1 to *e* equals 1.
- Number *e* is the base of natural logarithm, i.e. it is such number *e* for which *log(e,x) = area under the function's curve from 1 to x*.
- ...

*e* to 100 decimal digits is:

2.7182818284590452353602874713526624977572470936999595749669676277240766303535475945713821785251664274...

*e* to 100 binary digits is:

10.101101111110000101010001011000101000101011101101001010100110101010111111011100010101100010000000100...

Just as pi, *e* is a real transcendental number (it is not a root of any polynomial equation) which also means it is an irrational number (it cannot be expressed as a fraction of integers). It is also not known whether *e* is a normal number, which would means its digits would contain all possible finite strings, but it is conjectured to be so. As of 2023 *e* has been evaluated by computers to 35000000000000 digits.

A simple approximation of *e* is for example the fraction 1359/500.

## See Also

- pi
- golden ratio