User:Foxjwill/Proofs
Proof that , where is prime, is irrational
- Assume that is rational. Then such that is coprime to and .
- It follows that , and that .
- So, by the properties of exponents along with the unique factorization theorem, divides both and .
- Factoring out from (2), we have for some .
- Therefore divides .
- But this contradicts the assumption that and are coprime.
- Therefore .
- Q.E.D.
A theorem
DEFINITION. Let be a chord of some circle . Then the small angle of , denoted , is the smaller of the two angles cut by .
THEOREM. Let , and let be a circle. Then there exists a such that for every set A of chords of with lengths adding to ,