# 2.3 Complex numbers

A complex number may be written in the form:

Equation label:(3)

where and are real numbers and i is a special quantity with the property that .

Each complex number has a real part, Re, and an imaginary part, Im.

Complex numbers can be added,

and multiplied,

using the usual rules of algebra along with .

The complex conjugate of is (pronounced "z star").

This results in showing that is a positive real number, (unless ).

The modulus of the complex number is defined as

Equation label:(4)

which is a real, non-negative quantity.

Complex numbers can also be written in polar form,

Equation label:(5)

where and are real numbers. The relationship between and , and is shown in Figure 3. Here is the modulus of as defined in Equation (4); is known as the phase, and is a phase factor.

Figure 3 A diagram showing the relationship between Cartesian coordinates and , and polar coordinates and

## Exercise 6

Consider the complex number .

1. Write down its complex conjugate .

2. Calculate the modulus of .

3. Write in polar form.

### Answer

1. The complex conjugate is .

2. The modulus of is

3. In polar form where and , so radians. Therefore we can write .

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