Euler's identity: e^(iπ) + 1 = 0

Raising e to an imaginary power does not blow up. It rotates. Turn by half a circle and you land exactly on −1, which is what e^iπ + 1 = 0 says.

Why it is cool

Five constants that were discovered for unrelated reasons — 0, 1, e from growth, π from circles, i from solving x² = −1 — sit in one short equation. The equation is not a coincidence and not a trick of notation. It falls out of asking what compound growth does when the growth rate points sideways. The same fact is why every wave, every alternating current and every quantum phase can be written as a rotating exponential instead of separate sines and cosines.

The mechanism

First, what e is. Grow something by 100% in n tiny instalments, each adding a fraction 1/n of what you have. Multiply out (1 + 1/n)ⁿ and as n grows the answer settles on 2.718…, which we call e. So e^x means: apply growth x in many tiny steps, each multiplying by (1 + x/n).

Next, what multiplying by i does. Draw complex numbers as points on a plane, with the real part across and the imaginary part up. Multiplying a point by i turns it a quarter turn anticlockwise: 1 goes to i, i goes to −1. Multiplying by a small imaginary number i·θ/n therefore nudges a point sideways, at right angles to where it is.

Now put them together. e^iθ means: start at 1 and apply n tiny steps, each multiplying by (1 + iθ/n). Each step pushes the point sideways by a fraction θ/n of its distance from the centre. A push that is always at right angles to your position never changes your distance — it only changes your direction. So the point stays on the unit circle and walks around it. The total angle walked is θ, because each of the n steps turns by about θ/n.

Set θ = π. Half a turn around the unit circle from 1 lands on −1. So e^iπ = −1, and adding 1 gives 0.

In one breath

e^x is growth in tiny multiplicative steps. Make the step imaginary and each step turns instead of stretching. Turning by π radians from 1 reaches −1. That is the whole identity.