Comment by zozbot234
9 days ago
You need some notion of order or of metric structure if you want to talk about numbers being "close" enough to π. This is related to the property of completeness for the real numbers, which is rather important. Ultimately, the real numbers are also a rigorously defined abstraction for the common notion of approximating some extant but perhaps not fully known quantity.
The common metric on the reals one of only two that exist as completions of Q!
The other is the p-padics (basically, low-order bits matter more than high-order bits), which have distance but not ordering.
There are infinitely many different p-adic completions of the rationals for each prime p, so we have 2-adics, 3-adics, 5-adics etc, all different.