7.7

Complex Numbers

Imaginary Numbers

Previously, when we encountered square roots of negative numbers in solving equations, we would say “no real solution” or “not a real number”.

That’s still true. However, we will now introduce a new set of numbers.

Imaginary numbers which includes the imaginary unit i.

Imaginary Numbers

The imaginary unit, written i, is the number whose square is ‒1. That is,

2 1 and 1i i

Write using i notation.

a.

b.

c.

25

32

121

Examples

Multiply or divide as indicated.

a.

b.

3 5

125

5

Example

A complex number is a number that can be written in the form a + bi where a and b are real numbers.

a is a real number and bi would be an imaginary number.

If b = 0, a + bi is a real number.

If a = 0, a + bi is an imaginary number.

Standard Form of Complex Numbers

Adding and Subtracting Complex Numbers

Add or subtract as indicated.

a.

b.

(8 + 2i) – (4i)

(4 + 6i) + (3 – 2i)

Example

To multiply two complex numbers of the form a + bi, we multiply as though they were binomials. Then we use the relationshipi2 = – 1 to simplify.

Multiplying Complex Numbers

Multiply:

a. 8i · 7i b. -4i · 7

c. 3i ·3i ·3i d. 2i ·3i ·i ·5i

Example

Multiply.

a. 5i(4 – 7i) b. 4i(3i + 5)

Example

Multiply.

(6 – 3i)(7 + 4i)

Example

In the previous chapter, when trying to rationalize the denominator of a rational expression containing radicals, we used the conjugate of the denominator.

Similarly, to divide complex numbers, we need to use the complex conjugate of the number we are dividing by.

Complex Conjugate

The complex numbers a + bi and a – bi are called complex conjugates of each other.

(a + bi)(a – bi) = a2 + b2

Complex Conjugate

Divide.

Example

6 2

4 3

i

i

Divide.

Example

5

6i

i

2i

3i4i

5i

6i

7i

8i

Patterns of i

Find each power of i.

a. b.

c. d.

21i

32i

Example

53i

17i