2.12 Rational Representation

One big problem with the fractional representations we've seen is that they are not exact; that is, they provide a close approximation of real values, but they cannot provide an exact representation for all rational values. ^[5] For example, in binary or decimal you cannot exactly represent the value 1/3. You could switch to a ternary (base-3) numbering system and exactly represent 1/3, but then you wouldn't be able to exactly represent fractional values like 1/2 or 1/10. What we need is a numbering system that can represent any reasonable fractional value. Rational representation is a possibility in such situations.

Rational representation uses pairs of integers to represent fractional values. One integer represents the numerator (n) of a fraction, and the other represents the denominator (d) . The actual value is equal to n/d . As long as n and d are 'relatively prime' with respect to one another (that is, they are not both evenly divisible by the same value) this scheme provides a good representation for fractional values within the bounds of the integer representation you're using for n and d . In theory, arithmetic is quite easy; you use the same algorithms to add, subtract, multiply, and divide fractional values that you learned in grade school when dealing with fractions. The only problem is that certain operations may produce really large numerators or denominators (to the point where you get integer overflow in these values). Other than this problem, however, you can represent a wide range of fractional values using this scheme.

^[5] It is not possible to provide an exact computer representation of an irrational number, so we won't even try.