1. Complex Numbers 2
The Argand Diagram

2. Representing Complex Numbers
Real numbers are usually represented as positions on a
horizontal number line. -3 -2 -1 1 2 3 4 5
Real
Addition, subtraction, multiplication and division with real numbers takes place on this number line.

3. The Argand Diagram
Complex numbers also have an imaginary part so another dimension needs to be added to the number line 1 2 3 4 5 6 7 8 -2 -3 -4 -5 -6 -7 -8
Real
Imaginary -1
Complex numbers can be represented on the Argand diagram by straight lines. Putting complex numbers on an Argand diagram often helps give a feel for a problem.

4. Some examples Imaginary w u Real z v 1 2 3 4 5 6 7 8 -2 -3 -4 -5 -6 -7-8
Real
Imaginary -1 u v w z

5. Complex numbers and their conjugates1 2 3 4 5 6 7 8 -2 -3 -4 -5 -6 -7 -8
Real
Imaginary -1 w z w* z*

6. Addition 1 2 3 4 5 6 7 8 -2 -3 -4 -5 -6 -7 -8
Real
Imaginary -1 w z

7. Subtraction 1 2 3 4 5 6 7 8 -2 -3 -4 -5 -6 -7 -8
Real
Imaginary -1

8. The modulus of a complex number
Real O
Imaginary y x
x + yj

9. The argument of a complex number1 2 3 4 5 6 7 8 -2 -3 -4 -5 -6 -7 -8
Real
Imaginary -1 θ
z=2 + 3j
w=-3 - 5j α
between -180o and 180o

10. Radians

11. Loci using complex numbers1 2 3 4 5 6 7 8

12. The distance to a point Imaginary Real 1 2 3 4 5 6 7 8 -2 -3 -4 -5 -6-7 -8
Real
Imaginary -1

13. Loci using arguments Re Im Re Im Re Im