1. Complex Number Theory

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

3. Now for multiplication 6543210 2 x 3 = 6

4. Consider the following number square 9 3210-2-3 3 2 1 0 -2 -3 Complete the first four lines 63 0 -3 -6 -9 642 0 -2 -4 -6 321 0 -2 -3 000 0 0 0 0 -2 0 1 2 3 -6-4-2 0 2 4 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?

5. Add the negative numbers 3 + 2 - 1 = 4 -6-5-4-3-20654321 Multiplication ? Notice the –ve is in the opposite direction

6. Consider 4 x -1 = -4 -6-5-4-3-20654321 x -1 So multiplying by -1 rotates a vector by 180 0 x -1 - 4 x -1 = +4 It works !!

7. Consider 4 = 2 17 = 4.1231 -17 = ????? There is no simple number that will solve this We need a complex number !

8. Consider 4 = 4 Back to the number line 4 x 17 = x x = x x x = x So there must be a value to the square root of -1 ?

9. Consider - 4 -6-5-4-3-20654321 Clever !! 4 X -1 X -1 = multiplying by -1 x -1 = -1 But -1 rotates a vector by 180 0

10. But what about j4 -6-5-4-3-20654321 Very clever !! 4 X -1 = So multiplying by -1 rotates a vector by 90 0 j4 Remember -1 x -1 = -1 A rotation by 180 o We use a j to identify the vertical number line

11. Argand diagram Multiplication ? (-2 + j2) + (6 + j4) = -6-5-4-3-2 654321 -6 -5 -4 -3 -2 6 5 4 3 2 1 j -j 6 + j4 -2 + j2 4 + j6

12. Multiplication 2 x (3 + j2) = -6-5-4-3-2 654321 -6 -5 -4 -3 -2 6 5 4 3 2 1 j -j 3 + j2 6 + j4

13. Multiplication by j j x (4 + j3) = 4 + j3 4j + jxj3 = 4j - 3 = -3 + 4j Note the rotation by 90 o -6-6 -5-4-3-2 654321 -6 -5 -4 -3 -2 6 5 4 3 2 1 j -j -3 + j4

14. 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