College Math Resources - Topics in Precalculus

- The Real Number System -

What are 'Irrational' Numbers?

It may be surprising to you that such numbers exist - it certainly was to the ancient Greeks who first discovered this fact! And in common language, the word irrational has come to suggest something unreasonable or even mad, but whether we like it or not, it turns out that in some sense most real numbers are in fact irrational.

Examples of numbers that you may have been told are irrational include the square root of two, and the number pi giving the ratio of a circle's circumference to its diameter.(Here pi is defined as a ratio, but not a ratio of integers. In fact if the diameter of a circle is a whole number of units, then its circumference cannot be, and vice versa.)

Of course just having been told something is no good reason to believe it!

In order to be convinced that the square root of two is an irrational real number, we need to establish two things. First that there is in fact a real number whose square is exatly equal to two, and secondly that such a number cannot be the ratio of two integers.

The existence of root two as a real number can be understood intuitively on the basis of a geometrical interpretation of real numbers as all possible positions on a line. We can find an interval of length root two by using Pythagoras' theorem. In an analysis course it might also be established analytically in terms of a suitable set of axioms for the real number system.

The fact that no ratio of integers can have a square equal to two can be shown by assuming that a particular ratio, say m/n, does have a square of two, and then deducing that this must be false by showing that it leads to a self-contradiction. (Hint: If m/n is root 2, then (m^2)/(n^2)=2, so m^2 is even   ...  )

But actually the real situation is even stranger than that. Not only do the rationals not include all reals, but despite seeming to "fill" the number line they actually have a total length of zero!

The case of pi is perhaps a little trickier - and in fact pi is not just irrational, it is also an example of a transcendental number.

You might find how to complete these proofs in an introductory Analysis text, or by exploring our links to various web-based analysis course materials.

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