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Internet Resources for the Calculus Student - Topics in Precalculus

The Real Number System

Construction of the Real Numbers

One way to define the Reals is to construct them in terms of simpler concepts.

For example the properties of counting numbers may seem more familiar, or can be derived from the even more fundamental concept of sets. (Though noone has found a way of starting from nothing, and in this approach certain basic properties of sets are themselves taken for a starting point - as "Axioms").

Because they are so fundamental, the counting numbers (including zero) are known as the Natural numbers and other systems can be defined in terms of them. For example the Integers or signed numbers can be thought of as representations of all possible difference problems among Natural numbers (including "impossible" ones like 2-3), and similarly for Rational numbers as representations of division problems or "ratios" of integers. The limits defining Real numbers can then be identified with sequences of Rational numbers (at least those special ones which get closer and closer together  - like the decimal approximations to a fixed number for example).

This process may be elaborated in an Analysis course, but we'll not do that here.

Another approach is to take the Real Number System as fundamental and to define it axiomatically.

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.

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