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

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

Complex Numbers

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

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.

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

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

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

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

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

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

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

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

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 16+25 = 4 41 + 5 41 𝑖

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

Writing the Quotient of a Complex Number 2+3𝑖 4−5𝑖 (2+3𝑖) (4−5𝑖) (4+5𝑖) (4+5𝑖) = 8+12𝑖+10𝑖−15 16+25 = −7+22𝑖 41 =− 7 41 + 22 41 𝑖

Powers of 𝑖

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

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 = 2 3 +3 3𝑖 2 2 +3 3𝑖 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 = 3 2 +2∙2𝑖∙3+ (2𝑖) 2 =9+12𝑖−4=5+12𝑖

Homework Pg A67 9-46