2
Complex Numbers
2.1
The imaginary number
i
[see Riley 3.1, 3.3]
Complex numbers are a generalisation of real numbers. they occur in many branches of mathematics and
have numerous applications in physics.
The imaginary number is
i
=
√

1
⇔
i
2
=

1
The obvious place to see where we have already needed this is in the solution to quadratic equation. Eg.
finds the roots of
z
2
+ 4
z
+ 5 = 0
(
z
+ 2)
2
+ 1 = 0
(
z
+ 2)
2
=

1
and
z
1
,
2
=

2
±
√

1
.
So in this case we use the imaginary number and write the solutions as
z
1
,
2
=

2
±
i .
which is called a complex number. The general form of a complex number is
z
=
x
+
iy
where
z
is the conventional representation and is the sum of the real part
x
and
i
times the imaginary
part
y
: these are denoted as
Re
(
z
)
=
x
Im
(
z
)
=
y ,
respectively. The imaginary or real part can be zero, so if the imaginary part is, the number is real and
hence real numbers are just a subset of complex numbers.
Also when using the quadratic solutions formula, we had situations where there were no (real) roots
as
b
2

4
ac <
0. we could have solved the above quadratic to get the same results:
z
1
,
2
=

4
±
√
16

20
2
=

4
±
√

4
2
=

4
±
2
√

1
2
=

2
±
i .
A complex number may also be written more compactly as
z
= (
x, y
) where
x
and
y
are two real numbers
which define the complex number and may be thought of as Cartesian coordinates.
1
z=x+iy
q
r
Re(z
)
Im(z)
Argand diagram
Recall that in Cartesian coordinates
x
=
r
cos
θ
y
=
r
sin
θ
Therefore we can represent
z
in polar coordinates as
z
=
x
+
iy
=
r
(cos
θ
+
i
sin
θ
)
.
The number
r
is called the modulus of
z
, written as

z

or mod(
z
). This can be written in terms of
x
and
y
as

z

=
p
x
2
+
y
2
.
The angle
θ
is called the argument of
z
, written as arg(
z
) (or arg
z
) and is defined as
arg(
z
) = tan

1
y
x
.
so arg(
z
) is the angle that the line joining the origin to
z
on an Argand diagram makes with the positive
x

axis. The anticlockwise direction is taken to be positive by convention.
However,
θ
is not unique since
θ
+ 2
nπ
(
n
is zero or any integer) are also arguments for the same complex
number.
We therefore define a principal value of a complex number as that value of
θ
which satisfies

π < θ
≤
π
. (it could also be 0
< θ
≤
2
π
). Also, account must be taken of the signs of