College Math Resources - Topics in Precalculus

The Real Number System

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,
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