Adding, Subtracting and Multiplying Complex Numbers

Addition and subtraction of complex numbers are performed as if the complex numbers were algebraic expressions with i being a variable.

Example 1

Addition and subtraction of complex numbers

Find the sums and differences.

a) (2 + 3i) + (6 + i)

b) (-2 + 3i) + (-2 - 5i )

c) (3 + 5i) - (1 + 2i)

d) (-2 - 3i) - (1 - i )

Solution

a) (2 + 3i) + (6 + i) = 8 + 4i

b) (-2 + 3i) + (-2 - 5i ) = -4 - 2i

c) (3 + 5i) - (1 + 2i) = 2 + 3i

d) (-2 - 3i) - (1 - i ) = -3 -2i

We can give a symbolic definition of addition and subtraction as follows.

Addition and Subtraction of Complex Numbers

The sum and difference of a + bi and c + di are defined as follows:

(a + bi) + (c + di) = (a + c) + (b + d)i

(a + bi) - (c + di) = (a - c) + (b - d)i

Complex numbers are multiplied as if they were algebraic expressions. Whenever i² appears, we replace it by -1.

Example 2

Products of complex numbers

Find each product.

a) 2i(1 + i)

b) (2 + 3i)(4 + 5i)

c) (3 + i)(3 - i)

Solution

b) Use the FOIL method to find the product:

c) This product is the product of a sum and a difference.

We can find powers of i using the fact that i² = -1. For example, i³ = i² · i = -1 · i = -i.

The value of i⁴ is found from the value of i³:

i⁴ = i³ · i = -i · i = -i² = 1.

In the next example we find more powers of imaginary numbers.

Example 3

Powers of imaginary numbers

Write each expression in the form a + bi.

a) (2i)²

b) (-2i)²

c) i⁶

Solution

a) (2i)² = 2² · i² = 4(-1) = -4

b) (2i)2 (2)2  i2 4i2 4(1) 4

c) i⁶ = i² · i⁴ = -1 · 1 = -1

For completeness we give the following symbolic definition of multiplication of complex numbers. However, it is simpler to find products as we did in Examples 2 and 3 than to use this definition.

Multiplication of Complex Numbers

The complex numbers a + bi and c + di are multiplied as follows:

(a + bi)(c + di) = (ac - bd) + (ad + bc)i