The Most Beautiful Equation

Complex Analysis · Axiom Academy

One short equation ties together the five most important numbers in mathematics — and a single trip around a circle shows you why it's true. Mathematics grew up in separate rooms: e came out of calculus and continuous growth, i out of algebra's attempt to solve x^2 = -1 , out of the geometry of circles, and 1 and 0 out of plain arithmetic. Euler's identity, , gathers all five into a single true statement — no fudge factors, no extra numbers. Watch the engine behind it. The point rides the unit circle, and as the angle grows from 0 it sweeps halfway around. The play head leaves 1 on the right and travels until, at , it lands exactly on -1 . Halfway around the circle is exactly half a turn — and half a turn lands you on the opposite side, at -1 . That is all is saying. Where the point comes from: cos and sin Euler's formula says — the horizontal coordinate is , the vertical coordinate is . Drag the handle to set and watch the point trace the circle. Stop it at and read off -1 + 0i . Notice the point never leaves the circle — its distance from the center stays exactly 1 at every angle, because . Send the angle to and the identity falls out Push the angle toward (half a turn). When it lands, substitute into Euler's formula and the algebra writes itself: , , so . Add 1 to both sides and you have the identity. The circle is the proof: half a turn is exactly the trip from 1 to -1 , so can only be -1 .

This is the written version of the interactive lesson above. See the full Complex Analysis course.