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College Math Resources - Topics in Precalculus

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The Real Number System

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Axioms for the Real Numbers

In order to have absolute confidence in their results, mathematicians need
more than just a lot of confirming examples. This is the source of the
concept of a "proof" in which the required result is shown to be a logical
consequence of other better established facts.
Unfortunately there is no way to get something from nothing, and so
some basic assumptions are always needed as a starting point. These are
called "Axioms".

In order to prove results involving Real numbers, we could express the
reals and everything we want to say in terms of simpler objects - and maybe
start with a set of axioms for just counting numbers, or even for the theory
of sets. But another approach is to take as fundamental a set of basic
properties of the Real numbers themselves. These are then referred to as
Axioms for the Real Number System.

These ideas are elaborated more fully in an Analysis
course.

If you have come across any good web-based illustrations of these and
related concepts,

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