Is $e$ arbitrary? If not, how is it derived?

Question

Probably a stupid question but where did the constant $e$ come from? How did it come about? How is it derived mathematically other than $e^{i\cdot \pi} = -1$? What exactly does natural growth mean? Or is Euler's constant arbitrary?

Answer 1

Some definitions of $e$:

• The number such that $\int_1^e \frac{\mathrm d x}{x}=1$.
• $\sum_{n=0}^\infty \frac{1}{n!}$.
• $\lim\limits_{n \to \infty} \left(1+ \frac{1}{n}\right)^n$

Interesting topic is to prove that all those definitions lead to the same real number.

Answer 2

The number $e$ is the unique $a>0$ such that the function $f(x)=a^x$ has derivative $f'(x)=a^x=f(x).$