The

Naperian or natural base to logarithms; Naperian does not refer to

Napier having created it, in fact his concept of a

logarithm is different than the modern concept, but he is credited as creating logarithms, hence this extremely useful number is identified with him. e is approximately 2.718281828459045 to fifteen digits, and can be expressed in many ways:

it is the

limit of the slowly converging function

/ 1 \
lim | 1 + - | ^ n
n->oo \ n /

or the more quickly converging

series
oo
---
\ 1
| ---
/ n!
---
n=0

or as the number satisfying d/du (e^u) = (e^u).

Euler gave this number the letter it is associated with; some say it is because e is the first letter of his name, but Euler said that all others were taken up to e. Euler discovered the relationship e^(

i*

theta) = cos(theta) + i*sin(theta), which is one of the most important formulas in complex analysis, and is responsible for the amazing relationship between the five most important numbers in mathematics: e^(

i*

pi) +

1 =

.