Definition: The polar form of a complex number writes it as r(cos theta + i sin theta), using its distance r from the origin and its angle theta in the complex plane.
What is the polar form?
Plot a + bi at the point (a, b). Its distance from the origin is the modulus r = square root of (a2 + b2), and its angle is the argument theta. Then a + bi = r(cos theta + i sin theta).
Why use it?
Powers become easy with De Moivre’s theorem: zn = rn(cos n theta + i sin n theta).
Worked example: z = 1 + √3 i. Then r = square root of (1 + 3) = 2 and tan theta = √3, so theta = 60 degrees. To find z5, take r5 = 32 and angle 300 degrees. Then z5 = 32(cos 300 + i sin 300) = 32(1/2 – (√3/2)i) = 16 – 16√3 i.
What do students slip on?
Raising only the angle or only the modulus to the power. Mathbot’s reminder: the modulus gets the exponent, the angle gets multiplied by it.