Beginning Algebra, 11th Edition

Simplifying Square Roots of Negative Numbers

Write each number as a multiple of i.

EXAMPLE 1

SECTION 9.4 Complex Numbers 575

(a) (b)

= 2 i

=i# 2

=i 25 =i 24

2 - 5 2 - 4 (c)

=i# 2 # 22 ,or2i 22

=i# 24 # 22

=i 28

2 - 8

NOW TRY

Numbers that are nonzero multiples of iare pure imaginary numbers.The

complex numbersinclude all real numbers and all imaginary numbers.

CAUTION It is easy to mistake for with the iunder the radical. For

this reason, it is customary to write the factor ifirst when it is multiplied by a radical.

For example, we usually write i 22 rather than 22 i.

22 i 22 i,

NOW TRY

EXERCISE 1

Write as a multiple
of i.

2 - 12

NOW TRY ANSWER

1. 2 i 23

Complex Number

A complex numberis a number of the form where aand bare real num-

bers. If a= 0 and bZ0,then the number biis a pure imaginary number.

abi,

For example, the real number 2 is a complex number, since it can be written as

Also, the pure imaginary number is a complex number.

and Other complex numbers

In the complex number ais called the real partand bis called the imaginary

part.* A complex number written in the form (or ) is in standard

form.FIGURE 1shows the relationships among the various types of numbers. (Com-

pare this figure with FIGURE 7in Section 1.4.)

a+bi a+ib

a+bi,

3 - 2 i, 1 +i 22 , - 5 + 4 i

2 + 0 i. 3 i= 0 + 3 i

*Some texts refer to bias the imaginary part.

Pure

imaginary

numbers

4 i

i √ 7

–11i

i

Complex numbers

6 + 4i 3 – i √ 2 8 – i

Rational numbers

Real numbers

Irrational numbers

Integers

..., –3, –2, – 1

Whole

numbers

Natural

numbers

0

1, 2, 3,...

• 8
  15
  23

4

π

π

4

-^1
5
–3^2

• 0.125 1.5 0.18

9

4

7

11

√ 4

FIGURE 1