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Difference between revisions of "User:Foxjwill/Proofs"

(→‎A theorem)
(→‎A theorem)
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==A theorem==
 
==A theorem==
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'''THEOREM.''' Let <math>C</math> be a circle of radius <math>r</math>, let <math>A</math> be the set of chords of <math>C</math>, for all <math>p\in \mathbb{R}^+</math>, let <math>S_p\equiv \{B\subset A|B\mbox{ is finite}, \sum_{b\in B}|b|=p\}</math>. <!--Additionally, let <math>\Theta\equiv \{\mbox{angle }\theta| \mbox{for all }a\in</math>--> Then for all <math>p\in \mathbb{R}^+</math>, there exists an angle <math>\phi\in \mathbb{R}^+</math> such that for all <math>B\in S_p</math>, there exists a positive integer <math>k</math> such that for all sets <math>\Theta</math> consisting only of elements <math>\theta</math> such that for each <math>b\in B</math>, there is exactly one <math>\theta \in \Theta</math> cut by <math>b</math>,
+
'''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_{\theta\in \Theta}\theta = \phi
+
\sum_{a\in A}S(a) = \theta.
 
</cmath>
 
</cmath>

Revision as of 19:12, 13 January 2009

Proof that $p^{1/n}$, where $p$ is prime, is irrational

  1. Assume that $p^{1/n}$ is rational. Then $\exists a,b \in \mathbb{Z}$ such that $a$ is coprime to $b$ and $p^{1/n}={a \over b}$.
  2. It follows that $p = {a^n \over b^n}$, and that $a^n=pb^n$.
  3. So, by the properties of exponents along with the unique factorization theorem, $p$ divides both $a^n$ and $a$.
  4. Factoring out $p$ from (2), we have $a^n=p^{n-1}b'$ for some $b'\in \mathbb{Z}$.
  5. Therefore $p$ divides $a$.
  6. But this contradicts the assumption that $a$ and $b$ are coprime.
  7. Therefore $p^{1/n}\not\in \mathbb{Q}$.
Q.E.D.

A theorem

DEFINITION Let $a$ be a chord of some circle $C$. Then the small angle of $a$, denoted $S(a)$, is the smaller of the two angles cut by $a$.

THEOREM. Let $p\in \mathbb{R}^+$, and let $C$ be a circle. Then there exists a $\theta\in \mathbf{R}^+$ such that for every set A of chords of $C$ with lengths adding to $p$, \[\sum_{a\in A}S(a) = \theta.\]

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