Remember the Argand diagram in which the point (a, b) corresponds to the complex number z = a + ib

When the complex number is written as a + ib where a and b are real numbers, this is known as the Cartesian form.

This point (a,b) can also be specified by giving the distance, r, of the point from the origin and the angle, , between the line joining the point to the origin and the positive x-axis.

By some simple trigonometry it follows that a = r cos and b = r sin

Thus the complex number z can be written as r cos + i r sin

This is known as the polar form of a complex number.
r is called the modulus of z and is the argument of z.

The definitions of these terms are:

Modulus of a complex number The modulus r of a complex number z = a + ib is written | z | and defined by | z | =

Argument of a complex number The argument of a complex number is the angle between the positive x-axis and the line representing the complex number on an Argand diagram. It is denoted arg (z).

Polar form of a complex number The polar form of a complex number is z = r (cos + i sin ) where r is the modulus and is the argument.

In some cases calculations in polar form are much simpler so it is important to be able to work with complex numbers in both forms.
There will be times when conversion between these forms is necessary.

Given a modulus (r) and argument () of a complex number it is easy to find the number in Cartesian form.

Use the following steps to do this:

• Evaluate 'a' = r cos and 'b' = r sin
• Write down the number in the form a + ib

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Example 1

If a complex number z has modulus of 2 and argument of , express z in the form a + ib and plot the point which represents the number in an Argand diagram.

Answer:

• a = = = 3
• b = = = -1
• So a + ib = 3 - i

Check: 3 - i lies in the fourth quadrant and is also in the fourth quadrant.

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Example 2

Plot the complex number with modulus 2 and argument in an Argand diagram.

Answer:

NB: notice that this is exactly the same point as the previous example. This duplication demonstrates that different arguments can give the same complex number. To prevent confusion one argument is referred to as the principal value of the argument.

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Principal value of an argument The principal value of an argument is the value which lies between - and

The values of the principal argument for a complex number in each quadrant are shown on the following diagrams.

In the previous two examples the complex number is represented by a point in either the first or second quadrant and the principle argument is positive (lying between 0 and ).
In the next two examples when the complex number is represented in the third and fourth quadrants the principal argument is negative (lying between 0 and -).

NB: Taking without the modulus sign on a calculator may give a different value than required.

Now look at some examples:

Exercise in finding the modulus and argument

Further exercise in finding the modulus and argument