Stephen ARNOLD
HomeOnLine Mathematics Resources →Complex Numbers Review

Complex Numbers Review

Based on a presentation for the Newcastle Mathematical Association by M. Curran, May, 1996.

Curves and Regions on the Argand Diagram

You should instantly recognise the following:

You should be able to work out the following by putting z = x + i y in (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

We know that the complex number z = x + i y may be represented on the Argand diagram in one of two ways:

1. By the point Z with coordinates (x, y).
2. 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.

ADDITION, SUBTRACTION AND MULTIPLICATION BY VECTORS

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

• SUBTRACTION: the vector representing z1 - z2 is the vector from the point representing z2 to the point representing z1.

• MULTIPLICATION by a positive real number is represented by a scaling of the vector (with no change in direction).

• MULTIPLICATION by i: the vector representing i z is formed by the rotation of the vector representing z anticlockwise through 90º.

• MULTIPLICATION by a complex number: to multiply z by a complex number z1 involves both a rotation through arg z1 and a scaling by |z |.