What is the numeral system in which $\pi$ would have the lowest number of decimals possible?
closed as off-topic by Najib Idrissi, Peter Franek, Claude Leibovici, Daniel W. Farlow, Lord_Farin Sep 6 '15 at 9:37
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$\pi$ is an irrational number meaning it can never be expressed as a ratio of integers. In particular this means $\pi$ has infinitely many digits after the decimal point in any integer base $b>1$, for if not, we would have
where each of the coefficients $c_i$ are integers between $0$ and $b-1$. If $k\ge0$ this directly implies $\pi$ is an integer, while if $k<0$, we have
so $\pi$ would be rational.
Since $\pi$ is irrational, it has no finite decimal representation in any integer base $b>1$.
However, we can use non integer bases. In fact, we can represent numbers using base $\pi$ itself! $$\pi=10_\pi$$