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| The constant ''e'' ( |
The constant ''e'' (occasionally called ''Napier's constant'' in honor of the Scottish mathematician ] who introduced logarithms) is the base of the ]. It is approximately equal to | ||
| : ''e'' = 2.71828 18284 59045 23536 02874 ..... | : ''e'' = 2.71828 18284 59045 23536 02874 ..... | ||
| It is equal to exp(1) where exp is the ] and therefore it is the ] of (1 + 1/''n'')<sup>''n''</sup> as ''n'' goes to infinity and can | It is equal to exp(1) where exp is the ] and therefore it is the ] of (1 + 1/''n'')<sup>''n''</sup> as ''n'' goes to infinity and can | ||
Revision as of 13:49, 17 September 2002
The constant 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 (1 + 1/n) as n goes to infinity and can also be written as the infinite series
- e = 1 / 0! + 1 / 1! + 1 / 2! + 1 / 3! + 1 / 4! + ...
Here n! stands for the factorial of n.
The infinite continued fraction expansion of e contains an interesting pattern as follows:
- e = .
The number e is relevant because one can show that the exponential function exp(x) can be written as e; 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 the most remarkable formula in the world.