Polar Representation

Complex Analysis · Axiom Academy

Every complex number is an arrow — a length and a direction. Master converting between rectangular a+bi and polar . 1. A Complex Number Is an Arrow Picture z as an arrow from the origin to the point (a, b) . It has a length r = |z| — the modulus — and it points in a direction — the argument , measured from the positive real axis. Naming the arrow by instead of (a, b) is the polar form. the polar (trigonometric) form r is the length, is the direction Start with the coordinates a and b . They are the two legs of a right triangle whose hypotenuse is the arrow. So r is the hypotenuse (Pythagoras) and is the angle that leg-pair turns through. Everything here comes straight from the right triangle . a is the run along the real axis — how far right (or left) the arrow reaches. b is the rise along the imaginary axis — how far up (or down) it reaches. By Pythagoras, — the length of the arrow. , so — the direction it points. The legs are a = 1 and . Then and . So . Now go the other way. Given the arrow's length r and direction , drop its shadow onto each axis. The shadow on the real axis is ; the shadow on the imaginary axis is . Those two shadows are the rectangular coordinates. Why it works: the arrow, its horizontal shadow, and its vertical shadow form a right triangle. Trigonometry on that triangle gives and ; multiply by r to free a and b . Take r = 2 , back to rectangular: and . We land exactly back on — the conversion is a perfect round trip.

This is the written version of the interactive lesson above. See the full Complex Analysis course.