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Pythagoras's Theorem


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

 d^2=s^2+s^2=2s^2,

so

 (d/s)^2=(p/q)^2=2,

and p^2=2q^2, so p^2 is even. But if p^2 is even, then p is even. Since p/q is defined to be expressed in lowest terms, q must be odd; otherwise p and q would have the common factor 2. Since p is even, we can let p=2r, then 4r^2=2q^2. Therefore, q^2=2r^2, and q^2, so q must be even. But q cannot be both even and odd, so there are no d and s such that d/s is rational, and d/s must be irrational.

In particular, Pythagoras's constant sqrt(2) is irrational. Conway and Guy (1996) give a proof of this fact using paper folding, as well as similar proofs for phi (the golden ratio) and sqrt(3) using a pentagon and hexagon. A collection of 17 computer proofs of the irrationality of sqrt(2) is given by Wiedijk (2006).


See also

Irrational Number, Pythagoras's Constant, Pythagorean Theorem

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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.

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Pythagoras's Theorem

Cite this as:

Weisstein, Eric W. "Pythagoras's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PythagorassTheorem.html

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