Pythagoras's Theorem
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 inlowest 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 andOdd, so there are no and such that is Rational, and must beIrrational.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.
- Convex Optimization Theory
- Convex Polygon
- Convex Polyhedron
- Convolution
- Convolution Theorem
- Conway-Alexander Polynomial
- Conway Groups
- Conway Notation
- Conway Polyhedron Notation
- Conway Polynomial
- Conway Puzzle
- Conway's Constant
- Conway Sequence
- Conway's Game of Life
- Conway's Knot
- Conway's Knot Notation
- Conway's Life
- Conway Sphere
- Coordinate Geometry
- Coordinates
- Coordinate System
- Coordination Number
- Copeland-Erdös Constant
- Coplanar
- Coprime