Numbers & Functions · 8 Numbers That Shouldn't Exist

8.5 Making it rigorous: polar form, and why angles add

That claim is the heart of everything that follows, so let’s prove it. We’ve been writing a complex number by its across-and-up address, a + bi. But a point in a plane has a second address: how far it is from the origin, and which direction it points. Call the distance r and the direction θ. This is the polar form.

real imaginary r a + bi b a θ Figure 8.3. Polar form: the two addresses of one point. The across-and-up address (a, b) and the distance-and-direction address (r, θ) describe the same point, linked by a = r cos θ and b = r sin θ.

Numbers That Shouldn’t Exist Converting between the two addresses is pure Chapter 6. A point at distance r and angle θ has across-coordinate r cos θ and up-coordinate r sin θ — because cosine and sine are the coordinates of the unit-circle walker, scaled to radius r. So: a + bi = r cos θ + i r sin θ = r(cos θ + i sin θ).

Every complex number is (some distance) × (some direction).

♦Why it’s true — skip on a first read, but the real brain-training The fact we’re after is in the box below: multiplying complex numbers multiplies their distances and adds their angles. Take one at distance r, angle α, another at distance s, angle β. Pull the distances out front (r × s) and multiply the two direction-pieces, keeping i2 = −1: (cos α + i sin α)(cos β + i sin β) = (cos α cos β −sin α sin β) + i(sin α cos β + cos α sin β).

Now look at those two grouped pieces. We have seen them before — they are exactly the right-hand sides of the Chapter 6 addition formulas: cos α cos β−sin α sin β = cos(α+β), sin α cos β+cos α sin β = sin(α+β).

So the whole product collapses to rs cos(α+β)+i sin(α+β)
— a complex number at distance rs and angle α + β. (Notice we did not use Euler’s formula to get here — we’ll need that independence in a moment.) When you multiply two complex numbers, their distances multiply and their angles add.

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