College Math Resources - Topics in Precalculus

The Real Number System

What are the 'Rational' Numbers?

The rational numbers are just those numbers which can be expressed as ratios of integers.

Of course this begs the question of how to define a ratio or quotient when the division doesn't go exactly. But anyone who has shared a pizza has some intuition for the ideas of fractional arithmetic, and all of the basic rules (or "axioms") for the Rational Number System are motivated by the goal of constructing a model for the sharing process. If you ever have dificulty with the arithmetic or algebra of fractions, you will find that time spent on convincing yourself that the "rules" make sense is a lot more useful in the long run than time spent on memorizing them or practicing with their use (although the latter -ie practice - has its place, and is an important part of getting to the point of complete familiarity).

 The rationals include positive and negative fractions such as -2/3 or 3/4 or 117/9. They also include the integers since any integer n can be written as a fraction n/1 (or 2n/2, etc). But, perhaps surprisingly, they do not include all real numbers. In fact,  by any reasonable way of counting, most of the reals are irrational.

 If you have come across any good web-based illustrations of these and related concepts,
     please do let us know and we will add them here.