Unit 5 - Objective 2 - Basic Operations with Complex Numbers
Quick Review

We can add, subtract, multiply, or divide complex numbers.
Combine like terms when you add or subtract.

Example 1

If a+bj and c+dj are complex numbers then their sum is defined as:

(a+bj)+(c+dj)=(a+c)+(b+d)j

 1. (9 + 2j) + (8 + 6j) = (9 + 8) + (2 + 6)j = 17 + 8j 2. (6 + 3j) + (5 - 7j) = (6 + 5) + (3 - 7)j = 11 - 4j

Subtraction of complex numbers

If a+bj and c+dj are complex numbers, then their difference is defined as:

(a + bj) - (c + dj) = (a - c) + (b - d)j

(distribute as before a+bj-c-dj)

Example 2
 1. (3 + 4j) - (2 + j) = (3 - 2) + (4 - 1)j = 1 + 3j 2. (-8 + 4j) - (3 + 10j) = (-8 - 3) + (4 - 10)j = -11 - 6j

Multiplication of complex numbers

If a+bj and c+dj are any two complex numbers, then their product is defined as multiplying like binominals (FOIL) except replace j^2 with -1:

Example 3
 1. (2 + 5j) (3 - 4j) = (2) (3) + 2(-4)j + 5j(3) + 5(-4)j2 = 6 - 8j + 15j - 20(-1) = 6 - 8j + 15j + 20 = 26 + 7j 2. (5 + 3j)2 = (5 + 3j) (5 + 3j) = (5) (5) + 5(3j) + 3j(5) + (3j) (3j) = 25 + 15j + 15j + 9j2 = 25 + 15j + 15j + 9(-1) = 25 + 15j + 15j - 9 = 16 + 30j

Division of complex numbers

If a+bj and c+dj are complex numbers, then the quotient (a+bj)/(c+dj) is obtained by multiplying the numerator and denominator by the conjugate.

Example 4