but what is an infinite sum of terms — EnPassant

The limit of the series of partial sums. By definition.

When this limit exists, one says that the series is convergent or summable, or that the sequence is summable. In this case, the limit is called the sum of the series. — Wikipedia

When this limit exists, one says that the series is convergent or summable, or that the sequence is summable. In this case, the limit is called the sum of the series. — Wikipedia

You need to provide the link. As a working definition that may well be useful but there are still problems with infinite sums.

This doesn't exist, because there's no limit for the sequence 10, 100, 1000, ... (That is, the sequence {10^n}). — Andrey Stolyarov

It can be defined using 10-adics:
$\underset{n\to+\infty}{lim}{|10|^n_{10}}=0$

You need to provide the link — EnPassant

The word "Wikipedia" at the bottom of the quote is a clickable link to the article in question.

When this limit exists, one says that the series is convergent or summable, or that the sequence is summable. In this case, the limit is called the sum of the series. — Wikipedia

They don't say the infinite sum, ie the sum of all terms.

They're talking about infinite series, and saying that the limit is the sum of that series. I didn't quote the whole article, just the relevant part. Click the link and read for yourself.

Or I can quote the preceding few sentences too if you're not going to bother clicking:

The infinite sequence of additions implied by a series cannot be effectively carried on (at least in a finite amount of time). However, if the set to which the terms and their finite sums belong has a notion of limit, it is sometimes possible to assign a value to a series, called the sum of the series. This value is the limit as n tends to infinity (if the limit exists) of the finite sums of the n first terms of the series, which are called the nth partial sums of the series. — Wikipedia

They're talking about infinite series, and saying that the limit is the sum of that series. I didn't quote the whole article, just the relevant part. Click the link and read for yourself. — Pfhorrest

It hardly matters. An infinite sum is undefined because nobody has ever computed it. The problem is that while it is ok to apply the logic to finite quantities, nobody knows what an infinite sum is.

nobody knows what an infinite sum is — EnPassant

Actual mathematicians do.

The infinite sequence of additions implied by a series cannot be effectively carried on (at least in a finite amount of time). However, if the set to which the terms and their finite sums belong has a notion of limit, it is sometimes possible to assign a value to a series, called the sum of the series. This value is the limit as n tends to infinity (if the limit exists) of the finite sums of the n first terms of the series, which are called the nth partial sums of the serie — Wikipedia

Emphasis mine.

An infinite sum is undefined because nobody has ever computed it. — EnPassant

An infinite sum is defined because the mathematics community defined it; same as "twelve" is defined because English speakers defined it.

tends to infinity — Wikipedia

...that's a term of art. It means to increase without bound. You're choking on mathematical language that you think represents some ideal thing that it just doesn't represent.

ETA: Quite honestly, this whole discussion of infinite sums reminds me a lot of the maths video from Look Around You:

Narrator - What's the largest number you can think of?
Girl - uhm... hundred thousand?
Man - nine hundred and ninety nine thousand
Older man - A million
Narrator - In actual fact it's neither of these. The largest number is about forty five billion, although mathematicians suspect there are even larger numbers

...if I'm to understand correctly, the quibble you have is about what the "actual" infinite sum "actually" adds up to. Ironically, your quibble includes the notion that infinity is not a number. So I have no idea what you're talking about. Mathematicians define infinite sums differently.

Actual mathematicians do. — Pfhorrest

See next answer.

An infinite sum is defined because the mathematics community defined it; same as "twelve" is defined because English speakers defined it. — InPitzotl

That still begs the question what is an infinite sum if nobody has ever computed it? You can't jump from the finite to the infinite and expect finite rules to apply. And it is questionable that it has been defined. All that has been rigorously defined is a limit.

That still begs the question what is an infinite sum if nobody has ever computed it? — EnPassant

That begs the question of what the heck you mean when you're talking about an infinite sum. Mathematicians regularly compute what they mean by it. It's the thing you're talking about that's nonsense, not the thing mathematicians are talking about.

You can't jump from the finite to the infinite and expect finite rules to apply. — EnPassant

Case in point... what are you talking about?

And it is questionable that it has been defined. — EnPassant

No, it's factual that it has been defined. Definitions aren't handed to us from an abstract guy giving out tablets in some Platonic/Pythagorean plane of existence. They're established by people... in this case, it's technical definitions given by mathematicians. They define it. You question that they define it, but that doesn't erase the fact that they, indeed, defined it.

All that has been rigorously defined is a limit. — EnPassant

The infinite sum itself has been defined to be the limit... by mathematicians... who are the both the speakers of and designers of the language of math.

Mathematicians regularly compute what they mean by it. — InPitzotl

They compute limits which are not the same as sums. A limit is what a finite sum converges to.

