I love the idea... on the surface, it sounds like it would be easy, and you could choose how you would like to count. How many symbols do you want in your number system? 10? Then you would count: 0,1,2,3,4,5,6,7,8,9,10,11,12..... read more
One sense in which one can "count" in "base [math]\pi[/math]" is like so: any real number [math]\geq 0[/math] can be written as a Laurent series in [math]\pi[/math], where each coefficient is a natural number [math]<\pi[/math] (non-uniquely in general, but if there is a finite such series, it is unique). read more
The set $\{\pi^n : n\in\mathbb{Z}\}$ is linearly independent under $\mathbb{Q}$, so certainly it is independent over {0, 1, 2}. An element with a finite base expansion as above can be written as a vector with the independent $\pi$ power basis, so it's unique. read more