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