Polar Form Introduction

Complex Analysis · Axiom Academy

A second way to name a complex number — not by how far across and up, but by how far out and at what angle. The same point, named two different ways So far every complex number has been an address on a grid: a + bi means "go a across, then b up." But a point in the plane has another, equally honest description — how far it sits from the origin , and which direction you'd point to reach it. That distance-and-angle description is the polar form , and watching one turn into the other is the whole idea. Watch z = 2 + 2i first as a grid address — 2 across, 2 up — and then watch those two legs fold into a single arrow: its length is the modulus r = |z|, and its tilt off the positive real axis is the argument θ. Same point, new name: z = r(cos θ + i sin θ). Nothing about the point moved — only the way we describe it. r is how far out; θ is which way. Drag the point — watch r and θ fall out of a and b Drag the dot anywhere in the plane. You're setting the rectangular coordinates a and b directly — and the polar pair is forced on you: the modulus r = √(a² + b²) is just the distance to the origin, and the argument θ = atan2(b, a) is the angle the arrow makes. Move it to a flat spot, straight up, into the corners — and read both names at once. Two names, one point: a + bi tells you the grid address; r and θ tell you the distance and direction. The conversion is forced — you never get to pick both. Set r and θ — watch a + bi appear

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