You are here
Home$\pi$ and $\pi^2$ are irrational
Primary tabs
$\pi$ and $\pi^{2}$ are irrational
Theorem 1.
$\pi$ and $\pi^{2}$ are irrational.
Proof.
For any strictly positive integer $n$ ,$x\in(0,1)$ we define:
$f=f(x)=\frac{x^{n}(1x)^{n}}{n!}=\frac{1}{n!}\sum_{{m=n}}^{{2n}}c_{m}x^{m}$ 
where $c_{m}$ are integers. For $0<x<1$ we have
$0<f(x)<\frac{1}{n!}$  (1) 
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
$G(x)=b^{n}[\pi^{{2n}}f(x)\pi^{{2n2}}f^{{\prime\prime}}(x)+\pi^{{2n4}}f^{{(4% )}}(x)...+(1)^{n}f^{{(2n)}}(x)].$ 
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
$f^{{(m)}}(0)=\frac{m!}{n!}c_{m},$ 
an integer. Hence $f(x)$ and all its derivates take integral values at $x=0$.Since $f(1x)=f(x)$, the same is true at $x=1$
so that $G(0)$ and $G(1)$ are integers. We have
$\displaystyle\frac{d}{dx}[G^{{\prime}}(x)\sin{\pi x}\pi G(x)\cos{\pi x}]$  $\displaystyle=$  $\displaystyle[G^{{\prime\prime}}(x)+\pi^{2}G(x)]\sin{\pi x}$  
$\displaystyle=$  $\displaystyle b^{n}\pi^{{2n+2}}f(x)\sin{\pi x}$  
$\displaystyle=$  $\displaystyle\pi^{2}a^{n}\sin{\pi x}f(x).$ 
Hence
$\pi\int_{0}^{1}a^{n}\sin{\pi x}f(x)dx=[\frac{G^{{\prime}}(x)\sin{\pi x}}{\pi}% G(x)\cos{\pi x}]_{0}^{1}$ 
$=G(0)+G(1),$ 
witch is an integer. But by equation 1,
$0<\pi\int_{0}^{1}a^{n}\sin{\pi x}f(x)dx<\frac{\pi a^{n}}{n!}<1.$ 
For a large enough $n$, we obtain a contradiction.
For any integer $n$, if $a^{n}$ is irrational then a is irrational (proof), and since $\pi^{2}$ is irrational $\sqrt{\pi^{2}}=\pi$ is also irrational. ∎
The irrationality of $\pi$ was Proved by Lambert in 1761. The above proof is not the original proof due to Lambert.
References
 1 G.H.Hardy and E.M.Wright An Introduction to the Theory of Numbers, Oxford University Press, 1959
See also

The MacTutor History of Mathematics Archive, A history of Pi

The MacTutor History of Mathematics Archive, Johann Heinrich Lambert
Mathematics Subject Classification
5100 no label found1100 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
Recent Activity
new question: Prime numbers out of sequence by Rubens373
Oct 7
new question: Lorenz system by David Bankom
Oct 19
new correction: examples and OEIS sequences by fizzie
Oct 13
new correction: Define Galois correspondence by porton
Oct 7
new correction: Closure properties on languages: DCFL not closed under reversal by babou
new correction: DCFLs are not closed under reversal by petey
Oct 2
new correction: Many corrections by Smarandache
Sep 28
new question: how to contest an entry? by zorba
new question: simple question by parag
Corrections
Contains proof by mathwizard ✓
Last sentence by alozano ✓
portability by jac ✘
notation by Wkbj79 ✓
Comments
\pi is irrational
To make it clearer that \pi is irrational because \pi^2 is irrational, have a look at http://planetmath.org/?op=getobj&from=objects&id=5779
Re: \pi is irrational
Sincerely, what we won with this Mr. Gunnar? I think that anyway we must read the Lambert's hard paper.
$\pi$ and $\pi^2$ are irrational
In general,
$\cos{x} \neq \pi\cos{\pix}$
and
$\sin{x} \neq \sin{\pix}$
Re: \pi is irrational
Well of course it's interesting to read that paper. I'll start googling for it.