Lesson P.6 (pg:60)

In the beginning there were counting numbers (Natural Numbers)

And then we needed Integers

1

2

In the beginning there were counting numbers

And then we needed integers

1

2-1

-3

In the beginning there were counting numbers

And then we needed integers

And rationals

1

2-1

-3

0.41

In the beginning there were counting numbers

And then we needed integers

And rationals

And irrationals1

2-1

-3

0.41

2

In the beginning there were counting numbers

And then we needed integers

And rationals

And irrationals

And reals1

2-1

-3

0.41

2

0

1

2

8

9

By definition

Consider powers if i

10

21 1i i

2

3 2

4 2 2

5 4

1

1 1 1

1

...

i

i i i i

i i i

i i i i i

It's any number you can imagine

7

6

5

4

3

2

i

i

i

i

i

i

ii

7

6

5

4

3

22 11

i

i

i

i

i

i

ii

7

6

5

4

3

22

111

1

11

i

i

i

i

iii

i

ii

1

1

161284

151173

141062

13951

iiii

iiiii

iiii

iiiii

15

Now we can handle quantities that occasionally show up in mathematical solutions

What about

16

1a a i a

49 18

Combine real numbers with imaginary numbers◦ a + bi

Examples

17

Real partImaginary

part

3 4i

36

2i

4.5 2 6i

We have always used them. 6 is not just 6 it is 6 + 0i. Complex numbers incorporate all numbers.

1

2-1

-3

0.41

2

3 + 4i2i

0

A number such as 3i is a purely imaginary number

A number such as 6 is a purely real number

a + ib is the general form of a complex number

6 + 3i is a complex number

-2 + 7i is also a complex number

Note:

If x + iy = 6 – 4i

then x = 6 and y = -4

The ‘real part’ of 6 – 4i is 6

The ‘imaginary part’ of 6 – 4i is -4

20

3i = 0 + 3i

-7= -7 + 0i

0= 0 + 0i

Write these complex numbers in standard form a + bi

1. 2.

3. 4.

22

9 75 16 7

5 144 100

Complex numbers can be combined with ◦ Addition

◦ Subtraction

◦ Multiplication

◦ Division

23

3 8 2i i

9 12 7 15i i

2 4 4 3i i

3

5 2

i

i

Example 1:

24

2 4 4 3i i

Example 2:

25

3 8 2i i

Example 3:

(Use FOIL method)

26

9 12 7 15i i

We need to know Conjugate of a+bi= a-bi

(Conjugate of a complex number is obtained

by changing the sign of the imaginary part)

For example:

27

28

To solve

Multiply and divide by

its conjugate

29

Qs. Simplify

The trick is to make the denominator real:

2

3 7i

6. Simplify

The trick is to make the denominator real:

2

3 7i

2 3 7 2(3 7)

3 7 3 7 58

(3 7)

29

7 3

29

i i

i i

i

i

Example 4:

Division technique◦ Multiply numerator and denominator by the

conjugate of the denominator

32

3

5 2

i

i

2

2

3 5 2

5 2 5 2

15 6

25 4

6 15 6 15

29 29 29

i i

i i

i i

i

ii

3

5 2

i

i

1. Simplify4

Evaluate : 3 4i i

3. Simplify3 4i i

6. Simplify2

3 7i

)(23

2

1166

2

166

2

52366

0136.7 2

Conjugatessolutionscomplexix

x

x

x

xxSolve

Use the correct principles to simplify the following:

39

3 121

4 81 4 81

2

3 144

Lesson P.6

Page - 65

Exercises 1 – 32, 37, 38,

41, 42, 55-62

40