Math is about to get imaginary!

Complex Numbers

Exercise

3234

101010100

Simplify the following square roots:

12)1

32)2

45)3

1000)4

24216

5359

Consider the quadratic equations: x2-1 = 0 and x2+1= 0

Solve the equations using square roots.

Notice something weird?

Let’s look at their graphs to see what is going on…f(x) = x2 - 1 f(x) = x2 + 1

How many x-intercepts does this graph have? What are they?

How many x-intercepts does this graph have? What are they?

Imaginary Numbers

Simplify imaginary numbers

Ex: 4i7i 28i2 28 1 Remember i

2 1

28

Ex# 2: 8 5

i 8 i 5 Remember that 1 i

i2 40 12 10

2 10

Ex# 3: i19

i19 i18 i i18 i i2 9

i

i2 9i 1 9 i

Answer: -i

Complex Numbers: A little real, A little imaginary…

A complex number has the form a + bi, where a and b are real numbers.

a + bi

Real part Imaginary part

Adding/Subtracting Complex NumbersWhen adding or subtracting complex numbers,

combine like terms.

Ex: 8 3i 2 5i 8 2 3i 5i

10 2i

Try these on your own

1. (16 4i) (12 3i)

2. (3i 7) (34 6i)

3. ( 4 7i) (8 9i)

4. ( 1 i) ( 3 4i)

ANSWERS:

1. 4 i

2. 3i 41

3. 4 2i

4. 2 + 3i

Multiplying Complex NumbersTo multiply complex numbers, you use the same

procedure as multiplying polynomials.

3 12

36 1 36

6i

Lets do another example.

3 2i 5 3i

15 9i 10i 6i2

F O I L

15 9i 10i 6 i2 1Next

Answer:

Now try these:

21-i

1. 5 20

2. 4 5i 4 5i 3. (3 2i)2

Next

Answers:

1. 10i

2. 41

3. 5 12i

Now it’s your turn!

Do Now①What is an imaginary number?

①What is i7 equal to?

②Simplify:① √-32 *√2② (5 + 2i)(5 – 2i)

The Conjugate

Let z = a + bi be a complex number. Then, the conjugate of z is a – bi

Why are conjugates so helpful? Let’s find out!

= a2 + abi – abi –(bi)2

The Conjugate

What happens when we multiply conjugates

(a + bi)(a – bi)F O I L

= a2 – (bi)2

= a2 – b2i2 = a2 – b2(-1) = a2 + b2

Lets do an example:

Ex: 8i

1 3i

8i

1 3i1 3i

1 3iRationalize using the conjugate

Next

8i 24i2

1 98i 24

10

4i 12

5Reduce the fraction

Lets do another example

Ex: 4 i

2i4 i

2i

i

i

4i i2

2i2

Next

4i i2

2i2 4i 1

2

Try these problems.

1. 3

2 5i

2. 3 - i2 - i

1. 2 5i

9

2. 7 i

5

So why are we learning all this complex numbers stuff anyway?

Remember when we looked at this the other day??????f(x) = x2 - 1 f(x) = x2 + 1

How many x-intercepts does this graph have? What are they?

How many x-intercepts does this graph have? What are they?

Quadratic FormulaDo we remember it?

2

2

4ab

a

bx

c

• What does it do?

It solves quadratic equations!

Discriminant: The expression under the radical in the quadratic formula that allows you to determine how many solutions you will have before solving it.

Using the Discriminant

2

2

4ab

a

bx

c

Quadratic Equations can have two, one, or no solutions.

Discriminant

Why is knowing the discriminant important?

Find the discriminant of the functions below:

Put the functions into your graphing calculator:

Do you notice something about the discriminant and the graph?

452 xxy 442 xxy 742 xxy

Properties of the Discriminant

2 4 0b ac 2 Solutions

2 4 0b ac 1 Solutions

2 4 0b ac No Solutions

Discriminant is a positive number

Discriminant is zero

Discriminant is a negative number

2) 3 5 1a x x Ex. 1 Find the number of solutions of the following.

23 5 1 0x x 5

1

3

c

b

a

2 (3( 5 )4 )) ( 1

25 1237 0

2 solutions

2.) 2 3b x x

2

3

1

c

b

a

2 4(1( 2) 3))(

2 2 3 0x x

4 128

NO solutions

0

2.) 4 4 1c x x

4

1

4

c

b

a

2 4(4( 4) 1))(

24 4 1 0x x

16 160

1solution

0

Now it’s your turn!

Exit Slip!

① Simplify: (-4 + 2i) (3-9i)

① What is the conjugate of 2 – 3i?

② What type and how many solutions does the equations x2 + 2x + 5 =0 have?

③ What are the solution(s) to the equation in #3?