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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 'Real' Numbers?

The so-called 'Real Number System' is an attempt to extend the ideas of
number and arithmetic (which originate in counting) to describe more general
measurements such as length, weight and temperature.
In fact, the way we measure anything __is__ based on counting. When
we use a ruler or tape to measure a length, we do so by counting the number
of marked intervals of some specific length or "unit" (eg cm,m, or inch).
But this is often not exact - and we divide each unit interval into smaller
parts (or fractions) to get a more accurate (but still not perfect) measure
of the length.

It might be expected that if we take all possible fractional divisions
of the basic unit then we'd be able to give an exact description for any
concievable length and so that every length can be measured exactly by
a "rational" number of units.

But this expectation is FALSE! (Do you know why?)

In fact, we can get arbitrarily __close__ to any real length with
rationals, but we can't always match it exactly. (This idea of arbitrarily
close approximation is the source of the mathematical concept of a limit
which underlies many of the concepts of Calculus.)

On the other hand, if that was the bad news, the good news is that the
operations of arithmetic (defined for whole numbers in terms of combinations
of sets) can be extended in a natural way to fractions and also to those
"limits of fractions" that we need to account for all possible measurements.

One way to define the Real Number System is to "construct"
it by identifying real numbers with sequences of approximating rationals
and showing that the arithmetic operations do extend consistently. Another
is to specify the basic properties that the resulting system should satisfy
and to take these as "Axioms" from which less
obvious properties can be proved (or disproved as the case may be).

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.