Portal:Mathematics/Featured article/2007 12

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e is the unique number such that the slope of of y=ex (blue curve) is exactly 1 when x=0 (illustrated by the red tangent line). For comparison, the curves y=2x (dotted curve) and y=4x (dashed curve) are shown.
The mathematical constant e is occasionally called Euler's number after the Swiss mathematician Leonhard Euler, or Napier's constant in honor of the Scottish mathematician John Napier who introduced logarithms. It is one of the most important numbers in mathematics, alongside the additive and multiplicative identities 0 and 1, the imaginary unit i, and π, the circumference to diameter ratio for any circle. It has a number of equivalent definitions. One is given in the caption of the image to the right, and three more are:
  1. The sum of the infinite series
    \begin{align} e & = \sum_{n = 0}^\infty \frac{1}{n!} \\ & = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots \\ \end{align}
    where n! is the factorial of n.
  2. The global maximum of the function
    f(x) = x^{1 \over x}.
  3. The limit (which arises in the compound-interest problem):
    e = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n

The number e is also the base of the natural logarithm. Since e is transcendental, and therefore irrational, its value can not be given exactly. The numerical value of e truncated to 20 decimal places is 2.71828 18284 59045 23536.

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