Tangentially related: in modern math we do have alternative number systems to use for problems that are cumbersome with Arabic-Hindu numerals. Check out p-adic numbers on Wikipedia, for instance.
Yes, if I remember correctly, any n-ary number system is equivalent and you can translate any proof between them. It's just easier to work in one or another system.
p-adic is quite different though. It's not really that relevant to this conversation though as far as I can tell as it's not just about representing the integers or rationals in a different way.