Department of Mathematics --- College of Science --- University of Utah

Mathematics 1010 online

Irrational Numbers

Consider a right triangle whose two short sides have a length of $ 1 $ foot each. By the Pythagorean Theorem the long side has a length of $ \sqrt{2} $ feet. It turns out that $ \sqrt{2} $ 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 $ \sqrt{2} $ is in fact rational and then derive a contradiction. Since there are no contradictions in mathematics our assumption must be false and so $ \sqrt{2} $ must be irrational.

So let us suppose that $ \sqrt{2} $ can be written as a ratio of two integers $ p $ and $ q $:

$\displaystyle \sqrt{2} = \frac{p}{q}.\qquad\qquad(*) $

We don't know $ p $ and $ q $, and if they exist they may be very large. However, we may assume that $ p $ and $ q $ have no factors greater than $ 1 $ in common, because if they did we could cancel that factor in the expression $ \frac{p}{q} $. In particular, we may assume that $ p $ and $ q $ are not both even. (One may be, or the other, but not both.)

We start with the equation $ (*) $. Squaring on both sides gives

$\displaystyle 2=\frac{p^2}{q^2}. $

Multiplying with $ q^2 $ on both sides gives the new equation

$\displaystyle 2q^2 = p^2. $

Because of the factor $ 2 $ on the left side, the right side, i.e., $ p^2 $, must be even. For this to be true, $ p $ itself must be even. If $ p $ is even then $ p^2 $ is divisible by $ 4 $. Hence $ 2q^2 $ is also divisible by $ 4 $ which means that $ q^2 $ must be even. That implies in turn that $ q $ must be even. Thus $ p $ and $ q $ must be both even, which contradicts our (legitimate) assumption that $ p $ and $ q $ have no factor in common. Our (doubtful) assumption that $ \sqrt{2} $ is rational therefore can't be true -- the square root of $ 2 $ 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: