Pythagoras,s Theorem
المؤلف:
Conway, J. H. and Guy, R. K.
المصدر:
The Book of Numbers. New York: Springer-Verlag
الجزء والصفحة:
...
26-7-2020
2012
Pythagoras's Theorem
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.
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