nLab
e

Contents

Contents

Idea

The irrational number conventionally denoted ee (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 ee. One is

e n01n!=1+1+12!+13!+.e \coloneqq \sum_{n \geq 0} \frac1{n!} = 1 + 1 + \frac1{2!} + \frac1{3!} + \ldots.

This can be interpreted as the groupoid cardinality for core(FinSet)core(FinSet). Perhaps more important than the constant ee is the standard exponential function (defined for all complex numbers xx)

exp(x)= n0x nn!\exp(x) = \sum_{n \geq 0} \frac{x^n}{n!}

for which e=exp(1)e = \exp(1). This exponential function is especially convenient because it is uniquely characterized as a function f(x)f(x) equal to its own derivative such that f(0)=1f(0) = 1 (necessary in order that it satisfy the exponential law f(x+y)=f(x)f(y)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 pp, the tangent at pp meets the radial line at pp 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 pp to a second point qq, the distance from pp to the origin differs to the distance from qq to the origin by a factor of ee.

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 ee applies.

Irrationality

It is a simple matter to show that ee is irrational. For if on the contrary we have e=p/qe = p/q, then en!e \cdot n! would be an integer for any nqn \geq q. However,

en!=integer+1n+1+1(n+1)(n+2)+e \cdot n! = integer + \frac1{n+1} + \frac1{(n+1)(n+2)} + \ldots

where the nonzero tail after the integer part is bounded above by k=1 1/(n+1) k=1/n<1\sum_{k=1}^\infty 1/(n+1)^k = 1/n \lt 1 for n>1n \gt 1, giving a contradiction.

It is harder to show that ee is transcendental. An online proof (written up by David Richeson) may be found here.

References

  • Wikipedia

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

Last revised on September 2, 2019 at 19:20:54. See the history of this page for a list of all contributions to it.