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)

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