Infetisimal calculus. Take half, and half again. Keep getting closer but you never really get there.

It does though. It defines the sum of an infinite series as the limit that the partial sums approach. — Pfhorrest

I understand what you are saying but a literal infinite sum is not considered in calculus. If partial sums are added they approach the limit. If more terms are added it gets closer to the limit and so on. This is how people like Cauchy formulated calculus. They don't consider literal infinite sums. And this is the question I am raising: is the concept of an infinite sum coherent?
When infinitesimals were invented/discovered it was asserted that, they are so small, no matter how many of them are added together, you end up with another infinitesimal. This is why Berkeley balked at these 'Ghosts of departed quantities.' They were so small you couldn't do anything with them.

As I see it, the conclusion that was being reached was that an infinite sum of zeros add to 1. And this did not make sense. So infinitesimals were invented. This is why I have reservations about literal infinite sums.

Actually it is, that's why they use the equals sign. It's the entire essence of calculus — Pantagruel

I'd disagree since I can express the outcome of the limit of 1 divided by x with x approaching zero, but I cannot simply divide by zero. Further more depending whether I approach zero with x from infinity or from minus infinity provides two vastly different outcomes. Saying that the limit of something equals the equal sign is effectively saying that minus infinity = infinity in the provided example. Hence in mathematics we say that 1 divided by zero is unsolvable rather than saying that it's both infinity and minus infinity.

What a silly thing to say. .999, eighteen, XVI, and .999... all represent numbers. — InPitzotl

I take my definition of "number" from OED: "an arithmetical value representing a particular quantity and used in counting and making calculations". Notice specifically the criteria "particular quantity". This rules out the possibility that .999... is a number.

Show me a definition of "number" which allows that .999... is a number.

Show me a definition of "number" which allows that .999... is a number. — Metaphysician Undercover

Kummer did not believe in real numbers; "God made the integers and all the rest is the work of man". This is a bit extreme as real numbers - whatever they are - are cool. Without them we would not have calculus.

I think the system of real numbers allows that "number" remain undefined, indefinite, and this is why "the real numbers" is not a fixed system. Rigorous defining of "number" has been withdrawn for the sake of producing the real numbers.

Show me a definition of "number"... — Metaphysician Undercover

...and there's Meta's problem.

Family Resemblance.

Numbers are symbols that are only useful and meaningful when applied to real world situations. Does 0.999... in your calculations get your spaceship to the next start system, or does 1? If not then the correct solution lies somewhere in between. So if you really want to know if 0.999...=1 then apply it to the real world. If you can't, then the distinction is meaningless and useless.

"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.

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.

Notice specifically the criteria "particular quantity". This rules out the possibility that .999... is a number. — Metaphysician Undercover

Ah, I think I understand. This is a language barrier. In the language spoken by the mathematics community, .999... represents the same particular quantity that 1 does.

I think the system of real numbers allows that "number" remain undefined, indefinite, and this is why "the real numbers" is not a fixed system. Rigorous defining of "number" has been withdrawn for the sake of producing the real numbers. — Metaphysician Undercover

Yes exactly, calculus works in practice. You can sum $10^{1000000000000000000000000}$ terms and that's fine because it is a finite sum.

But what is $10^{\infty}$?

that 0.999... is taken as another representation for 1 — Andrey Stolyarov

It may well be that the infinite sum is 1 but mathematicians were suspicious about such a concept because infinity is not a number. This is why calculus is formulated in terms of limits, not infinite sums.

What matters to the present discussion is that .999... does not represent a number. Nor does .111... — Metaphysician Undercover

Well. fair enough. For you, real numbers are not numbers. Just for the heck of it, what are they, then? And never mind numeral or representation, we've covered that. That which the rest of the world calls a number you say is not. What do you say it is?

The diagonal of a square, for example, measured in the units that the sides are measured in, is how long? Is that length not a number? Or did something magic happen?

The diagonal of a square, for example, measured in the units that the sides are measured in, is how long? Is that length not a number? Or did something magic happen? — tim wood

In geometry the length of the line - in this example $\sqrt{2}$ - is exact. But the decimal expansion representing it is not, unless we go to an infinite number of places.

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. — Andrey Stolyarov

Sounds like mathematical poetry.

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.

Sounds like mathematical poetry. — Harry Hindu

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

No, you're not getting the point. There's no need for any "infinite sum" — Andrey Stolyarov

But this is what is being asserted: 1/10 + 1/100 + ...taken to an infinite sum of terms. Let S be this literal infinite sum. What is S? Is it 1 or $\infty$? That is what the problem is, we don't know what a literal infinity is because finite algebra does not apply to literal infinities.

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. — Andrey Stolyarov

Earlier on someone wrote a very convincing 'proof':
x = 0.999...
10x = 9.999...
10x = 9 + 0.999...
10x = 9 + x
9x = 9
x = 1

All well and good. But what does x = 0.999... mean? In terms of infinite sums let the sum be S.
The last two lines give:
9S = 9
S = 1.

But what if S is infinite? That is, what if an infinite sum of terms is infinite?
Then we have $9\infty = 9$
$\infty = 1$

That's the thing, we don't really know what S is because you can't apply finite arithmetic to infinite sums.

Numbers are symbols that are only useful and meaningful when applied to real world situations. — Harry Hindu

Is strong encryption a real world situation?

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.

