Euler's formula e^{iθ} explained
In the last lesson we saw that multiplying complex numbers is rotation, and that when rotation matters the polar form r∠θ is convenient. Yet the notation ∠θ feels like a makeshift. The real formula is something else: eiθ. Putting an imaginary number into an exponential sounds absurd at first, but the result is remarkably clean. eiθ is the point on the unit circle at angle θ, nothing more, and it equals cos θ + i sin θ. Why an exponential? Because exponentials add their exponents when multiplied, so the rule that rotation angles add becomes a single line of exponent law. This eiθ is the heart of the phasors that electrical engineering uses for AC, and the language of damped oscillations and signals. In this lesson you turn eiθ around the unit circle yourself and see why it is the most natural way to write rotation.
First, see what eiθ is. Grab the gold point and turn it around the circle: the point always stays on the circle of radius 1, the unit circle. The angle θ fixes its position. How far it goes horizontally is cos θ and how far it goes vertically is sin θ; the dashed green and orange lines dropped to the two axes mark those lengths, and their values appear below. So eiθ = cos θ + i sin θ, which is Euler's formula. There is nothing to memorize: remember the picture, eiθ is the point on the unit circle at angle θ. Set θ to 0 and the point is 1 on the right; at 90 degrees it is i at the top; at 180 degrees it is −1 on the left. The r∠θ of the last lesson with r=1 is exactly this eiθ, and a general complex number multiplies in its size and is written r·eiθ.
Now let the angle flow with time. Put ωt in place of θ and you get eiωt: an arrow that goes around the circle at a steady speed ω as time passes. That is a phasor. Press "Play" and the arrow spins; the speed buttons change how fast it turns. The key here is the arrow's shadow. Its shadow on the real axis, Re(eiωt) = cos ωt, becomes the green dot that swings back and forth along the horizontal axis. That is exactly the waveform of an AC voltage or a signal. The 60Hz AC at a wall outlet and a radio signal are both, underneath, the shadow of a spinning arrow like this. That is why electrical engineering handles AC with one rotating arrow instead of messy sines and cosines. Rotating is easier to work with than shaking.
Here it becomes clear why the exponential is the natural notation for rotation. Drag the two points eiθ₁ and eiθ₂ on the unit circle, and the gold arrow shows where their product lands. Apply the most basic property of exponentials, that multiplying adds the exponents, and you get eiθ₁·eiθ₂ = ei(θ₁+θ₂). The product is the point at angle θ₁+θ₂. The rule we checked by hand in the last lesson, that multiplying adds angles, falls out here as one line of exponent law, with nothing to prove. This property, which turns multiplication into the addition of angles, is the real power of the eiθ notation. No need to memorize the trigonometric addition formulas; just add the exponents.
Now real engineering begins. So far the rotation was pure, so the size stayed fixed at 1. Add a real part to the exponent, though, and you get e(σ+iω)t. Here ω is still the turning speed, and the newcomer σ is the rate at which the size changes, because the size goes as eσt. Drag σ along the track below and the gold curve changes shape. When σ is 0, the size does not change and the point simply circles. When σ is negative, the size shrinks and the curve winds inward in a spiral: an oscillation that dies out, a decay. When σ is positive, the spiral spreads outward: an oscillation that keeps growing. A bell that rings and fades, the transient response of a circuit, and whether a suspension bridge is stable or unstable are all decided by the sign of this σ. e(σ+iω)t packs rotation ω and decay σ into a single exponent.
Finally, gather all of this into one point. Drag the point until θ is exactly half a turn, π, or press the "Send θ to π" button. eiπ is the point half a turn around the unit circle, which is −1. When the point reaches it, that mark on the circle lights up in gold. So eiπ = −1, and adding 1 to both sides gives eiπ + 1 = 0. This is Euler's identity, often called the most beautiful equation in mathematics. It is beautiful because Euler's number e, the imaginary unit i, the circle constant π, and 1 and 0, five constants from entirely different places, sit together in one line without a single extra symbol. For us, though, it is not a mystical incantation but simply a picture. Turn half a turn and you get −1; that is all. Once you see eiθ as rotation, the equation that looked hardest becomes the most natural single step.