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

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.

**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.

© 1996-9

1999-05-26