# Mathematics 1010 online

## 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.)

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!