Complex Multiplication

A complex number can be drawn as an arrow from the origin. The arrow has a length, called the modulus, and a direction, called the argument, and the plane of all such arrows is the complex plane. Adding two complex numbers is adding the arrows tip to tail, the same as adding any two vectors. Multiplying them is the less obvious operation: the product’s angle is the sum of the two angles, and the product’s length is the product of the two lengths. The rules of complex arithmetic all follow from that.

z: 0.500 + 0.866i; stretch 1.000, turn π/3
w: 0.866 + 0.500i; stretch 1.000, turn π/6
z·w — the product: i; stretch 1.000, turn π/2
the turns add: π/3 + π/6 = π/2
the stretches multiply: 1.000 × 1.000 = 1.000

Both stretches are 1, so the product’s stretch is 1 as well and multiplying comes down to adding the two turns. The product stays on the unit circle.

Drag either arrowhead to change its turn and its stretch. Angles snap to multiples of 15°, and the stretch snaps to 1 near the unit circle. Arrow keys turn z, up and down stretch it; hold shift for w; Home resets. Each number is printed twice: as a + bi, and as a stretch and a turn.

One arrow, two coordinates

e^iθ = cos θ + i sin θan arrow of length 1, turning at a steady rate

The sine and cosine waves of the previous page are the two coordinates of this one turning arrow, each recorded over time.

The previous page gave a point on the unit circle a height (sine) and a horizontal reach (cosine). The same two numbers give an arrow its rectangular form $a + bi$ ; its length and angle give its polar form $r\,e^{i\theta}$ ; and the conversion between them is $a = r\cos\theta$ , $b = r\sin\theta$ . Seen from here, trigonometry is the conversion between two ways of writing down a point in the plane.

Minus times minus

The number $-1$ is the arrow of length 1 pointing backward: its angle is half a turn. Multiplying by it adds half a turn to whatever it multiplies, so multiplying by $-1$ twice turns any arrow through a whole turn and leaves it where it started. That is the content of the rule minus times minus is plus. The instrument has a button that walks through it.

The number $i$ is the arrow of length 1 pointing straight up: a quarter turn. Two quarter turns make a half turn, so $i \cdot i = -1$ . The name “imaginary” dates from a time when it was unclear what such a number could mean; on the plane it is the unit along the second axis, and “the square root of $-1$ ” means a quarter turn, which done twice is a half turn.

Two forms for one number

Rectangular form makes addition simple — add the real parts, add the imaginary parts — and makes multiplication the four-term expansion $(a+bi)(c+di) = (ac - bd) + (ad + bc)i$ , in which the turn and the stretch are hard to see. Polar form makes multiplication simple and addition awkward. Much of the work of using complex numbers is choosing the form in which the operation you need is simple, and converting; each conversion costs a sine and a cosine.

Powers and roots

Turn on powers of z and drag. Each successive power turns by the same angle and multiplies the length by the same factor, so the powers step around the origin by equal angles while their lengths grow or shrink geometrically. With the length set to exactly 1 the powers stay on the unit circle, stepping around it. With the length below 1 they spiral inward, and above 1 outward; the curve they trace is a logarithmic spiral.

Going the other way, the $n$ -th roots of 1 are the $n$ arrows of length 1 spaced evenly around the unit circle, because raising any of them to the $n$ -th power turns through a whole number of turns and lands on 1.

Euler's formula

An arrow of length 1 at angle $\theta$ is written $e^{i\theta}$ . The second panel shows why the exponential is the right notation: as $\theta$ increases the arrow’s rate of change is the arrow itself turned by a quarter turn, that is, multiplied by $i$ — and a quantity whose rate of change is a fixed multiple of itself is an exponential. Its two coordinates are

e^{i\theta} = \cos\theta + i\sin\theta,

so the sine and cosine waves of the previous page are the vertical and horizontal coordinates of one turning arrow. The equation $e^{i\pi} + 1 = 0$ says that the arrow turned through half a turn is $-1$ .

A sum of arrows turning at different rates, each with its own length, traces a periodic curve, and any periodic curve can be built this way; reading the arrows off a given curve is Fourier analysis. The next page takes a single turning arrow and treats its rate of turning as a musical pitch.