2.2 Rational Numbers

The rational number system is an extension of the integer number system [42]. The set of all rationals is denoted by   :

Each rational number is a ratio of two integers: a numerator and a non-zero denominator. The rationals extend the integers since the integers are homomorphic to the rationals. An injective mapping is a homomorphism if all the properties of are preserved in . The mapping (abbreviated as ),

is a homomorphism from to . In order to verfiy this one must show that:

and

When dealing with different   number systems in close mutual proximity it will be useful to precisely specify operators as was done above. With a formal definition of rational addition and multiplication, showing that is a homomorphism is straightforward. Because of the simple homomorphism , the integers are commonly viewed as a subset of the rationals.

The nice properties of the integers extend to the rationals. The rationals are closed under addition and multiplication. Rational addition has a total inverse as did integer addition. Rational addition and multiplication are associative and commutative; together they obey the distributive law. Rational multiplication has an ``almost'' total inverse:

The inverse is not total because zero is not allowed in the denominator of a rational. So the rationals are closed under division, except for division by zero. As with integers, comparisons between rational numbers result in one of three orderings. Most mathematicians do not view the lack of a total multiplicative inverse as a major failing of the rationals.

Consider the squaring operator defined by:

The inverse of the squaring operator is the square root operator, which satisfies the following:

There is no total square root operator over the rationals: consider or . This is one reason to extend the rationals. Many popular operators are not total over the rationals.

Common Practice

There are computer facilities, in the form of software libraries or hardware assist, for performing operations on rationals. These facilities store each rational as a pair of integers [51], or as a continued fraction [36]. Although arithmetic operations (+, -, , and ) can be performed with these libraries, many other popular operations cannot be performed directly since many popular operations are not total over the rationals. It is also difficult to predict the computational resources required for a string of operations since the time required to perform an operation is dependent on the numbers involved. Limiting the computational requirements usually results in a system like floating point, which will be discussed shortly.

Jeff Tupper

March 1996