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':
x = 0.999...
10x = 9.999...
10x = 9 + 0.999...
10x = 9 + x
9x = 9
x = 1

All well and good. But what does x = 0.999... mean? In terms of infinite sums let the sum be S.
The last two lines give:
9S = 9
S = 1. — EnPassant

This standard "proof" is of course bullpucky. It's true, but not actually a proof at this level. Why? Well, as you yourself have pointed out, the field axioms for the real numbers say that if $x$ and $y$ are real numbers, then so is $x + y$. By induction we may show that any finite sum is defined. Infinite sums are not defined at all.

To define infinite sums, we do the following:

* We accept the axiom of infinity in ZF set theory, which says that there is an infinite set that models the Peano axioms. We call that set $\mathbb N$. The axiom of infinity is a very powerful assumption that allows us to get higher mathematics off the ground. However, the axiom of infinity is manifestly false in the physical world. It's precisely at this point that mathematics diverges from physics. It doesn't matter how useful math is for physics. One must realize that no mathematical truth can have ontological significance in the physical world. The fact that .999... = 1 is in the end no more meaningful than asking why the knight moves as it does in chess. It's just a consequence of the rules of a formal game.

* Having modeled $\mathbb N$ within set theory, we use equivalence relations to build up the sets $\mathbb Z$ and $\mathbb Q$ of integers and rationals, respectively.

* Using Dedekind cuts (or any of a number of other constructions) we define the real numbers $\mathbb R$ and show that they are a complete, totally ordered, Archimedean field. Such a field is categorigal, meaning that any other such field is isomorphic to the one we've constructed. Complete in this context means that the reals have the least upper bound property, which says in effect that there are no "holes" in the real numbers. This is the property that characterizes the reals and distinguishes them from the other famous densely ordered set, the rationals.

* Having rigorously defined the real numbers and shown that they are essentially unique, we then define the limit of a sequence of real numbers via the usual epsilon definition.

* Having defined the limit of a sequence, we then define (as you have pointed out) the sum of an infinite series as the limit of the sequence of partial sums.

* Having done all that, we then prove a theorem that says that if we have a convergent sequence, we can multiply each of its terms by a constant, and the resulting sequence converges to that constant times the sum of the original sequence.

That last theorem is the ONLY WAY to justify $10 \times .999... = 9.999...$

So using that proof requires mathematical principles and reasoning far more sophisticated than the mere fact that .999... = 1. This "proof" is at best a heuristic for beginners. I wouldn't object to it if it were presented this way. But every time someone writes $10 \times .999... = 9.999...$, they are implicitly invoking the theorem on term-by-term multiplication of a convergent infinite series by a constant; and they are leading students into confusion.

As you have noted, addition in the real numbers is only defined for finite sums. To define infinite sums requires a whole lot of technical machinery based ultimately on the axiom of infinity. We must in fact make a powerful conceptual leap, one that contradicts everything we know about the real world, to get a satisfactory theory of the real numbers.

Same remarks for the 1 = 3 x 1/3 = 3 x .333... = .999... proof. A heuristic for beginners, but hopelessly bogus as an actual mathematical proof. Not because it's wrong, but rather because it is secretly invoking mathematical principles that are deeper and more sophisticated than the fact claiming to be proved.

But what if S is infinite? — EnPassant

Then it wouldn't be representable by a repeating decimal. Only series that converge (not diverge to infinity) can be represented by repeating decimals. So you'll never have this problem with a proof of that sort.

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. — Andrey Stolyarov

You need to read back a few pages to see what I'm saying. It is like this-
It is being asserted that 1/10 + 1/100 + ...taken to an infinity of terms is 1.
If we take a finite number of terms they converge to a limit. But a limit is not a sum. It is what a finite sum converges to. We can't know what an infinite sum is. It may well be 1 but it may also be infinity. We don't know because you can't apply finite arithmetic to infinity. You can't jump to an infinite sum and assume it is 1.

A limit is by definition something that will not be exceeded. We can be absolutely sure that 1/1 + 1/2 + 1/4 + 1/8 ... will never add up to infinity, because the limit of the partial sums is 2, which means it will never ever ever add up to more than 2, and only "at infinity" will even add all the way up to 2.

A limit is by definition something that will not be exceeded. We can be absolutely sure that 1/1 + 1/2 + 1/4 + 1/8 ... will never add up to infinity, because the limit of the partial sums is 2, which means it will never ever ever add up to more than 2, and only "at infinity" will even add all the way up to 2. — Pfhorrest

No. Calculus is formulated in terms of finite sums and limits. You can't jump to infinity and expect the rules of finite arithmetic to apply. Jumping from the finite to the infinite is an infinite distance and we don't know what happens there.

I'm only talking about finite sums and limits.

The limit of the series of finite sums represented by 1/1 + 1/2 + 1/4 + 1/8 ... is 2. You do agree that you can calculate a limit, right? We can know that the limit of that series is not infinity, right? What do you think "the limit is 2" means, if not "this will never add up to more than 2, no matter how many terms you add"? And given that, we know that it will never, ever add up to infinity.

A limit is by definition something that will not be exceeded. — Pfhorrest

By a finite number of terms.

What do you think "the limit is 2" means, if not "this will never add up to more than 2, no matter how many terms you add"? — Pfhorrest

Yes, you are correct but what is an infinite sum of terms? What happens when you sum an infinity of terms? Calculus does not account for this.