- Complex Numbers Made Easy

- Properties of Mods and Args

- Properties of Conjugates

- Curves and Regions in the Argand Diagram

- Interactive Complex Arguments

- Representation of Complex Numbers by Vectors

- Addition, Subtraction and Multiplication by Vectors

- Some Review Questions

## Properties of Mods and Args

Table of Contents

## Properties of Conjugates

Table of Contents

## Curves and Regions on the Argand Diagram

Table of Contents

You should instantly recognise the following:

You should be able to work out the following by putting

z = x + i yin (or by using properties of mods, args, conjugates).

The following are worth knowing also (play with this interactive figure to help!):

## Representation of Complex Numbers by Vectors

Table of Contents

We know that the complex number

z = x + i ymay be represented on the Argand diagram in one of two ways:

- By the point Z with coordinates (
x, y).- By ANY vector whose length equals |
z| and whose direction is parallel to OZ.

NB: a complex number z is represented by the length and direction of a vector, and, if the vector begins at O, then its tip will actually be at the point that represents.z.Table of Contents

ADDITION, SUBTRACTION AND MULTIPLICATION BY VECTORS

Table of Contents• ADDITION by drawing the diagonal of the parallelogram (or by completing the polygon).

• SUBTRACTION: the vector representing

zis the vector from the point representing_{1}- z_{2}zto the point representing_{2}z._{1}• MULTIPLICATION by a positive real number is represented by a scaling of the vector (with no change in direction).

• MULTIPLICATION by

i: the vector representingi zis formed by the rotation of the vector representingzanticlockwise through 90º.• MULTIPLICATION by a complex number: to multiply

zby a complex numberzinvolves both a rotation through arg_{1}zand a scaling by |_{1}z|.

For comments & suggestions, please e-mail Steve Arnold.