modified roman numerals

documented from this post

start by breaking roman numerals into the simplest possible rules:

• there is an arbitrary number of symbols
• each symbol has a given value, each of which is part of a ring (wiki page: ring). the ring restriction is overkill but it will make sense later
• each symbol has a given order
• a symbol placed before a symbol of higher order is worth the additive inverse of its value (in numbers, 1 before 5 is worth negative 1)
• a sequence of symbols denotes the value acquired by summing the value of each symbol

now we have a pretty versatile system, right? like you can represent just about anything from a group that can add and subtract into itself (like the integers, for example) [edit: untrue! i'm not sure for what exactly this doesn't hold, but we'll start with division-rings, most of those pretty obviously don't work out, since we can't get, for lack of a better term, "finer resolution" values than the finest one defined in our set of symbols. this is the whole reason i implemented a system for the rationals later in this post and i'm pissed i missed on this detail when i proofread this post the first time]

but it gets better, because we can also impose a couple more symbols. a pair of symbols for grouping a sequence of symbols such that it acts as one, and a symbol to take the multiplicative inverse of another symbol. this is why i defined the elements to be part of a ring earlier, it guarantees this part works. we also have to tweak our original system to, instead of defining an order for each symbol, allow to compute the order of symbols as a bijective function of the ring containing all of their values to real numbers so that we can decide whether groups of symbols should be added or subtracted from other symbols

now we can do fun things like represent every rational number in our generalized roman numerals and it's only a little bit miserable

here's a tiny set of symbols you need to do that:
• i (a symbol with a numeric value of 1)
• [] (opening and closing group symbols)
• ~ (takes multiplicative inverse of next symbol or grouped symbols)

now we can represent any rational number:
1/2 = ~[ii]
2/3 = ~[iii]i (subtracts 1/3 from 1)
-4/7 = ~[iiiiiii]~[iiiiiii]~[iiiiiii]~[iiiiiii]ii[ii] (subtracts 4 copies of 1/7 and 2 copies of 1 from 2)

you can also write everything a basically infinite number of ways. for instance the following are also true:
1/2 = ~[iiii]~[iiii]i
-4/7 = iii[ii]~[iiiiiii]~[iiiiiii]~[iiiiiii]

thanks for reading my post about an exceedingly pointless number system. if you would like to expand on my system (possibly by adding some subset of irrational numbers through additional syntax, or even higher dimensional numbers, oh boy) or have some fun with ridiculous quirky things you can do using it, please do not hesitate to @ me. i am also interested in factually incorrect things i have written here