SearchOnMath

Euler's identity, \(e^{i\pi}+1=0\), relates the imaginary unit and two famous irrational numbers in a way that is so simple it has been called beautiful.

A definition of the exponential function that has found application in numerous approximation schemes is \(e^x = \lim_{y\to\infty} \left(1+\frac{x}{y}\right)^y\).

The \(k\)th Fibonacci number can be written in closed form as \(F(k) = \frac{\varphi^k - (-\frac{1}{\varphi})^k}{\sqrt{5}}\), where \(\varphi\) is the celebrated golden ratio.