This reminds me of being in secondary school and procrastinating/nerd sniping our teacher for A-level Further Maths (so I was roughly 17-18) by arguing that a number system in base e (so 1, 2, 2.1, ..., 2.7, 2.71, ..., 10) would simplify a lot of the maths we were studying.
Except that's not how bases work. The digits would be of the form e^n. See [1]. Since e is not an integer, its number system uses 3 symbols (0, 1, 2) as 3 is bigger than e. Thus, 20 = 2e^1 + 0e^0 = 5.43656...
I don't know about simplify. Subtracting 1 from the above would yield an irrational number. So certainly not useful for arithmetic. How about for mathematical proofs? Well, any integer above 3 is also irrational. Makes Taylor series annoying. Even mundane things like the factorial would not have a nice representation.
> a number system in base e (so 1, 2, 2.1, ..., 2.7, 2.71, ..., 10)
I don't think I understand the significance of those dots. How would one write, say, 100 in this number system?
With that said, the usual "base" notation works perfectly well for non-integer radices, though it can behave in unexpected fashion. Thus, I might write 12012 for e^4 + 2e^3 + e + 2, which is approximately 99.49, and so can be regarded as the "e-adically integer" part of 100.
