multiplying complex numbers rotation
Meet the complex number a+bi, and the first question is what on earth i is. Move complex numbers off the number line and onto a plane, though, and suddenly they make sense. The horizontal axis holds the real numbers and the vertical axis the imaginary ones, the ones carrying an i. Then a+bi is simply the point at (a, b). The real magic is in multiplication. On the real line, multiplying means stretching; on the complex plane, it means rotating while stretching. Multiply by i in particular and the point turns exactly 90 degrees. This one fact becomes the language of AC circuits, signal waves, and everything that rotates. In this lesson you see complex numbers on the plane and turn them by hand to see why multiplication is rotation.
First, seat a complex number on the plane. Grab the gold point and drag it anywhere: how far it goes horizontally is the real part a, and how far it goes vertically is the imaginary part b. The dashed green and orange lines dropped from the point to the two axes mark those values, which also appear below. Upper right means a and b are both positive; lower left means both are negative. One complex number is one point on the plane, and one point on the plane is one complex number. The real numbers are the special case living only on the horizontal axis, and i is one step up from the origin, the direction of the vertical axis. This single step, seeing a complex number as a position on the plane rather than as a number, is the key to everything that follows.
Now see what multiplying by i does. Place a point and press the "Times i" button: the point turns exactly 90 degrees counterclockwise about the origin. Its length stays the same; only its direction turns by 90 degrees. Start at 1 and you go 1 → i → −1 → −i, and then back to 1. The four positions are marked on the dashed circle, and only the one where the point sits lights up in gold. Four presses bring you back where you started, because multiplying by i is a 90-degree turn and four of them make 360 degrees, one full turn. Multiplying by i twice is 180 degrees, and that is exactly i² = −1. The question of why −1 should be the square of i turns into a picture: half a turn lands you on the opposite side. Nothing in this lesson shows more cleanly than i that multiplication is rotation.
It is not just i: multiplying by any complex number rotates and scales. Drag the green arrow and the orange arrow, and their product is drawn as the gold arrow. The rule takes exactly two lines. Angles add, and lengths multiply. The two lines below show the current angles added together and the current lengths multiplied. For example, if the first is at 30 degrees with length 2 and the second at 40 degrees with length 1.5, the product is at 70 degrees with length 3. Multiplying means turning one arrow further by the other's angle and stretching it further by the other's length. Since i has length 1 and angle 90 degrees, multiplying by i keeps the length (×1) and only adds 90 degrees to the angle. The 90-degree turn from before was one case of this general rule. Do not memorize multiplication; see it as a turn and a stretch.
If angle and length are the real stars of multiplication, it is better to write complex numbers in those terms from the start. That is the polar form. Drag the point and two numbers follow it: the distance r from the origin to the point, and the angle θ measured from the horizontal axis. In the figure, the radius of the dashed circle is that distance, and the gold arc next to the origin is that angle. The same point is called r∠θ instead of a+bi, and the two forms convert through a=r cos θ and b=r sin θ, whose values appear in the bottom line. The reason to switch is that multiplication becomes very easy in polar form. r₁∠θ₁ times r₂∠θ₂ is simply (r₁r₂)∠(θ₁+θ₂). The rule from before, lengths multiply and angles add, becomes a one-line formula. For problems where rotation matters, r∠θ is far more convenient than a+bi. The eiθ of the next lesson packs exactly this polar form into a single symbol.
Finally, pick up two more things: the conjugate and the magnitude. Drag the gold point and a blue point, flipped top to bottom with the horizontal axis as a mirror, moves with it; that is the conjugate z̄. The conjugate of a+bi is a−bi, with only the sign of the imaginary part changed. In the picture, it is a mirror reflection across the real axis. The distance from the origin to the point is the magnitude |z|, the radius of the dashed circle, and by the Pythagorean theorem |z|=√(a²+b²). The conjugate and the magnitude come as a pair. Multiply z by z̄ and the angles θ and −θ add to 0, while the lengths give |z|·|z|, so z·z̄=|z|². The rotations cancel exactly, leaving only the square of the length. You can read that value directly in the bottom line, and this |z|² reappears unchanged when you measure AC power or signal strength later on.