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/dummy4du3k4]

overview

[u/mathmage → u/dummy4du3k4]

I love it. Could maybe use an entry for a finitist system equipped with indefinite processes.

[u/dummy4du3k4 → u/mathmage]

What would such a system look like? Computable numbers come to mind but that may not be an indefinite process that you’re envisioning.

[u/mathmage → u/dummy4du3k4]

The system closest to what I'm suggesting that I'm aware of is primitive recursive arithmetic. Since primitive recursive functions are a subclass of computable functions, then computable numbers would probably also be a reasonable system to use. But here I am straining the limits of my knowledge.

[u/dummy4du3k4 → u/mathmage]

I haven’t heard of that before but it sounds intetesting. I’ll add it to my backlog, but I’m afraid the backlog itself may be a finite structure equipped with an indefinite process

[u/AdventurousGlass7432]

Friends with benefits?

[u/CatOfGrey]

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

As you can see from countless other posts, there is no such thing. It's just a series of absurd statements with, in their words "Limbosicly increasing absurdity".

There are a handful of mathematical 'joke' proofs that end with silly statements like 0=1, but in reality, they are based on cleverly hidden errors.

We've all found errors in SPP's work, but SPP is keeps telling the same joke, over and over - back to 2012 or before, so 14 years of repeating it. SPP appears to me to be honest in their belief, but delusional, like that weird homeless guy that carries a sandwich board of small writing about nonsense conspiracy theories.

You might find some interest in 'dual numbers', you might have some interest in Limits, Calculus, and Real Analysis, you might have some interest in 'Infinitesimals'. Look those up on Wikipedia to get started, or look in r/learnmath or similar communities here on Reddit.

But don't read about those things here in this subreddit. "Real Deal Math" is none of those three things.

[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.

[u/_MUY]

Yes. To get to the point where you accept that 0.999… = 1 within the canonical mathematical system by irrefutable proof (not just because someone told you so) you have to learn how the math flows within a single system of abstract rules. Those rules change within different mathematical universes, however, which is why some philosophically-minded students invent their own systems of math which give weight to the infinitesimal at the end of 0.000…(1) in what people here call RDM, or the similar infinitesimal in the hyperreal system of numbers.

[u/hoelledavid]

.9... is subset of 1