E (mathematical constant)
The constant <math>e</math> (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
- <math>e = \lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n</math>
and can also be written as the infinite series
- <math>e = \sum_{n=0}^\infty {1 \over n!} = {1 \over 0!} + {1 \over 1!}
+ {1 \over 2!} + {1 \over 3!}
+ {1 \over 4!} + \cdots</math>
Here <math>n!</math> stands for the factorial of <math>n</math>.
The number e is relevant because one can show that the exponential function exp(x) can be written as <math>e^x</math>; 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 was the first number to be proved transcendental without having been specifically constructed by Charles Hermite[?] in 1873. It is conjectured to be normal. It features (along with a few other fundamental constants) in Euler's identity:
- <math>e^{i\pi}+1=0</math>
which was described by Richard Feynman as "The most remarkable formula in mathematics"!
The infinite continued fraction expansion of <math>e</math> contains an interesting pattern as follows:
- <math>e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, \ldots]. </math>
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