Game Development Reference
In-Depth Information
Variations on Rational Numbers
When you carry out division, you end up with different types of quotients. To
repeat, a rational number is a number that you can express as a ratio. You can
express an integer as a ratio, as you can a whole number. In some cases, you can
reduce the quotient form of the number to a whole number. In other cases, the
quotient form of the number cannot be reduced to a whole number, and
therefore has a decimal representation. A number is rational if its decimal
representation is terminating or repeating.
Terminating Rational Numbers
If the decimal representation of the quotient can be expressed by an exact
number of digits, then it is a terminating rational number. Consider the fol-
lowing numbers and their decimal representations:
1
2 ¼ 0
1
4 ¼ 0
3
4 ¼ 0
3
6 ¼ 0
:
5,
:
25,
:
75,
:
5
In each case, the result of the division is exact. A melon can be divided into two
equal parts. It can also be divided into equal fourths. Likewise, you can have
exactly three fourths of a melon, as you can have five sixths of a melon.
Repeating Rational Numbers
In some cases, when you establish a ratio between two numbers, the quotient is
not terminating. You do not end up with exact pieces or proportions (i.e., the
decimal representation does not have an exact number of digits). The most you
can do is approximate the piece or proportion. Consider, for example, what
happens when you divide 1 by 3 or 2 by 3. The result is a repeating rational
number:
1
=
3 ¼ 0
:
3333333333 . . .
and
2
=
3 ¼ 0
:
6666666666 . . .
Mathematicians indicate the repeating part of the fraction by placing a bar above
the number that repeats:
=
3 ¼ 0
:
=
3 ¼ 0
:
1
3or2
6
When a single number repeats, you need only one number bearing a bar. In other
cases, the pattern of repetition involves several numbers. In that case, you use the
bar to designate the sequence of numbers that repeat. Consider the pattern that
 
Search WWH ::




Custom Search