| it seems inevitable to me that someone would invent binary, or ternary, or likely nonary bases are just the number of symbols you’re allowed to use to represent a number. so base 10 has ten symbols: 0123456789. base 2 has two symbols: 01, base 3 three: 012, etc etc. someone will eventually look at a system with 10 symbols and think “what if we have x symbols instead of 10?” once you realise the basic mechanism through which decimal represents a number - i.e. you can write any natural number as “some y multiples of x + [a number z between 0 and x-1]” - with a bit of nesting you can derive any number of any base take 1327. let’s write it in base 9. the way I would do this is to write 1327 in the form: xy + z = 9y + [0,8] then if y != 0, we rewrite y itself in this form, then again with each new y, until y = 0. each z we produce is the next most significant digit in the answer. when y=0, the z produced is the most significant digit 1327 = 9147 + 4. so our number ends with 4 y=147 is not 0, so we do 147 = 169 + 3. so our number ends with 34 y=16 is not 0, so we do 16 = 19 + 7. our number ends with 734. y=1 is not 0, so we do 1 = 0*9 + 1. our number ends with 1734 y=0 is 0 so we terminate and our number is 1734 you can do this with any number and any base as long as you have the symbols for it |