Euler’s Identity

eiπ + 1 = 0

Move the angle. See the rotation. Reveal the identity.

Walk the unit circle

Begin at 1. Change the angle θ and watch eiθ rotate. At θ = π, the point reaches −1. Add 1, and the result is 0.

Interactive complex unit circle A point and radius vector rotate counterclockwise according to the selected angle. 1 −1 i −i 0

Use the arrow keys for fine movement. Page Up and Page Down move faster.

Angle
0.000 rad (0)
Real part
1.000
Imaginary part
0.000
Complex value
1.000 + 0.000i
Magnitude
1.000

What is happening?

Start at 1 on the real axis. Move the slider or choose a special angle.

ei·0 = 1

Evidence boundary

The rotation and identity are proven mathematics. Philosophical or spiritual meanings may be contemplated separately, but they are not mathematical or empirical proof.