e^{iθ} Is Rotation on the Unit Circle

First, let's see what e^iθ even is. Grab the point and spin it around the circle. The point always sits on the circle of radius 1 — the unit circle. The angle θ is where the point is. How far it went horizontally is cos θ; how far vertically is sin θ. So e^iθ = cos θ + i sin θ — that's Euler's formula. Nothing to memorize. Just remember the picture: "e^iθ is the point on the unit circle at angle θ." Set θ to 0 and the point is 1 (right); at 90° it's i (up); at 180° it's −1 (left). Last lesson's r∠θ with r = 1 is exactly this e^iθ, and a general complex number is just that scaled by r, written r·e^iθ.

Now let the angle flow with time. Put ωt in place of θ and you get e^iωt: an arrow circling at a steady rate ω as time runs. This is the phasor. Press play. The arrow spins round and round. The key here is the arrow's shadow. Its shadow cast on the real axis — Re(e^iωt) = cos ωt — slides back and forth, oscillating. That, right there, is the waveform of an AC voltage or a signal. The 60 Hz alternating current in your wall socket, a radio signal — under the hood they are the shadow of an arrow turning like this. This is exactly why electrical engineering handles AC with a single spinning arrow instead of fussy sines and cosines: turning is easier to work with than shaking.

Why the exponential is the natural notation for rotation becomes clear right here. Drag the two points e^iθ₁ and e^iθ₂ on the unit circle. Where does their product land? Apply the most basic property of exponentials, "multiplying adds the exponents," and you get e^iθ₁·e^iθ₂ = e^{i(θ₁+θ₂)}. The product is the point at angle θ₁+θ₂. The "multiplying adds angles" rule you checked by hand last lesson now comes out, with nothing to prove, as a single line of exponent algebra. This power to turn rotation into addition is the real strength of the e^iθ notation. No need to memorize the awkward trig addition formulas — you just add the exponents.

Now the real engineering begins. So far it's been pure rotation, magnitude fixed at 1. But add a real part to the exponent and you get e^(σ+iω)t. Here ω is still the rate of turning; the newcomer σ is the rate at which the size changes — because the magnitude is e^σt. Drag σ and watch. When σ is 0 the size never changes, so it just goes round the circle. When σ is negative the size shrinks steadily, a spiral winding inward — a damped oscillation, ringing as it dies away. When σ is positive it's a spiral spreading outward, an oscillation that keeps growing. A struck bell ringing and fading, a circuit's transient response, whether a wobbling bridge is stable or not — all of it is decided by the sign of this σ. Packing rotation (ω) and damping (σ) into one exponent is exactly what e^(σ+iω)t does.

Finally, let's gather all of this into a single point. Drag the point so θ lands exactly on π — a half turn. You can also press the "send θ to π" button. e^iπ is the point half a turn around the unit circle, which is −1. So e^iπ = −1, and adding 1 to both sides gives e^iπ + 1 = 0. This is Euler's identity, often called the most beautiful equation in mathematics. Why beautiful? Because the natural constant e, the imaginary unit i, the circle constant π, and 1 and 0 — five constants from utterly different places — gather in one line with not a scrap to spare. But to us this isn't a mystical incantation; it's just a picture. "A half turn lands on −1," that's all it is. Once you have the eye to see e^iθ as rotation, the equation that looked hardest becomes the most obvious single step.