Question: The comments section of [this post](http://www.ehow.com/how_5157582_calculate-pi.html) says that $\pi$ does repeat itself if done under base 11... and that it somehow defines the universe.

* Can anyone expand on the idea that irrational numbers may repeat if a different base is used?
* Does $\pi$ in fact repeat under base 11?
* Is there any known value in numbers that have this property? (or conversely the ones that don't have this property?)

Answer: 
The comment is mistaken.

A number is rational if and only if its expansion *in any base* is eventually recurring (possibly with a recurring string of $0$s). Thus $\pi$ has no recurring expansion in any base.

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**Why?** Well suppose it did. Then
$$\pi = n . d\_1d\_2 \cdots d\_k \overline{r\_1 r\_2 \cdots r\_{\ell}}$$
in some base $b$ say, where $r\_1 \cdots r\_{\ell}$ is the recurring part. Thus
$$b^k\pi = m . \overline{r\_1 r\_2 \cdots r\_{\ell}}$$
where $m$ is some integer, it doesn't really matter what it is. Also notice
$$b^{k+\ell}\pi = mr\_1 r\_2 \cdots r\_{\ell}. \overline{r\_1 r\_2 \cdots r\_{\ell}}$$

Subtracting one from the other gives
$$b^{k+\ell}\pi - b^k\pi = mr\_1 r\_2 \cdots r\_{\ell} - m = M$$
where $M$ is again some integer, and hence
$$\pi = \dfrac{M}{b^{k+\ell} - b^k}$$
so $\pi$ is rational.

...this ain't true -- we know that $\pi$ is irrational -- so our assumption that $\pi$ has a recurring expansion to base $b$ must have been mistaken.

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My notation above was a bit sloppy. The $d\_i$ and $r\_i$s refer to *digits* whilst the other letters refer to *numbers*. I hope it's understandable, but if it isn't then let me know and I'll clarify.
> 
> **Zach Lynch**  
> 
>  actually, pi repeats if you use base 11 arithmetic and use a computer to search for patterns. It describes the universe as we know it.
> 
> 
> 

*What* computer can tell whether a pattern will repeat? This means not knowing the basics of mathematics. The second statement is simply ridiculous. Please, be careful in believing to what can be found on the net.

A definition of an irrational number that uses base ten expansion would be ridiculous as well and indeed nobody seriously interested in mathematics would accept it.

A number is rational if some integer multiple of it is integer. It is irrational otherwise. The fact that $\pi$ is irrational has been established by Joahann Heinrich Lambert in 1761, more than 250 years ago. Can you believe that in 250 years nobody would have caught a flaw in the proof of such an important statement? 

About a century later, Lindemann proved the fact that $\pi$ not only is irrational but also transcendental. The proof by Lindemann closed, with a negative answer, a 22 century old problem, that is, squaring the circle with ruler and compass. Can you believe that in more than 100 years nobody would have caught a flaw in Lindemann's proof or in the one of a more general theorem by Weierstraß?

Of course, this might be a conspiration involving all mathematicians, who don't want that the rationality of $\pi$ is known by the general public, because the NSA folks base on it their algorithms for decoding instantaneously all encrypted communications.