Game Development Reference
In-Depth Information
Appendix A
Complex Numbers
A.1 Definition
The set of complex numbers  is a field containing the set of real numbers  and
the “imaginary” number i . The number i is defined to be the square root of 1
:
i
=−
1
.
(A.1)
Thus, the square root of any negative number n
can be written as
−=
nin
.
(A.2)
A complex number z is one of the form
za i
=+
,
(A.3)
where a and b are real numbers. The number a is called the real part of z , denot-
ed by
()
Re z , and the number b is called the imaginary part of z , denoted by
()
Im z . If
=
0
, then the number z is purely real. If
a
=
0
, then the number z is
b
purely imaginary.
A.2 Addition and Multiplication
The sum of two complex numbers abi
+
and c
+
di
is given by
(
)
(
)
(
)
(
)
abi
+++=+++
cdi
ac
bdi
.
(A.4)
The product of two complex numbers can be calculated by using the distributive
property and the fact that
2
. The product of abi
+
and c
+
di
is given by
i
=−
1
(
)(
)
(
)
(
)
a
+
bi
c
+
di
=
ac
bd
+
ad
+
bc i
.
(A.5)
499
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