This combines many of the fundamental numbers with mathematical beauty

The number \(0\), the additive identify

The number \(1\), the multiplicative identity

The number \(\pi\), the ratio between a circle’s circumference and its diameter

The number \(j\), used to find the roots of polynomials defined as \(j=\sqrt{-1}\)

The number \(e\), from continuous compounding interest and from \(\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{e}^x=\mathrm{e}^x\)

Cartesian and Polar coordinates

Euler’s formula traces out a unit circle in the complex plane as a function of \(\varphi\). Here, \(\varphi\) is the angle that a line connecting the origin with a point on the unit circle makes with the positive real axis, measured in radians.

A point in the complex plane can be represented by a complex number written in cartesian coordinates. Euler’s formula lets you convert between cartesian and polar coordinates. The polar form simplifies the mathematics when used in multiplication or powers of complex numbers. [wiki]

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