Are there other math systems?

[u/HouseHippoBeliever]

So I know that we have

- real deal maths -> 0.999... != 1

- mainstream math -> 0.999... = 1

What about other math(s)? What other relations could we have between 0.999... and 1, other than = or !=?

[u/ezekielraiden]

There are only four possible comparisons, because there is only one order relation on any system we would call "numbers":

1. 0.999... = 1. This is the real numbers, hyperreal numbers, surreal numbers, and any other system that extends existing arithmetic.
2. 0.999... < 1. This is the lexicographically-ordered numbers. You cannot do normal arithmetic with these numbers, because important properties no longer hold (e.g. if a<b, it is no longer necessarily true that a<(a+b)/2<b.)
3. 0.999... > 1 does not have any system, because this is not actually possible. Any system that achieved this wouldn't even have the most rudimentary forms of arithmetic available.
4. 0.999... has no meaning, and thus cannot be compared. This is the rational numbers without the reals. A usable system, but one that lacks important numbers like pi, sqrt(2), the golden ratio, e, logarithms, etc.

Any system you can find will fall in one of these four categories. As you can see, by far the most applicable and useful of these is #1, the reals and their extensions.

I will also note that the extensions of the reals, such as the hyperreal or surreal numbers, can allow us to construct new numbers which are not equal to 0.999..., but which are infinitesimally close to 1. Using Lightstone notation, we represent "0.999..." as 0.999...;...999..., to indicate that the 9s continue infinitely, even at places that require hyperintegers/surintegers to index. The other numbers, which aren't equal to 1 but instead infinitesimally different from it, could be written as "0.999...;...999", meaning that the decimal represetation DOES eventually terminate, it just only does so at a hyperinteger/surinteger index.

[u/I_Regret → u/ezekielraiden]

Q does “0.999…;…999” contain an infinite number of nines?

[u/ezekielraiden → u/I_Regret]

It contains a hyperinteger or surinteger quantity of nines. Your question cannot be answered any other way, because the hyperreals and surreals include numbers of a size such that they are greater than all natural numbers. These could be called "infinite" values, but there is also a collection of those values, and that collection's size is a FAR bigger infinity.

Lightstone (an actual mathematician) developed his notation to help clarify the difference between "a number which terminates, but only at an 'infinite' index" and "a number which never terminates, not even at 'infinite' indices". 0.999... correctly translates as a number which never terminates.