We can't know what an infinite sum is — EnPassant

Hmm, have you ever asked yourself a simple question "why"? Definitely we can, and we do, for centuries (since Leonard Euler's time). However, if you postulate it this way - something like "when we see the infinity sign, we can't know anything" - then (despite you thereby postulate math mostly doesn't exist) perhaps no one can argue. This just means you don't believe in math (again, despite that, actually, math doesn't need to be believed in). With precisely the same effect you can deny to believe anything at all, e.g., me or other people on this forum, postulating we simply don't exist. Or, you can postulate that the Earth is flat. It is impossible to disprove postulates, because any proof must be based on postulates, too, and if you postulate something, then every postulate that contradicts becomes false in your own universe.

By the way, if you state that infinity is something that can't be researched and/or known of, then these damn infinite decimal fractions will disappear in fear. They are based on the assumption we can work with infinities.

You can't jump to infinity and expect the rules of finite arithmetic to apply — EnPassant

Definitely we can, and we always do. Actually, this is all higher math (in contrast to elementary math) is all about. Correctness of such a "jump" is shown and proved centuries ago. I can tell you even more. There's beautiful piece of math named "Functional analysis" (https://en.wikipedia.org/wiki/Functional_analysis), which works with spaces that have infinite number of dimensions (I believe this should impress more than just an infinity in a single miserable dimension), and, BTW, this piece of math has a lot of practical applications. May be I surprise you if I say that Pythagorean theorem perfectly works in some of these (3D? 4D? 5D? infinite-D!) spaces.

If we return to the topic, 0.999... doesn't need any higher math or any magic to be 1. It is simply a practical fact which, as I mentioned before, I was told about in elementary school.

But what is 10∞? — EnPassant

This doesn't exist, because there's no limit for the sequence 10, 100, 1000, ... (That is, the sequence {10^n}). If you take, e.g., 1.1 instead of 10 (or actually any number between 1 and 2, but strictly lesser than 2), the limit will exist, so we'll be able to say what it is. Actually, as far as I remember, the limit will be the number of 1. There will even be a finite sum of the series (like 1.1 + 1.1*1.1 + 1.1*1.1*1.1 + ...).

Earlier on someone wrote a very convincing 'proof': — EnPassant

It is not a proof at all, there's no theorem here. This is standard and well-known technique to convert a periodic decimal fraction back to rational form. Yes, for 0.(9) it works as expected.

But what if S is infinite? — EnPassant

It is not. For any decimal fraction, the limit of both the corresponding sequence and the corresponding series is always finite, which is obvious enough: it is always between 0 and 1, there can be no infinity here.

Furthermore, the equality 9S = 9 has the obvious solution S = 1, and it doesn't have any other solutions. In most math theories, infinity is not considered as a number so we can't multiply 9 and infinity; however, if we add infinity to the domain (which is rarely done in math, but is still possible), then 9 times infinity will be again infinity, not 9. So infinity is not a solution for the equation, even if we agree to use infinity as a number.

BTW, when it comes to limits, including the limits of "infinite sequences", we don't need the infinity as such. If you recall the definition of the notion of limit, the symbol "infinity" (I can't figure out how do you make them appear here) is not used in that definition at all. In this context, the word "infinity" simply means that we can take as many members of a sequence as we want or need, and no one will stop us from getting more of them. Actually, the word "infinity" in math is not as complicated and scary as you can expect.

Sounds like mathematical poetry. — Harry Hindu

Exactly. They enjoy what they do just like poets enjoy writing their poems.

It may well be that the infinite sum is 1 — EnPassant

No, you're not getting the point. There's no need for any "infinite sum", schoolchildren don't learn such complicated math that early. I don't really remember what school year it was, may be 6th (that is, 12-13 years old children were learning this). The phrase "0.(9) is another representation of 1" came without any explanations in that textbook; only several years later, I understood why it is true.

This is why calculus is formulated in terms of limits, not infinite sums — EnPassant

And so what, actually? What do you think an "infinite decimal fraction" is? Well, by definition, it is exactly that famous Limit. For this particular case, you can either take the limit of the sequence 0.9, 0.99, 0.999, ..., or you also can work in terms of so called series (see https://en.wikipedia.org/wiki/Series_%28mathematics%29) and consider the series of 9/(10^n), that is, 9/10 + 9/100 + 9/1000 +..., the result will be exactly the same. Mathematicians are good with both. Furthermore, I'd say mathematicians are "suspicious" (well, this is the right word) about infinite decimal fractions as such. Math is not done on decimal fractions; physics is, but physics is different. For a mathematician, rational number is a fraction n/m (where n is integer, m is natural). The problem is that there exist sequences of rationals that obviously have limit, but the limit can not be represented as n/m, so in order to "close up" the set of numbers, we need more numbers. That's how "real" numbers appear. Infinite decimal fractions are just one simple case of these sequences and series.

Again, do you have a system in mind where wages are paid on the basis of effort? — Isaac

As far as I remember, there were multiple attempts to build such a system, and they failed. There's a simple reason for it. The same amount of effort made by people of different abilities make different amount of value. And the things are even worse than that: actually, no one can tell whose effort "really" costs more or less, there's no way to measure.

Folks, I'm a bit surprised with all the discussion. I even wanted to show that 0.(9) is the same as 1 several ways (okay, the number as such is the same, 1, and 0.(9) is simply its alternative representation) but I read thru the discussion and noticed that every way of showing it (of what I know) is already mentioned by someone. But this is not what surprised me.

When I was a schoolboy, and we learned infinite (both periodic and non-periodic) decimal fractions in the class, there was a clear phrase in the textbook we used, that 0.999... is taken as another representation for 1. Surely the situation may be different in other countries, but I'm still a bit surprised anyway.

"Numbers are symbols that are only useful and meaningful when applied to real world situations."

That's true for physics. That's even true for the so-called applied math. But this is not true for math in general. There exist huge math theories that have no applications at all, and math people often tend to be proud of such math that can not be used for any practical needs. Most (not all) of tensor-related math is a good example.