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.
**A irrational number is a number that cannot be expressed as an integer over another integer.**

That is the *definition*, and it does not say anything at all about repeating or terminating decimals. It is a **theorem, not a definition** that when expanded in a base-$b$ numeral system, where $b\ge 2$, it does not terminate or repeat.

The erroneous definition, that rationality and irrationality are defined via terminating or repeating expansions, is a very very persistent meme, lasting many decades, without ever being taught in classrooms. Everyone who's graded homework assignments knows there are lots of memes that persist that way without ever being taught.