Honors Pre-Calculus

Appendix A7Complex Numbers

Objectives

• Add, Subtract, Multiply, and Divide Complex Numbers

• Graph Complex Numbers• Solve Quadratic Equations in the Complex

Number System

Complex Numbers

Complex Numbers

• does not have any real solutions because when any number is multiplied by itself we get a positive number

• To remedy this situation we can introduce a number, called the imaginary unit, which we will denote by , whose square is -1; that is,

Complex Numbers

• Complex numbers are numbers of the form where and are real numbers. The real number is called the real part of the number ; the real number is called the imaginary part of .

• Examples:•

3 is the real part, 2 is the imaginary part.•

7.2 is the real part, is the imaginary part.

Comparing, Adding and Subtracting Complex Numbers

• We can only compare complex numbers in terms of equality.

• is true if and only if , and

• Sum of Complex Numbers

• Difference of Complex Numbers

Comparing, Adding and Subtracting Complex Numbers

• ComparingIf then , and

• If then:

• Adding

Comparing, Adding and Subtracting Complex Number (continued)

• Subtracting:

Multiplying Complex Numbers

Proof:

Multiplying Complex Numbers (continued)

• Examples:

Complex Conjugate

If is a complex number, then its conjugate, denoted by is defined as

The product of a complex number and its conjugate is a nonnegative number. That is, if , then

Complex Conjugate (continued)• Examples:If its complex conjugate is

If its complex conjugate is

If its complex conjugate is

Properties of Conjugates

• =

Writing the Reciprocal of a Complex Number

Writing the Quotient of a Complex Number

Writing the Quotient of a Complex Number

Powers of

1

2

3 2

4 2 2

1

( 1)( 1) 1

i i

i

i i i i

i i i

5 4

6 4 2

7 4 3

8 4 4

1

1

i i i i

i i i

i i i i

i i i

Evaluating Powers of

Evaluating Powers of a Complex Number

Homework

• Pg A67 9-46