Polar Multiplication Example

Complex Analysis · Axiom Academy

Multiply two complex numbers in polar form by multiplying the moduli and adding the arguments. Calculate the product , leaving your answer in polar form. In the complex plane, (blue) and (orange). Their product (purple) has length and points at — straight up the imaginary axis. Nice work — you multiplied two complex numbers in polar form without a single FOIL or i^2 . Multiply the moduli: the new length is . Add the arguments: the new angle is . Result: , which is the pure-imaginary number 6i . That is the whole appeal of polar form: multiplication becomes one product and one sum, and the angles literally add — here landing the answer straight up the imaginary axis.

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