E

The number $e$ can be defined in the following way: let $f$ be the unique solution of the initial value problem $$y'=y;y(0)=1,$$ then $e=f(1)$.

Properties

Theorem: The folllowing formula holds: $$e=\displaystyle\lim_{k \rightarrow \infty} \left( 1 + \dfrac{1}{k} \right)^k,$$ where $e$ denotes Euler's constant.

Proof: █

Theorem: The real number $e$ is irrational.

Proof: proof goes here █

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 4.1.16

Who proved $e$ is irrational?