Number systems and ring Theory

Question

Are different base number systems defined as separate rings or single ring e.g Z? asking question in other words decimal numeral system different fundamentally from unary or binary or ternary or any other? wiki says that even fractions can be used as bases. https://en.wikipedia.org/wiki/List_of_numeral_systems

If yes what does the different number systems vary in fundamentally when "10" written numerally in

• base 2 = 2 in base 10(decimal system)
• base 3 = 3 in base 10(decimal system)
• base 4 = 4 in base 10(decimal system)
• base 5 = 5 in base 10(decimal system)
• base 6 = 6 in base 10(decimal system)
• ...
• base 10 = 10 in base 10(decimal system)

does the numeral "here 10" differ fundamentally in value or something else?

Answer 1

The Peano axioms characterize the set $\mathbb{N}$ of natural numbers as an infinite set with certain properties. Then the integers $\mathbb{Z}$ are constructed by some set theoretical procedure as, in down to earth terms, the smallest set containing the additive inverses of all natural numbers, thus making it a ring.

For the above it is irrelevant how you actually write the numbers.

The number "five" is the successor of the successor of the successor of the successor of the successor of "zero" which, according to the axioms, is the only natural number not a successor of any other natural number, no matter if you write it

• $5$ (decimal notation),
• $101$ (binary notation),
• V (roman numerals),

pretty much in the same way as you may call it cinq, cinco, funf or whatever language you choose to speak.