Difference between revisions of "User:Foxjwill/Proofs"
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==A theorem== | ==A theorem== | ||
− | ''' | + | '''DEFINITION.''' Let <math>a</math> be a chord of some circle <math>C</math>. Then the ''small angle'' of <math>a</math>, denoted <math>S(a)</math>, is the smaller of the two angles cut by <math>a</math>. |
+ | |||
+ | '''THEOREM.''' Let <math>p\in \mathbb{R}^+</math>, and let <math>C</math> be a circle. Then there exists a <math>\theta\in \mathbf{R}^+</math> such that for every set A of chords of <math>C</math> with lengths adding to <math>p</math>, | ||
<cmath> | <cmath> | ||
− | \sum_{ | + | \sum_{a\in A}S(a) = \theta. |
</cmath> | </cmath> |
Latest revision as of 19:13, 13 January 2009
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
,