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College Math Resources - Topics in Precalculus

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The Real Number System

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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.