MATHEMATICAL BEAUTY

Euler's Identity

One of mathematics' most remarkable equations brings together five fundamental constants: 0, 1, p, e and i. Explore why they appear together.

e^ip + 1 = 0

Angle

Real Part

-1.000

Imaginary Part

0.000

Magnitude

1.000

Euler's Formula

The key to Euler's identity is Euler's formula. It describes a beautiful connection between exponential functions and rotations in the complex plane.

e^i? = cos(?) + i sin(?)

The slider controls the angle ?. As the angle increases, the point moves around the unit circle. Its horizontal position is cos(?), while its vertical position is sin(?).

Now set the angle to p radians. The point reaches the far left of the circle:

e^ip = cos(p) + i sin(p) = -1

Adding 1 to both sides gives the famous identity.

e^ip + 1 = 0

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