Multiplying Complex Numbers Rotates and Scales
A complex number a+bi — at first you think, "what even is i?" But move a complex number off the number line and onto a point in the plane, and suddenly it makes sense. Horizontal axis: the real part. Vertical axis: the imaginary part, the one carrying i. Then a+bi is just a point sitting at (a, b). And here's the real magic: multiplication. On the real line, multiplying was "stretching." On the complex plane, multiplying is "rotating while stretching." Multiply by i in particular and you turn exactly 90 degrees. This one fact becomes the language of alternating current, of waves, of everything that spins. In this lesson you'll see complex numbers as points on a plane and spin them yourself to feel why multiplication is rotation.
First, let's seat a complex number on the plane. Grab the point and drag it anywhere. How far it went horizontally is the real part a; how far vertically is the imaginary part b. Dashed lines drop to the two axes and read off a and b for you. Upper right: a and b both positive. Lower left: both negative. One complex number is one point on the plane, and one point on the plane is one complex number. The real numbers are just the special case that live on the horizontal axis. Read i as one step up — the vertical direction. This single step, seeing a complex number not as a "number" but as a "position on a plane," is the key that unlocks everything else.
Now watch what multiplying by i does. Drop a point and press the "times i" button. Each press turns the point exactly 90 degrees counterclockwise about the origin. The length stays the same; only the direction turns by 90. Start at 1 and you get 1 → i → −1 → −i, then back to 1. Four presses and you're home — multiplying by i is a 90-degree turn, and four of them make 360, one full circle. Multiply by i twice and that's 180 degrees, which is exactly i² = −1. "Why is −1 the square of i?" turns into the picture "a half turn lands you on the opposite side." The one-line claim of this lesson, that multiplication is rotation, is shown most cleanly by i itself.
It's not just i. Multiplying by any complex number rotates and scales. Drag the two arrows z₁ and z₂ around. Their product z₁z₂ is drawn along with them. The rule is just two lines: angles add, lengths multiply. If z₁ is at 30° with length 2, and z₂ is at 40° with length 1.5, the product is at 70° with length 3. So "to multiply" means "turn one arrow further by the other's angle, and stretch it by the other's length." Since i has length 1 and angle 90°, multiplying by i keeps the length (×1) and adds 90° to the angle. The 90-degree turn from the last block was one case of this general rule all along. You don't memorize multiplication; you watch it as a turn-and-stretch.
If angle and length are the real stars of multiplication, we may as well write a complex number with those two from the start. That's the polar form. Drag the point and two numbers follow: the distance r from the origin to the point, and the angle θ measured from the horizontal axis. The same point you called a+bi is now called r∠θ. They convert by a = r cos θ, b = r sin θ. Why bother switching? Because multiplication becomes effortless in polar form. r₁∠θ₁ times r₂∠θ₂ is just (r₁r₂)∠(θ₁+θ₂). Lengths multiply, angles add — that rule from before collapses into one line. Whenever rotation is what matters, r∠θ beats a+bi by a mile. And the next lesson's eiθ is exactly this polar form compressed into a single symbol.
Finally, two more to pocket: the conjugate and the magnitude. Drag the point z, and a point flipped across the horizontal axis moves with it. That's the conjugate z̄. The conjugate of a+bi is a−bi — only the sign of the imaginary part flips. In the picture, it's a mirror reflection across the real axis. And the distance from the origin to z is the magnitude |z|, the radius of the dashed circle. By Pythagoras, |z| = √(a² + b²). Conjugate and magnitude are a pair: multiply z by z̄ and the angles are θ and −θ, which add to 0, while the lengths are |z|·|z| — so z·z̄ = |z|². The rotation cancels exactly, leaving pure length squared. This |z|² shows up later, unchanged, when you measure AC power or signal strength.