Imaginary & Complex Numbers

Once upon a time…

no real solution

-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.

i

 

1

-

The

imaginary numbers

consist of all numbers bi, where b is a real number and i is the imaginary unit, with the property that i² = -1.

-The first four powers of i establish an important pattern and should be memorized.

i

1 

i

Powers of i

i

2   1

i

3  

i i

4  1

i

4  1

i

3  

i i i

2   1

Divide the exponent by 4 No remainder: answer is 1.

remainder of 1: answer is i.

remainder of 2: answer is –1.

remainder of 3:answer is –i.

Powers of i 1.) Find i 23 2.) Find i 2006 3.) Find i 37 4.) Find i 828     1

i



i

 1

Complex Number System Reals Rationals (fractions, decimals) Integers (…, -1, -2, 0, 1, 2, …) Whole (0, 1, 2, …) Natural (1, 2, …) Irrationals (no fractions) pi, e Imaginary i, 2i, -3-7i, etc.

Simplify.

-Express these numbers in terms of i.

 5   1*5   1 5 

i

5   7    1* 7    1 7  

i

7  99   1*99   1 99 

i



You try… 7.

6.

 7 

i

7   36   6

i

8.

 160 

To multiply imaginary numbers or an imaginary number by a real number, it is important first to express the imaginary numbers in terms of i.

Multiplying 9.

 94

i

10.

2

i

  5  2

i

2 11.

    7  

i

3 

i

7  

i

2 5 5   2 5 21  21

Complex Numbers real

a + bi

imaginary The

complex numbers

consist of all sums a + bi, where a and b are real numbers and i is the imaginary unit. The real part is a, and the imaginary part is bi.

Add or Subtract 7.) 7

i

 9

i

 16

i

13.

   (4    3  5

i

   

i i

Multiplying & Dividing Complex Numbers

Part of 7.9 in your book

REMEMBER: i² = -1 1) Multiply 3 4

i

 12

i

2    12 2)   2  7 2 2

i

   49

3) 4) You try…  

i

  11

i

 2    84

i

2

   

2     84 ( 84 121 (  1 )  1 )   121

5) Multiply   4  28 3

i

 8

i

 7   21

i

2 

i

 6

i

2   28  28  29

i

29

i

 6

i

2   28  29 22 29

i i

 6

6)   6 2  

i

 20 3

i

  10

i

3

i

  10

i

    6 6 6    17

i

17 17

i i

16    10

i

 10 10 2   17

i

2

7) You try…  5  7

i

 5  7

i

  25  35

i

 35

i

 49

i

2  25   25  49  74

Conjugate

-The

conjugate

of a + bi is a – bi -The

conjugate

of a – bi is a + bi

8)

Find the conjugate of each number…

3 4

i

3 4

i

9)  

i

10) 5

i

 

i

 5

i

11) 6

Because

6  6 0

i

is the same as 6  0

i

Divide… 12) 5 9

i

1 

i

1 

i

1 

i

 1 

i

2 4

i

 1   

i

2 2 9

i

2 7 2

i

You try… 13) 2 3 

i



i

 

i

9 

i

25

i

2 

i

  6 10

i i i

 15

i

15

i

2  25

i

2  

i

34