Euler's Formula

Complex Analysis · Axiom Academy

Raising e to an imaginary power lands you on the unit circle — and that single fact ties the exponential to sine and cosine. 1. Is a Point on the Unit Circle Pick an angle (in radians) and the formula hands you a single complex number, . Plotted in the complex plane it isn't scattered anywhere — it lands exactly on the unit circle , at the point you reach by rotating counterclockwise from the positive real axis. As grows, that point sweeps around the circle. The point sits at angle on the circle Its distance from the origin is always 1 Every point in the complex plane is "(real part) (imaginary part)". Drop the point straight down onto the real axis and you read off its horizontal coordinate; slide it across onto the imaginary axis and you read off its vertical one. Those two shadows are exactly and — which is the whole formula. The horizontal shadow of the point. It runs from +1 (at ) to -1 (at ). The vertical shadow. It runs from 0 up to +1 (at ) and back down. is the corner where the two shadows meet — coordinates . is the Pythagorean theorem on that right triangle. The point is at , so . The real shadow is and the imaginary shadow is — and indeed . At the quarter-turns the point lands squarely on an axis, so comes out as a tidy number. Watch it step around: . The standout is the half-turn — at the point is at -1 , giving Euler's identity .

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