| Version 11 |
Version 10 |
| \newtheorem{thm}{Theorem} |
\newtheorem{thm}{Theorem} |
| \begin{thm} |
\begin{thm} |
| $\pi$ and $\pi^2$ are irrational. |
$\pi$ and $\pi^2$ are irrational. |
| \end{thm} |
\end{thm} |
|
|
| \begin{proof} |
\begin{proof} |
| For any strictly positive integer $n$ ,$x\in (0,1)$ we define: |
For any strictly positive integer $n$ ,$x\in (0,1)$ we define: |
| $$f=f(x)=\frac{x^n(1-x)^n}{n!}=\frac{1}{n!}\sum_{m=n}^{2n}c_mx^m$$ |
$$f=f(x)=\frac{x^n(1-x)^n}{n!}=\frac{1}{n!}\sum_{m=n}^{2n}c_mx^m$$ |
| where $c_m$ are integers. For $0<x<1$ we have |
where $c_m$ are integers. For $0<x<1$ we have |
|
|
| \begin{equation} |
\begin{equation} |
| \label{firseq} |
\label{firseq} |
| 0<f(x)<\frac{1}{n!} |
0<f(x)<\frac{1}{n!} |
| \end{equation} |
\end{equation} |
|
|
| For a contradiction, suppose $\pi^2$ is rational, so that $\pi^2=\frac{a}{b}$, where $a,b$ are positive integers. |
For a contradiction, suppose $\pi^2$ is rational, so that $\pi^2=\frac{a}{b}$, where $a,b$ are positive integers. |
|
|
| For $x\in (0,1)$ let us define |
For $x\in (0,1)$ let us define |
| $$G(x)=b^n[\pi^{2n}f(x)-\pi^{2n-2}f''(x)+\pi^{2n-4}f^{(4)}(x)-...+(-1)^nf^{(2n)}(x)].$$ |
$$G(x)=b^n[\pi^{2n}f(x)-\pi^{2n-2}f''(x)+\pi^{2n-4}f^{(4)}(x)-...+(-1)^nf^{(2n)}(x)].$$ |
| We have that $f(0)=0$ and $f^(m)(0)=0$ if $m<n$ or $m>2n$. But, if $n \leq m \leq 2n$,then |
We have that $f(0)=0$ and $f^(m)(0)=0$ if $m<n$ or $m>2n$. But, if $n \leq m \leq 2n$,then |
| $$f^{(m)}(0)=\frac{m!}{n!}c_m,$$ |
$$f^{(m)}(0)=\frac{m!}{n!}c_m,$$ |
| an integer. Hence $f(x)$ and all its derivates take integral values at $x=0$.Since $f(1-x)=f(x)$, the same is true at $x=1$ |
an integer. Hence $f(x)$ and all its derivates take integral values at $x=0$.Since $f(1-x)=f(x)$, the same is true at $x=1$.
|
|
|
| so that $G(0)$ and $G(1)$ are integers. We have |
so that $G(0)$ and $G(1)$ are integers. We have |
|
|
| \begin{eqnarray*} |
\begin{eqnarray*} |
| \frac{d}{dx}[G'(x)\sin{\pi x}-\pi G(x)\cos{\pi x}] |
\frac{d}{dx}[G'(x)\sin{\pi x}-\pi G(x)\cos{\pi x}] |
| &=& [G''(x)+\pi^2G(x)]\sin{\pi x} \\ |
&=& [G''(x)+\pi^2G(x)]\sin{\pi x} \\ |
| &=& b^n\pi^{2n+2}f(x)\sin{\pi x} \\ |
&=& b^n\pi^{2n+2}f(x)\sin{\pi x} \\ |
| &=& \pi^2a^n \sin{\pi x}f(x). |
&=& \pi^2a^n \sin{\pi x}f(x). |
| \end{eqnarray*} |
\end{eqnarray*} |
|
|
| Hence |
Hence |
| $$\pi\int_0^1a^n \sin{\pi x}f(x)dx=[\frac{G'(x)\sin{\pi x}}{\pi}-G(x)\cos{\pi x}]_0^1$$ |
$$\pi\int_0^1a^n \sin{\pi x}f(x)dx=[\frac{G'(x)\sin{\pi x}}{\pi}-G(x)\cos{\pi x}]_0^1$$ |
| $$=G(0)+G(1),$$ |
$$=G(0)+G(1),$$ |
| witch is an integer. But by equation \ref{firseq}, |
witch is an integer. But by equation \ref{firseq}, |
| $$0<\pi\int_0^1a^n \sin{\pi x}f(x)dx<\frac{\pi a^n}{n!}<1.$$ |
$$0<\pi\int_0^1a^n \sin{\pi x}f(x)dx<\frac{\pi a^n}{n!}<1.$$ |
| For a large enough $n$, we obtain a contradiction. |
For a large enough $n$, we obtain a contradiction. |
|
|
|
For any integer $n$, if $a^n$ is irrational then a is irrational \PMlinkexternal{(proof)}{http://planetmath.org/?op=getobj&from=objects&id=5779},
|
Whe have for an integer $n$ that if $a^n$ is irrational then a is irrational \PMlinkexternal{(proof)}{http://planetmath.org/?op=getobj&from=objects&id=5779},
|
| and since $\pi^2$ is irrational $\sqrt{\pi^2}=\pi$ is also irrational. |
and since $\pi^2$ is irrational $\sqrt{\pi^2}=\pi$ is also irrational. |
| \end{proof} |
\end{proof} |
|
|
| The irrationality of $\pi$ was Proved by Lambert in 1761. The above proof is not the original proof due to Lambert. |
The irrationality of $\pi$ was Proved by Lambert in 1761. The above proof is not the original proof due to Lambert. |
| \begin{thebibliography}{99} |
\begin{thebibliography}{99} |
| \bibitem a G.H.Hardy and E.M.Wright \emph{An Introduction to the Theory of |
\bibitem a G.H.Hardy and E.M.Wright \emph{An Introduction to the Theory of |
| Numbers}, Oxford University Press, 1959 |
Numbers}, Oxford University Press, 1959 |
| \end{thebibliography} |
\end{thebibliography} |
|
|
| \subsection*{See also} |
\subsection*{See also} |
| \begin{itemize} |
\begin{itemize} |
| \item The MacTutor History of Mathematics Archive, |
\item The MacTutor History of Mathematics Archive, |
| \PMlinkexternal{A history of |
\PMlinkexternal{A history of |
| Pi}{http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Pi_through_the_ages.html} |
Pi}{http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Pi_through_the_ages.html} |
| \item The MacTutor History of Mathematics Archive, |
\item The MacTutor History of Mathematics Archive, |
| \PMlinkexternal{Johann Heinrich |
\PMlinkexternal{Johann Heinrich |
| Lambert}{http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Lambert.html} |
Lambert}{http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Lambert.html} |
| \item \PMlinkexternal{Irrationality proofs}{http://numbers.computation.free.fr/Constants/Miscellaneous/irrationality.html} |
\item \PMlinkexternal{Irrationality proofs}{http://numbers.computation.free.fr/Constants/Miscellaneous/irrationality.html} |
| \end{itemize} |
\end{itemize} |
