The comments section of this post 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?)
On
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.
On
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.
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.
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.
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.