Complex Number Theory. First look at the number line 6543210 2 + 3 = 5.

Complex Number

Theory

First look at the number line

6543210

2 + 3 = 5

Now for multiplication

6543210

2 x 3 = 6

Consider the following number square

9 3210-1-2-3

3

2

1

0

-1

-2

-3

Complete the first four lines

6 3 0 -3

-6

-9 6 4 2 0 -2

-4

-6 3 2 1 0 -1

-2

-3 0 0 0 0 0 0 0

-3

-2

-1

0 1 2 3 -6

-4

-2

0 24 6 -9

-6

-3

0 3 6 9

Can you see the pattern, complete the square.

Rule:-

+ + = x +Ve - + = x -Ve + - = x -Ve - - =

x +Ve

Will this work on the number line?

Add the negative numbers

3 + 2 - 1 = 4

-6 -5 -4 -3 -2 -1 0 654321

Multiplication ?

Notice the –ve is in the opposite direction

Consider

4 x -1 = -4

-6 -5 -4 -3 -2 -1 0 654321

x -1

So multiplying by -1 rotates a vector by 1800

x -1

- 4 x -1 = +4

It works !!

Consider

4 = 2

17 =

4.1231

-17 = ?????

There is no simple number that will solve this

We need a complex number !

Consider

4 = 4

Back to the number line

4x

17 = 1717x

x = xxx

-1 = -1-1x

So there must be a value to the square root of -1 ?

Consider

- 4

-6 -5 -4 -3 -2 -1 0 654321

Clever !!

4 X -1 X -1 =

multiplying by -1 x -1 = -1

But -1 rotates a vector by 1800

But what about j4

-6 -5 -4 -3 -2 -1 0 654321

Very clever !!

4 X -1 =

So multiplying by -1 rotates a vector by 900

j4 Remember

-1 x -1 = -1

A rotation by 180o

We use a j to identify the vertical number line

Argand diagram

Multiplication ?

(-2 + j2) + (6 + j4) =

-6 -5 -4 -3 -2 -1 654321

-6

-5

-4

-3

-2

-1

6

5

4

3

2

1

j

-j

6 + j4

-2 + j2

4 + j6

4 + j6

Multiplication

2 x (3 + j2) =

-6 -5 -4 -3 -2 -1 654321

-6

-5

-4

-3

-2

-1

6

5

4

3

2

1

j

-j

3 + j2

6 + j4

6 + j4

Multiplication by jj x (4 + j3) =

4 + j3

4j + jxj3

= 4j - 3

= -3 + 4j

Note the rotation by 90o

-6 -5 -4 -3 -2 -1 654321

-6

-5

-4

-3

-2

-1

6

5

4

3

2

1

j

-j

-3 + j4

Simplify the following

a) 5 + 3j + 6 - 2j – 4 =

b) 2 + 2j - 7 + 2j + 8 - j =

c) 2j + 5j - 7 - 6j + 8 + 9j =

d) (3 + 4j) + (2 + 2j) + (3 - 3j) =

e) (5 + 2j) - (2 + 2j) + (3 + 2j) =

f) (3 + 4j) - (8 + 2j) - (3 - 6j) =

2 + 2j