Entry task- Solve two different ways

4.8Complex Numbers

Target: I can identify and perform operations with

complex numbers

-In the set of real numbers, negative numbers do not have square roots.

-Imaginary numbers were invented so that negative numbers would have square roots and certain equations would have solutions.

-These numbers were devised using an imaginary unit named i.

1i

Imaginary numbers:

i 1

i2 1

i is not a variable it is a symbol for a specific

number

With your a/b partner determine the values for the cycle of i

i

-i

1

i

1

-1 -1 -1

i

-i

1

1

Definition of Imaginary Numbers

Any number in form a+bi, where a and b are

real numbers and i is imaginary unit.

Definition of Pure imaginary numbers:

Any positive real number b,

where i is the imaginary unit and bi is called the pure

imaginary number.

b2 b2 1 bi

Simplify the expression.

1. 81 81 1 9i

4. 8i 3i 24i2 241

Remember i2 1

Simplify each expression.

24

5. 5 20 i 5i 20Remember that 1 i

i2 100 110Remember i

2 1

10

When adding or subtracting complex numbers, combine like terms.

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

10 2i

8 7i 1211i

8 12 7i 11i

418i

Simplify.

Simplify.

9 12 6i 2i

3 8i

9 6i 122i

Multiplying complex numbers.

To multiply complex numbers, you use the

same procedure as multiplying polynomials.

Simplify.

8 5i 2 3i

16 24i 10i 15i2F O I L

16 14i 15 31 14i

Simplify.

3018i 10i 6i2F O I L

3028i 6 2428i

62i 5 3i

-Express these numbers in terms of i.

1.) 5 1*5 1 5 5i

2.) 7 1*7 1 7 7i

3.) 99 1*99 1 99

9*11i

3 11i

Conjugates

In order to simplify a fractional complex number, use a conjugate.

What is a conjugate?

a b c d and a b c d

are said to be conjugates of each other.

Ex: 3 2i 5 and 3 2i 5

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

MULTIPLYING COMPLEX NUMBERS

2

1. 4( 2 3 )

2. ( )( 3 )

3. (2 )(4 3 )

4. (3 2 )(3 2 )

5. (3 2 )

Multiply

i

i i

i i

i i

i

1. 4( 2 3 ) 8 12i i

ANSWERS

2. ( )( 3 )i i 23 3( 1) 3i

3. (2 )(4 3 )i i 28 6 4 3i i i 28 2 3i i

(-1)

11 2i

4. (3 2 )(3 2 )i i (-1)

1325. (3 2 ) (3 2 )(3 2 )i i i

29 6 6 4i i i =9 12 4( 1)i

5 12i

249 i

Use the quadratic formula to solve the following:

22. 3 2 5 0x x a=3, b= -2, c=5

22 ( 2) 4(3)(5)x

2(3)

6

4

14

2 2 14i

6

1 14i

3

6042 x6

562 x

Let’s Review

You need to be able to:– 1) Recognize what i, i2, i3 ect. is equal to (slide 5)– 2) Simplify Complex numbers– 3) Combine like terms (add or subtract)– 4) Multiply (FOIL) complex numbers– 5) Divide (multiply by complex conjugates)

Assignment Pg.253 -254

Homework – #9-43 odds, skip 13,15,17

Challenge - 70