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
,