Section 2.1

Complex Numbers

The Imaginary Unit i

2

The Imaginary Unit

The imaginary unit is defined as

= -1, where 1.

i

i

i i

Complex Numbers and Imaginary Numbers

The set of all numbers in the form

a+b

with real numbers a and b, and i, the imaginary unit,

is called the set of complex numbers. The real number

a is called the r

i

eal part and the real number b is called

the imaginary part of the complex number a+b . If b 0,

then the complex number is called an imaginary number.

An imaginary number in the form b is called a p

i

i

ure

imaginary number.

Equity of Complex Numbers

a+b =c+d if and only if a=c and b=d.i i

Example

Express as a multiple of i:

2

16

7i

Operations with

Complex Numbers

Adding and Subtracting Complex Numbers

1. a+b d = a+c b+d

This says that you add complex numbers by adding their real

parts, adding their imaginary parts, and expressing the sum as

a complex number.

2

i c i i

. a+b c+d a-c -d

This says that you subtract complex numbers by subtracting

their real parts, subtracting their imaginary parts, and

expressing the difference as a complex number.

i i b i

Example

Perform the indicated operation:

7 4 9 5

8 3 17 7

i i

i i

Multiplication of complex numbers is

performed the same way as multiplication

of polynomials, using the distributive

property and the FOIL method.

Example

Perform the indicated operation:

3 5 6 2i i

Complex Conjugates

and Division

2 2

Conjugate of a Complex Number

The complex conjugate of the number a+bi is a-bi,

and the complex conjugate of - is . The

multiplication of complex conjugates gives a real

number.

a bi a bi

a bi a bi a b

a bi

2 2a bi a b

Example:

Find the product: (2 + 7i)(2 – 7i)

Using complex conjugates to divide complex numbers

Example

Divide and express the result in standard form:

7 6

5 9

i

i

Example

Divide and express the result in standard form:

2 3

4 5

i

i

Roots of Negative Numbers

Because the product rule for radicals only

applies to real numbers, multiplying radicands

is incorrect. When performing operations

with square roots of negative numbers, begin

by expressing all square roots in terms of i.

Then perform the indicated operation.

Principal Square Root of a Negative Number

For any positive real number b, the principal square

root of the negative number -b is defined by

-b i b

Example

Perform the indicated operations and write the

result in standard form:

54 7 24

Example

Perform the indicated operations and write the

result in standard form:

2

4 7

Example

Perform the indicated operations and write the

result in standard form:

8 48

4

(a)

(b)

(c)

(d)

Find the product.

9 5 3i i

45 27

45 27

45 27

45 27

i

i

i

i

(a)

(b)

(c)

(d)

20 3 45

Perform the indicated operation.

2 5 9

11 5

5 5

7 5

i

i