E

We define the irrational transcendental real number $e$ to be the number such that $$\displaystyle\int_1^e \dfrac{1}{t} \mathrm{d}t=1.$$ By the definition of the logarithm, we have $\log(e)=1$. The value of $e$ is $$e=2.71828182846\ldots.$$

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?