Numbers & Functions · 8 Numbers That Shouldn't Exist

8.3 Where do these numbers live? (The complex plane)

Here’s the question that turns i from a clever trick into something beautiful: where does 3 + 2i sit? The real numbers all live on a line. But a complex number has two independent parts, a real part and an imaginary part. Two numbers, two directions. So complex numbers don’t live on a line; they live on a plane. This changes everything. A complex number isn’t a weird hybrid of a “real” thing and an “imaginary” thing — it’s just a point in a plane, as concrete as a dot on a map. And once complex numbers are points in a plane, every arithmetic operation becomes a motion in that plane — and that is where the real magic begins.

Numbers That Shouldn’t Exist real imaginary 3 + 2i 2i the old number line 1 2 3 Figure 8.1. The complex plane. A complex number is just a point: 3 + 2i is 3 across and 2 up. The reals are the horizontal axis (the old number line); the imaginaries are the vertical axis.

8.4 Adding is sliding; multiplying is turning

Addition is the easy one: add the across-parts and up-parts separately, (3 + 2i) + (1 + 4i) = 4 + 6i. On the plane, that’s just sliding, like steps on a map. Multiplication is where the universe opens up.

Multiply two complex numbers like any algebra, remembering the one rule i2 = −1: (a + bi)(c + di) = (ac −bd) + (ad + bc)i.

The formula is forgettable. What it does on the plane is not. Take any complex number and multiply it by i. Try the real number 3: 3 × i = 3i, which is 3 up the vertical axis. Multiplying by i took a point on the rightward axis and rotated it a quarter turn, 90◦counterclockwise. Do it again: 3i × i = 3i2 = −3, now pointing left. Again: −3i, pointing down. Again: back to 3. Four multiplications by i, four quarter-turns, all the way around and home.

Multiplying by i is a 90◦rotation. And multiplying by any complex number is a rotation combined with a scaling.

Multiplication and turning are the same act — the bridge we’ve been building toward since Chapter 2.

real imaginary 3 3i −3 −3i ×i ×i ×i ×i Figure 8.2.

Multiplying by i is a 90◦rotation. Starting from 3, each multiplication by i turns a quarter circle counterclockwise; four turns return home. The number that fixes quadratics is the operation of turning.

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