Proves 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
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.
See also Irrational Number, Pythagoras's Constant, Pythagorean Theorem
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 183-186, 1996.
Pappas, T. ``Irrational Numbers & the Pythagoras Theorem.'' The Joy of Mathematics.
San Carlos, CA: Wide World Publ./Tetra, pp. 98-99, 1989.