Graphics Reference
In-Depth Information
where 2 is the real part and 3
i
is the imaginary part. The following are all complex
numbers:
2
,
2
+
2
i,
1
−
3
i,
−
4
i,
17
i.
Note the convention to place the real part first, followed by
i
. However, if
i
is as-
sociated with a trigonometric function such as
sin
or
cos
, it is usual to place
i
in
front of the function:
i
sin
θ
or
i
cos
θ
, to avoid any confusion that it is part of the
function's angle.
All that we have to remember is that whenever we manipulate complex numbers,
the occurrence of
i
2
is replaced by
−
1.
2.2.1 Axioms
The axioms defining the behaviour of complex numbers are identical to those asso-
ciated with real numbers. For example, given two complex numbers
z
1
and
z
2
they
obey the following rules:
Addition:
Commutative
z
1
+
z
2
=
z
2
+
z
1
Associative
(z
1
+
z
2
)
+
z
3
=
z
1
+
(z
2
+
z
3
) .
Multiplication:
Commutative
z
1
z
2
=
z
2
z
1
Associative
(z
1
z
2
) z
3
=
z
1
(z
2
z
3
)
Distributive
z
1
(z
2
+
z
3
)
=
z
1
z
2
+
z
1
z
3
(z
1
+
z
2
) z
3
=
z
1
z
3
+
z
2
z
3
.
2.3 The Modulus
bi
is defined as
√
a
2
+
+
b
2
. For example, the
The
modulus
of a complex number
a
modulus of 3
+
4
i
is 5. In general, the modulus of a complex number
z
is written
|
z
|
:
=
+
z
a
bi
a
2
b
2
.
|
z
|=
+
We'll see why this is so when we cover the polar representation of a complex num-
ber.
