Game Development Reference
In-Depth Information
The mathematical symbol for rational numbers is Q. Here is how to mathe-
matically represent the set of rational numbers:
p
q
Q
¼
p
,
q 2
Z
q 6¼
0
The vertical bar means ''such that,'' and the
means ''is a member of.'' The
hashed equal sign means ''not equal to.'' The definition reads, ''A rational
number (Q) is any number
p/q
, such that
p
and
q
are elements of the set of
integers (Z) and
q
is not equal to 0.''
[
Here is a set of rational numbers:
f
1
=
5,
3
=
4,
2
=
3,
1
=
2,
1
=
6, 0, 6
=
7, 4
=
2, 21
g
Notice that the elements in this set are negative, include zero, and positive. Also
note that 0 and 21 can be expressed as ratios: 0/0 and 21/1.
No Division by Zero
When you formally define a rational number, the two numbers you include in
the quotient,
p
and
q
, are both integers. The lower number,
q
, cannot be equal to
0. You cannot divide by 0.
We know that
2
¼
3 because 2
3
¼
6. If we consider
0
, the answer must be a
number that, when multiplied by zero, results in 6 (i.e.,
0
¼ x
where
x
0
¼
6),
but
x
0
¼
0 for all x's. Therefore,
0
has no value. Mathematicians say that such
a relation where any number is divided by zero is
undefined
.
Staying Clear on Your Numbers
Just because you can represent a number as a quotient does not mean that it is only a rational
number. The fact is that you can represent counting numbers, whole numbers,
integers, and
rational numbers as quotients. Consider the following representations of numbers:
2
4
2 ¼
2
All of the representations denote 2, which is a counting number, a whole num-
ber, an integer, and a rational number. When the quotient provides a ''rational'' way to
represent the number, whether in its quotient or non-quotient form, the number remains
a rational number.
1
¼
n
n
2 ¼
1
¼
3
The representations of 2 on each side of the equal sign designate
integers. The number 2 can be represented as
3
,
2
,
4
,or
12
6
. Therefore, it is a
rational number as well as an integer.
