Pythagoras's theorem states that the diagonal  of a square with sides of integral length  cannot be rational. Assume  is rational and equal to  where  and  are integers with no common factors. Then

so

and , so  is even. But if  is even, then  is even. Since  is defined to be expressed in lowest terms,  must be odd; otherwise  and  would have the common factor 2. Since  is even, we can let , then . Therefore, , and , so  must be even. But  cannot be both even and odd, so there are no  and  such that  is rational, and  must be irrational.

In particular, Pythagoras's constant  is irrational. Conway and Guy (1996) give a proof of this fact using paper folding, as well as similar proofs for  (the golden ratio) and  using a pentagon and hexagon. A collection of 17 computer proofs of the irrationality of  is given by Wiedijk (2006).

REFERENCES:

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 183-186, 1996.

Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 70, 1984.

Pappas, T. "Irrational Numbers & the Pythagoras Theorem." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 98-99, 1989.

Wiedijk, F. (Ed.). The Seventeen Provers of the World. Berlin: Springer-Verlag, 2006.

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.