Complex Numbers in Polar Form

Pre-Calculus · Axiom Academy

LESSON Complex Numbers in Polar Form One point, two descriptions: trade the parts a + bi for a reach and an angle. 1. A Point, Two Ways to Describe It Plot on the complex plane: go 1 right (the real part) and up (the imaginary part). But that same point also has a reach from the origin and an angle off the positive real axis. Watch both descriptions appear for the one point. The reach is the modulus r = |z| , and the angle is the argument . Drop the point to the real axis and you get a right triangle with legs a and b — so r is just its hypotenuse, and is the angle at the origin. modulus = hypotenuse (Pythagoras) argument = angle from the positive real axis 3. Polar Form — and Why It Earns Its Keep Reading and straight off the triangle and substituting into a + bi gives the polar form of any complex number. Written compactly as , where "cis" is shorthand for " plus ". Here is the payoff. To multiply two numbers in polar form you multiply the reaches and add the angles — multiplication becomes a rotation plus a scaling. Watch get multiplied by : same reach, swung 90° further around. multiply the reaches, add the angles Adding angles spins the result around the origin. Multiplying reaches stretches or shrinks it. (Multiplying by i , reach 1 , only rotates.) You can now read a complex number as a reach and an angle — and you know why that view makes multiplication easy. Scroll up to revisit any step.

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