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Difference between revisions of "User:Foxjwill/Proofs"

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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\equiv \{\mbox{angle }\theta| \theta \in \Theta\mbox{ iff there exists a } b\in B \mbox{ that cuts },\theta\mbox{ and for all }b\in B,b\mbox{ cuts exactly one element in }\Theta\},</math>
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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>,
 
<cmath>
 
<cmath>
 
\sum_{\theta\in \Theta}\theta = \phi
 
\sum_{\theta\in \Theta}\theta = \phi
 
</cmath>
 
</cmath>

Revision as of 13:18, 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

THEOREM. Let $C$ be a circle of radius $r$, let $A$ be the set of chords of $C$, for all $p\in \mathbb{R}^+$, let $S_p\equiv \{B\subset A|B\mbox{ is finite}, \sum_{b\in B}|b|=p\}$. Then for all $p\in \mathbb{R}^+$, there exists an angle $\phi\in \mathbb{R}^+$ such that for all $B\in S_p$, there exists a positive integer $k$ such that for all sets $\Theta$ consisting only of elements $\theta$ such that for each $b\in B$, there is exactly one $\theta \in \Theta$ cut by $b$, \[\sum_{\theta\in \Theta}\theta = \phi\]

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