Example 4

Write the complex number z = -2 + i2 in polar form.

The absolute value of z is

|z | = (2² + (-2)²)^1/2 = √8.

To find the argument θ, we use

tanθ = 2/(-2) = -1.

Since z is in the second quadrant (x negative and y positive), θ must be 3π/4. Thus

cis θ is helpful in computing products, quotients, and powers of complex numbers. Using the addition formulas for sines and cosines, we can prove the product formula

(2)

(r cis θ) · (s cis φ) = rs cis (θ + φ).

In words, this formula states: To multiply two complex numbers, multiply the absolute values and add the arguments. There is a similar formula for quotients:

(3)

To divide two complex numbers, divide the absolute values and subtract the arguments.