Review of Complex Numbers

Abstract

Complex numbers can be written as z = a + bi, where a and b are real numbers, and i = √ −1. This form, a + bi, is called the standard form of a complex number. When graphing these, we can represent them on a coordinate plane called the complex plane. It is a lot like the x-y-plane, but the horizontal axis represents the real coordinate of the number, and the vertical axis represents the imaginary coordinate. Examples Graph each of the following numbers on the complex plane: 2 + 3i, −1 + 4i, −3 − 2i, 4, −i (Graph sketched in class) The absolute value of a complex number is its distance from the origin. If z = a + bi, then |z| = |a + bi| = a 2 + b 2 Example Find | − 1 + 4i|. | − 1 + 4i| = √ 1 + 16 = √ 17 2 Trigonometric Form of a Complex Number The trigonometric form of a complex number z = a + bi is z = r(cos θ + i sin θ), where r = |a + bi| is the modulus of z, and tan θ = b a. θ is called the argument of z. Normally, we will require 0 ≤ θ < 2π. Examples 1. Write the following complex numbers in trigonometric form: (a) −4 + 4i To write the number in trigonometric form, we need r and θ. r = √ 16 + 16 = √ 32 = 4 √ 2 tan θ = 4 −4 = −1 θ = 3π 4 , since we need an angle in quadrant II (we can see this by graphing the complex number). Then, −4 + 4i = 4 √ 2 cos 3π 4 + i sin 3π 4