Complex Numbers 2 The Argand Diagram

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.

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.

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

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

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

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

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

The argument of a complex number 1 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

Radians

Loci using complex numbers 1 2 3 4 5 6 7 8

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

Loci using arguments Re Im Re Im Re Im