1. Appendix A7 Complex Numbers
Honors Pre-Calculus
Appendix A7
Complex Numbers

2. Objectives Add, Subtract, Multiply, and Divide Complex Numbers
Graph Complex Numbers
Solve Quadratic Equations in the Complex Number System

3. Complex Numbers

4. Complex Numbers
𝑥 2 =−1 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 𝒊, whose square is -1; that is, 𝒊 𝟐 =−𝟏

5. Complex Numbers
Complex numbers are numbers of the form 𝒂+𝒃𝒊 where 𝒂 and 𝒃 are real numbers. The real number 𝒂 is called the real part of the number 𝒂+𝒃𝒊; the real number 𝒃 is called the imaginary part of 𝒂+𝒃𝒊.
Examples:
𝟑+𝟐𝒊
3 is the real part, 2 is the imaginary part.
𝟕.𝟐+𝝅𝒊
7.2 is the real part, 𝜋 is the imaginary part.

6. Comparing, Adding and Subtracting Complex Numbers
We can only compare complex numbers in terms of equality.
𝑎+𝑏𝑖=𝑐+𝑑𝑖 is true if and only if 𝑎=𝑐, and 𝑏=𝑑
Sum of Complex Numbers
𝑎+𝑏𝑖 + 𝑐+𝑑𝑖 = 𝑎+𝑐 + 𝑏+𝑑 𝑖
Difference of Complex Numbers
𝑎+𝑏𝑖 − 𝑐+𝑑𝑖 = 𝑎−𝑐 + 𝑏−𝑑 𝑖

7. Comparing, Adding and Subtracting Complex Numbers
If 7+𝑥𝑖=𝑦+2𝑖 then 𝑦=7, and 𝑥=2
If 3𝑥+4𝑖=12+2𝑦𝑖 then:
𝑥=4, 𝑦=2
Adding
3+2𝑖 + 4+3𝑖 =7+5𝑖
2+5𝑖 +(4+2𝑖)
=6+7𝑖

8. Comparing, Adding and Subtracting Complex Number (continued)
3+5𝑖 −(2+2𝑖)
(1+3𝑖)
2+5𝑖 − 1+5𝑖 1
2+4𝑖 − 2−2𝑖 6𝑖

9. Multiplying Complex Numbers
𝑎+𝑏𝑖 𝑐+𝑑𝑖 = 𝑎𝑐−𝑏𝑑 + 𝑎𝑑+𝑏𝑐 𝑖 Proof: 𝑎+𝑏𝑖 𝑐+𝑑𝑖 =𝑎 𝑐+𝑑𝑖 +𝑏𝑖(𝑐+𝑑𝑖) =𝑎𝑐+𝑎𝑑𝑖+𝑏𝑐𝑖+𝑏𝑑 𝑖 2 =𝑎𝑐+𝑏𝑑 −1 + 𝑎𝑑+𝑏𝑐 𝑖 = 𝑎𝑐−𝑏𝑑 + 𝑎𝑑+𝑏𝑐 𝑖

10. Multiplying Complex Numbers (continued)
Examples:
5+2𝑖 2+3𝑖
10+15𝑖+4𝑖+6 𝑖 2 =4+19𝑖
(5−2𝑖)(2+3𝑖)
10+15𝑖−4𝑖−6 𝑖 2 =16+11𝑖
2+𝑖 3+2𝑖
6+4𝑖+3𝑖+2 𝑖 2 =4+7𝑖
2+3𝑖 2−3𝑖
4−6𝑖+6𝑖−9 𝑖 2 =4+9=13

11. Complex Conjugate
If 𝒛=𝒂+𝒃𝒊 is a complex number, then its conjugate, denoted by 𝑧 is defined as 𝒛 =𝒂−𝒃𝒊 The product of a complex number and its conjugate is a nonnegative number. That is, if 𝒛=𝒂+𝒃𝒊, then 𝒛 𝒛 = 𝒂+𝒃𝒊 𝒂−𝒃𝒊 = 𝒂 𝟐 + 𝒃 𝟐

12. Complex Conjugate (continued)
Examples:
If 𝒛=𝟐+𝟑𝒊 its complex conjugate is
𝒛 =𝟐−𝟑𝒊
𝒛 𝒛 = 𝟐+𝟑𝒊 𝟐−𝟑𝒊 = 𝟐 𝟐 + 𝟑 𝟐 =𝟏𝟑
If 𝒛=𝟏−𝟐𝒊 its complex conjugate is
𝒛 =𝟏+𝟐𝒊
𝒛 𝒛 = 𝟏−𝟐𝒊 𝟏+𝟐𝒊 = 𝟏 𝟐 + 𝟐 𝟐 =𝟓
If 𝒛=𝟐−𝒊 its complex conjugate is
𝒛 =𝟐+𝒊
𝒛 𝒛 = 𝟐−𝒊 𝟐+𝒊 = 𝟐 𝟐 + 𝟏 𝟐 =𝟓

13. Properties of Conjugates
( 𝑧) =𝑧
𝑧+𝑤 = 𝑧 + 𝑤
𝑧∙𝑤 = 𝑧 ∙ 𝑤

14. Writing the Reciprocal of a Complex Number
1 2+3𝑖 1 2+3𝑖 ∙ 2−3𝑖 2−3𝑖 = 1(2−3𝑖) (2+3𝑖)(2−3𝑖) = 2−3𝑖 4+9 = 2 13 − 3 13 𝑖 1 4−5𝑖 1 4−5𝑖 4+5𝑖 4+5𝑖 = 4+5𝑖 = 𝑖

15. Writing the Quotient of a Complex Number
3+2𝑖 2+3𝑖 3+2𝑖 2+3𝑖 ∙ 2−3𝑖 2−3𝑖 = 6+4𝑖−9𝑖 = 12−5𝑖 13 = − 5 13 𝑖

16. Writing the Quotient of a Complex Number
2+3𝑖 4−5𝑖 (2+3𝑖) (4−5𝑖) (4+5𝑖) (4+5𝑖) = 8+12𝑖+10𝑖− = −7+22𝑖 41 =− 𝑖

17. Powers of 𝑖

18. Evaluating Powers of 𝑖
𝑖 37 = 𝑖 36 ∙𝑖= 𝑖 4 9 ∙𝑖= 1 9 ∙𝑖=𝑖 𝑖 111 = ( 𝑖 4 ) 108 ∙ 𝑖 3 = ∙ 𝑖 2 ∙𝑖=−𝑖 𝑖 23 = ( 𝑖 2 ) 11 ∙𝑖= −1 11 ∙𝑖=−𝑖

19. Evaluating Powers of a Complex Number
(2+3𝑖) 3 (𝑥+𝑎) 3 = 𝑥 3 +3𝑎 𝑥 2 +3 𝑎 2 𝑥+ 𝑎 3 (2+3𝑖) 3 = 𝑖 𝑖 2 2+ (3𝑖) 3 =8+36𝑖−54+27 −𝑖 =−46+9𝑖 (3+2𝑖) 2 (𝑥+𝑎) 2 = 𝑥 2 +2𝑎𝑥+ 𝑎 2 (3+2𝑖) 2 = ∙2𝑖∙3+ (2𝑖) 2 =9+12𝑖−4=5+12𝑖

20. Homework
Pg A