Graphics Reference
In-Depth Information
part is 3
i
. There is no requirement for a complex number to have a real part, in fact, even the
imaginary part could be zero, which implies that the set of real numbers
R
. is a subset of the
complex number set
.
What is interesting is that even if we ignore what complex numbers are or what they can do,
they are extremely well behaved when subjected to the normal laws of algebra. For example, we
can add, subtract, multiply, divide, raise to a power, take logs, etc. of a complex number.
Z
7.3.1 Complex number operations
Briefly, here are the rules associated with complex numbers:
Equivalence
: Two complex numbers are equal only if both real and imaginary parts are the
same. For example, if
a
+
b
i
=
r
+
s
i
then
a
=
r and b
=
s
Addition and subtraction
: To add or subtract two complex numbers, the addition or subtraction
is performed individually on the real and imaginary parts:
a
+
b
i
±
r
+
s
i
=
a
±
r
+
b
±
s
i
For example,
12
+
3
i
−
9
−
4
i
=
3
+
7
i
Multiplication
: To multiply two complex numbers together, we expand the multiplication using
the conventional rules of algebra:
bs
i
2
a
+
b
i
×
r
+
s
i
=
ar
+
as
i
+
br
i
+
Collecting like terms, we obtain
a
+
b
i
×
r
+
s
i
=
ar
−
bs
+
as
+
br
i
Note that the
i
2
term makes the product bs negative.
For example,
12
i
2
12
+
3
i
×
9
−
4
i
=
108
+
27
i
−
48
i
−
=
120
−
21
i
Division
: To divide a complex number by another, we invoke the following subterfuge:
a
+
b
i
a
+
b
i
r
−
s
i
ar
−
as
i
+
br
i
−
bs
i
2
ar
+
bs
+
br
−
as
i
s
i
=
s
i
=
=
r
+
r
+
s
i
r
−
r
2
−
rs
i
+
rs
i
−
s
2
i
2
r
2
+
s
2
a
+
b
i
ar
+
bs
br
−
as
s
i
=
+
i
r
+
r
2
+
s
2
r
2
+
s
2
