Drill #81:Solve each equation or inequality

24.3

15.2

15510.1

x

xx

xx

Drill #83:Simplify each expression

)93()41(.5)46()23(.4

)5(4.3.2.1 432517

ii

iiii

5-9 Complex Numbers

Objective: To simplify square roots containing negative numbers and to add, subtract, and multiply complex numbers

(1.) i

Definition: i is called the imaginary unit.

What is the value of I squared?

1i

(2.) Pure Imaginary Numbers

Definition: For any positive number b,

where i is the imaginary unit, and bi is called a pure imaginary number.

Example:

bibb 122

i39

Evaluating the Square Root of Negative Numbers*

To find the square root of negative numbers:

1. First separate the negative

2. Evaluate each root separately and multiply818

228

2281

i

i

More Examples

Ex1.

Ex2.

12

327x

Powers of I *

1

1

1

4

3

2

i

ii

i

i

Finding Powers of i*

Powers of i are cyclical. They repeat after

To find : 1. Divide n by 4 and keep only the

remainder r2. , where r is the remainder of n/4 Note:

4i

ni

10 irn ii

Example (Powers of i)

Find the following:

Ex1. Ex2. Ex3.

)4(6 73387 iiii

(3.) Complex Numbers

Definition: A number in the form of

a + biwhere a and b are real numbers and i is the

imaginary unit.

a is called the real part.b is called the imaginary part.

Adding Complex Numbers*

To add complex numbers:1. add the real parts together (this is the real

part of sum)2. add imaginary parts together (this is the

imaginary part of the solution).

(a + bi) + (c + di) = (a + c) + (b + d)i

Ex: (5 + 6i) + (2 + 3i) = 7 + 9i

Adding Complex Numbers

Examples:Ex1.(2 + 9i) + (3 + 4i)Ex2.(5 + 6i) - (2 + 3i)Ex3.

)2413()85(

Multiplying Complex Numbers*

Definition: To multiply imaginary numbers you need to FOIL.

(a + bi)(c + di) = = ac (first) + adi (outside) + bci (inside) + bd

(last)= (ac - bd) + (ad + bd)i

Ex: (2 + 3i)(4 + 5i) = 2(4) + 2(5i) + 3i(4) + 3i(5i)= 8 + 10i + 12i – 15 = -7 + 22i

2i

Multiplying Complex Numbers

Ex1.(2 + 3i)( 1 + 4i)Ex2.(6 + 2i)( 3 – 2i)Ex3.(3 – 5i)(3 + 5i)

(4.) Complex Conjugates

Definition: Numbers of the forma + bi and a – bi are called complex

conjugates.

The product of complex conjugates is:

Example: 3 + 2i and 3 – 2i are complex conj.

2222 )())(( babiabiabia

The Product of Complex Conjugates*

)34)(34(2

)27)(27(1

iiex

iiex

Dividing by Complex Numbers*Rationalizing Complex Denominators

To rationalize a complex denominator you need to multiply the numerator and denominator by the complex conjugate.

Example: Simplify

1369

)3()2(96

3232

323

22

2 iiiii

ii

ii323

Solving 2nd Equations* (no 1st degree term)

To solve equations: 1. Isolate the square term.2. Take the (+/-) square root of both sides.

Example:

6

6

6

06

2

2

2

ix

x

x

x

(5.) Equal Complex Numbers

Definition: a + bi = c + diif and only if a = c and b = d.

The real parts must be equal and the imaginary parts must be equal.

Equal Complex Numbers*

Find values of x and y for which each equation is true:

iyixex

iyixex

8652.2

181264.1