2 Euler's Identity

By combining three infinite series, Euler revealed a profound connection between the exponential function and trigonometry.

The Taylor series for the exponential, sine, and cosine:

ex=1+x+x22!+x33!+x44!+⋯e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots
cos⁡x=1−x22!+x44!−⋯\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots
sin⁡x=x−x33!+x55!−⋯\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots

Substitute ix into the exponential series:

eix=1+ix−x22!−ix33!+x44!+⋯e^{ix} = 1 + ix - \frac{x^2}{2!} - \frac{ix^3}{3!} + \frac{x^4}{4!} + \cdots

Separate real and imaginary parts:

eix=cos⁡x+isin⁡xe^{ix} = \cos x + i\sin x

Set x = pi:

∴
  eiπ+1=0  \boxed{\;e^{i\pi} + 1 = 0\;}

Five fundamental constants (e, i, pi, 1, 0) and three basic operations (addition, multiplication, exponentiation) united in a single equation.