2.3 Complex numbers
A complex number may be written in the form:
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
which is a real, non-negative quantity.
Complex numbers can also be written in polar form,
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.
and , and polar coordinates and Exercise 6
Consider the complex number 
.
-
Write down its complex conjugate
. -
Calculate the modulus of
. -
Write
in polar form.
Answer
-
The complex conjugate is
. -
The modulus of
is
-
In polar form
where 
and 
, so 
radians. Therefore we can write 
.
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