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# Contents

## Idea

The irrational number conventionally denoted $e$ (a notation credited to Euler, hence also called the Euler number) is the base of the natural logarithm; it is approximately 2.71828182845 in decimal notation.

## Definition

There are numerous ways of defining $e$. One is

$e \coloneqq \sum_{n \geq 0} \frac1{n!} = 1 + 1 + \frac1{2!} + \frac1{3!} + \ldots.$

Perhaps more important than the constant $e$ is the standard exponential function (defined for all complex numbers $x$)

$\exp(x) = \sum_{n \geq 0} \frac{x^n}{n!}$

for which $e = \exp(1)$. This exponential function is especially convenient because it is uniquely characterized as a function $f(x)$ equal to its own derivative such that $f(0) = 1$ (necessary in order that it satisfy the exponential law $f(x + y) = f(x)f(y)$).

## Lay geometric description

Construct a polar coordinate grid (consisting of radial lines through a point called the origin, and concentric circles centered at the origin). Draw a curve starting at any point except the origin in such a way that at each of its points $p$, the tangent at $p$ meets the radial line at $p$ in a 45 degree angle. This curve is called a logarithmic spiral. Then, following the trajectory of the spiral inward (towards the origin, so to speak) through one radian from $p$ to a second point $q$, the distance from $p$ to the origin differs to the distance from $q$ to the origin by a factor of $e$.

Equivalently, imagine four ants situated at the corners of a square, and imagine that at some instant each begins crawling toward its neighbor looking clockwise from above, each at the same speed. The trajectory of each ant is a logarithmic spiral as described above, and the same description of $e$ applies.

• Wikipedia

• Eli Maor, e: The Story of a Number, Princeton University Press (1994). ISBN 0-691-05854-7.