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.
- Small Stellated Triacontahedron
- Small Stellated Truncated Dodecahedron
- Small Triakis Octahedron
- Small Triambic Icosahedron
- Small World Problem
- Smarandache Ceil Function
- Smarandache Constants
- Smarandache Function
- Smarandache-Kurepa Function
- Smarandache Near-to-Primorial Function
- Smarandache Paradox
- Smarandache Sequences
- Smarandache-Wagstaff Function
- Smith Brothers
- Smith Conjecture
- Smith Normal Form
- Smith Number
- Smith's Markov Process Theorem
- Smith's Network Theorem
- Smooth Manifold
- Smooth Number
- Smooth Surface
- Snake
- Snake Eyes
- Snake Oil Method