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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?

Sage
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    No, they're all $\mathbb{Z}$, just written in different ways. – Qiaochu Yuan Oct 11 '23 at 07:59
  • Base $b$-systems are only natural if $b\ge 2$ is an integer. Already the unary system is quite artificial and barely of any use. – Peter Oct 11 '23 at 08:40
  • @peter what would be 37th digit in base 37 numeral system be? Kindly ponder. – Sage Oct 11 '23 at 08:44
  • PARI/GP uses a vector with integer entries in the range $[0,b-1]$. Alternatively , you could invent arbitary new symbols as digits , this does not matter. Of course , this symbol method will at some point arrive at limits. But who wants to work in , say , a base $2^{25}$-system ? – Peter Oct 11 '23 at 08:54
  • @Peter thanks sir, but my emphasis was how the things originated in first place and philosophy of number in different cultures and time periods. – Sage Oct 11 '23 at 08:59

1 Answers1

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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.

Andrea Mori
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  • my basic question is about https://www.youtube.com/watch?v=R9m2jck1f90 Base 60 (sexagesimal) because it appears actually babylonians are also using base 10 system even you see the notation used in video of reference. – Sage Oct 11 '23 at 08:39
  • @Sage : we use a disguised sexagesimal base when we divide an hour in 60 minutes, a minute in 60 seconds, or an angle of $2\pi$ in $360$ degrees. Yet, this has nothing to do with the number themselves as I tried to explain. – Andrea Mori Oct 11 '23 at 08:43
  • @ Andrea Mori thanks plz be kind to explain the "disguised". it seems that any base numeral system more than base 36 would be disguised as there be no symbol per se in English alphabet and Arabic numerals to represent the last number e.g., in base 37 what would be the last digit? – Sage Oct 11 '23 at 08:49
  • also in sexagesimal base 36 would not be a digit. but a numeral based on decimal base. Is this meaning of disguised that we are using notation of decimal number system to represent a number in sexagesimal number system? – Sage Oct 11 '23 at 08:53
  • @Sage: disguised means that although we actually use decimals to write minutes and seconds, the system works so that a unit is divided into $60$ equal parts and not $100$ as it would be more natural with a base $10$ notation. Also: there's no problem, technically, about having a base bigger than 36: we are not bound to use letters and digits, we can use any set of symbols. – Andrea Mori Oct 12 '23 at 13:26