- 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 ArgsTable of Contents
Properties of ConjugatesTable of Contents
Curves and Regions on the Argand DiagramTable of Contents
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 VectorsTable of Contents
We know that the complex number z = x + i y may 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.
ADDITION, SUBTRACTION AND MULTIPLICATION BY VECTORSTable of Contents
Table of Contents
• 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 |.
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