### College Math Resources - Topics in Precalculus

###
- The Real Number System -

##
What are 'Irrational' Numbers?

Irrational numbers are just real numbers that cannot be expressed as
ratios of integers.

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,

please do let
us know and we will add them here.