Essential Question: What are the two complex numbers that have a square of -1? Chapter 4: Polynomial...

Essential Question: What are the two complex numbers that have a square

of -1?

Chapter 4: Polynomial and Rational Functions

4.5: Complex Numbers

4.5: Complex NumbersProperties of the Complex Number System

The complex number system contains all real numbersAddition, subtraction, multiplication, and division of complex

numbers obey the same rules of arithmetic that hold in the real number system with one exception:The exponent laws hold for integer exponents, but not necessarily for

fractional onesWe don’t need to worry about this for now, I just needed to list the

exceptionThe complex number system contains a number, denoted i, such

that i2 = -1Every complex number can be written in the standard form:

a + bia + bi = c + di if and only if a = c and b = d

Numbers of the form bi, where b is a real number, are called imaginary numbers. Sums of real and imaginary numbers, numbers of the form a + bi, are called complex numbers

4.5: Complex NumbersExample #1: Equaling Two Complex

NumbersFind x and y if 2x – 3i = -6 + 4yiThe real number parts are going to be equal

2x = -6x = -3

The imaginary number parts are going to be equal-3i = 4yi-3/4 = y

4.5: Complex NumbersExample #2: Adding, Subtracting, and

Multiplying Complex Numbersa)(1 + i) + (3 – 7i)

Combine like terms4 – 6i

b)(4 + 3i) – (8 – 6i)Distribute, then combine terms4 + 3i – 8 + 6i = -4 + 9i

c)4i(2 + ½ i)Distribute and simplify8i + 2i2 = 8i + 2(-1) = -2 + 8i

d)(2 + i)(3 – 4i)FOIL and simplify6 – 8i + 3i – 4i2 = 6 – 8i + 3i – 4(-1) = 6 – 8i + 3i + 4 =

10 – 5i

4.5: Complex NumbersExample #3: Products and Powers of

Complex Numbersa)(3 + 2i)(3 – 2i)

FOIL9 – 6i + 6i – 4i2 = 9 – 4(-1) = 9 + 4 = 13

b)(4 + i)2

16 + 4i + 4i + i2 = 16 + 4i + 4i + (-1) = 15 + 8i

4.5: Complex NumbersPowers of i

i1 = ii2 = -1i3 = i2 • i = -1 • i = -ii4 = i2 • i2 = -1 • -1 = 1i5 = i4 • i = 1 • i = iAnd we keep repeating from there…

Example #4: Powers of iFind i54

The remainder when 54 / 4 is 2, so i54 = i2 = -1

4.5: Complex NumbersComplex Conjugates

The conjugate of the complex number a + bi is the number a – bi, and the conjugate of a – bi is a + bi

Conjugates multiplied together yield a2 + b2

(a – bi)(a + bi) = a2 + abi – abi – b2i2 = a2 – b2(-1) = a2 + b2

The conjugate is used to eliminate the i from the complex number, and is used to remove the use of i in the denominator of fractions

4.5: Complex NumbersExample #5: Quotients of Two Complex

NumbersSimplify

multiply top & bottom by the conjugate of the denominator

3 4

1 2

i

i

2

2

3 4 1 2 3 6 4 8

1 2 1 2 1 2 2 43 2 8( 1)

1 4( 1)

3 2 8

1 411 2

511 2

5 5

i i i i i

i i i i ii

i

i

i

4.5: Complex NumbersAssignment

Page 300Problems 1-35 & 55-57, odd problems

Show work where necessary (e.g. FOILing, converting to i)

Due tomorrow

Essential Question: What are the two complex numbers that have a square

of -1?

Chapter 4: Polynomial and Rational Functions

4.5: Complex Numbers (Part 2)

4.5: Complex NumbersSquare Roots of Negative Numbers

Because i2 = -1, In general,

Take the i out of the square root, then simplify from there

Example #6: Square Roots of Negative Numbers

1i b i b

) 3 3

1 7 1 7 1 7)

3 3 3 3

a i

ib i

2

) (7 4)(5 9) (7 2 )(5 3 )

35 21 10 6

35 11 6( 1)

35 11 6

41 11

c i i

i i i

i

i

i

4.5: Complex NumbersComplex Solutions to a Quadratic Equation

Find all solutions to 2x2 + x + 3 = 0

22

2, 1, 3

(1) (1) 4(2)(3)4

2 2(2)

1 1 24

4

1 23

4

1 23

4

1 23

4 4

a b c

b b ac

a

i

i

4.5: Complex NumbersZeros of Unity

Find all solutions of x3 = 1Rewrite equation as x3 - 1 = 0Use graphing calculator to find the real roots (1)Factor that out(x – 1)(x2 + x + 1) = 0x = 1 or x2 + x + 1 = 0

22 (1) (1) 4(1)(1)4

2 2(1)

1 1 4 1 3

2 2

1 3 1 3

2 2 2

b b ac

a

ii

4.5: Complex NumbersAssignment

Page 300Problems 37-71 (odd) (skip 55/57, you did that last

night)Due tomorrowYou must show work