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 =
0.