## Irrational Numbers

Consider a right triangle whose two short sides have a length of
foot each. By the
Pythagorean Theorem the long side has a length of
feet. It turns out that is not a
rational number .

This remarkable fact can be seen by a classic argument usually attributed to
* Eudoxus of Cnidus* (approx. 406-355BC).
We assume that is in fact rational and then derive a
contradiction. Since there are no contradictions in mathematics our
assumption must be false and so must be irrational.

So let us suppose that can be written as a ratio of two
integers and :

We don't know
and , and if they exist they may be very large.
However, we may assume that and
have no factors greater than in common, because if they
did we could cancel that factor in the expression
. In
particular, we may assume that and are ** not both
even**. (One may be, or the other, but not both.)
We start with the equation . Squaring on both sides gives

Multiplying with on both sides gives the new equation
Because of the factor on the left side, the right side, i.e.,
, must be even. For this to be true, itself must be
even. If is even then is divisible by . Hence
is also divisible by which means that must be
even. That implies in turn that must be even. Thus and
must be ** both** even, which contradicts our (legitimate)
assumption that and have no factor in common. Our
(doubtful) assumption that is rational therefore can't be
true -- the square root of is not a rational number.

The following facts regarding irrational numbers are beyond the scope
of this class, but you can read about them in the book "What is
Mathematics" by Courant Robbins, or I would be pleased to tell you
more if you are interested:

- The square roots of prime numbers are irrational.

- is irrational.

- A rational number can be written as a repeating (or
terminating) decimal.

- A repeating or (or terminating) decimal can be written as a
fraction of two integers, i./e., it's rational.

- In a well defined sense, most real numbers are irrational.
In other words, there are more irrational numbers than rational numbers.

- In the same sense there are as many (but in particular no more)
rational numbers than there are integers. This is true despite the
fact that all integers are rational, and some rational numbers aren't
integers. Infinity is different!