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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