The infinite sum itself has been defined to be the limit... by mathematicians — InPitzotl

I'm aware of that. But it has not been explicitly defined. It has been defined in terms of limits which are limits of finite sums.

https://en.wikibooks.org/wiki/Calculus/Formal_Definition_of_the_Limit

They compute limits which are not the same as sums. A limit is what a finite sum converges to. — EnPassant

The sum of an infinite series is defined as the limit of the sequence of partial sums of the series.

I don't understand your objection. You are entirely right that there is no such thing as an infinite sum until we define it. And how do we define it? We first define the limit of a sequence, then we define the sum of an infinite series as the limit of sequence of the partial sums. To me this seems rather clever. We have something that at first makes no sense, then we MAKE it make sense with a clever definition. What exactly are you objecting to?

They compute limits which are not the same as sums — EnPassant

Given that the sum is by definition the limit, then by definition it's the same. You keep tripping over this same point.

I'm aware of that. — EnPassant

Apparently not... see the underlined as evidence for your continued confusion of the same point. The sum is by definition the same as the limit.

But it has not been explicitly defined. — EnPassant

Right after the citation @Pfhorrest gave:

That is,
$\displaystyle\sum_{i=1}^{\infty}{a_i} = \lim_{n\to\infty}\sum_{i=1}^{n}{a_i}$. — wikipedia

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.

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

Family Resemblance. — Banno

Sorry Banno, but in logic definitions are prerequisite. Family resemblance might suffice as a description of meaning in common vernacular, but mathematics is logic.

This is a language barrier. In the language spoken by the mathematics community, .999... represents the same particular quantity that 1 does. — InPitzotl

If that's the case, then why have two distinct representations for one and the same thing?

Just for the heck of it, what are they, then? — tim wood

I believe shenanigans is an apt word for a description of "real numbers". Modern mathematics contains a lot of sophistry, of which some is used for deception. Mathematics is loaded with tricks which the mathemgjicians have designed for the purpose of hiding contradictions.

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

A true square does not admit to a diagonal, the two sides are incommensurable, making the square an irrational figure, just like the circle. There is no such thing as the diagonal of a square, because there is no such thing as a square, just like there's no such thing as a circle. These items were designed as ideals, but the irrationality of the ratios demonstrates that this effort was a failure. Space cannot be represented as distinct dimensions, as the irrationality of these two dimensional figures demonstrates. One dimension is incommensurable with another, whether you represent the relationship between them as a curved line or as a right angle.

If that's the case, then why have two distinct representations for one and the same thing? — Metaphysician Undercover

I'd point out that 2 + 2 = 4, but we've previously determined that you don't even believe that.

If that's the case, then why have two distinct representations for one and the same thing? — Metaphysician Undercover

What do you mean "If that's the case, then"? There seems to be an implicit assumption that every thing should have exactly one name. Who exactly is making that assumption? It's not me, and it can't be you... does it say "Metaphysician Undercover" on your birth certificate? (And isn't 1 also equal to the fractions 1/1, 2/2, 3/3, and so on anyway?)

Any reason whatsoever for there being two distinct representations for the same thing would do. What's it to you that there are two of them? Is there supposed to be an objection here?

What matters to the present discussion is that .999... does not represent a number. Nor does .111... represent a number, and that's the problem with the op. — Metaphysician Undercover

As a matter of representing numbers, wouldn't most be fine with 9/9 = 9 × (1/9) = 9 × (0.111...) ? — jorndoe

So 1/9 is a number, even for Meta, but 0.111... is not? And this despite their being equal?

Is this Meta's claim?

Is this Meta's claim? — Banno

It looks to go beyond this. Not only is 0.111... not a number, but there's no such thing as squares, because dimensions are incommensurable (@tim wood asked the question I was thinking before I got to it... and that was his response). Circles aren't real, so maybe trigonometry is a lie. Looks to me like Meta's a strange sort of Pythagorean?

Hmm, have you ever asked yourself a simple question "why"? Definitely we can, and we do, for centuries (since Leonard Euler's time) — Andrey Stolyarov

No, we know what a limit is. But as I keep saying, a limit is not a literal infinite sum. Weierstrass did not formulate calculus in terms of literal infinities. He formulated it in terms of finite sums converging to a limit. I'm not arguing that limits are not as they are defined. I'm saying a literal infinite sum is qualitatively different to a finite sum. The definition of a limit only makes sense in terms of finite sums converging to a limit. This is how the limit was rigorously defined by Weierstrass, Cauchy...That definition says that the finite sum can get ever closer to the limit. But you are jumping from the finite to the infinite and saying finite calculations still apply to infinite sums.

The limit of 1/2 + 1/4 + 1/8... is 1. But what is the infinite sum, as a literal infinity? We cannot assume it is 1 just because finite arithmetic points in that direction.

A true square does not admit to a diagonal, the two sides are incommensurable, making the square an irrational figure, just like the circle. There is no such thing as the diagonal of a square, because there is no such thing as a square, just like there's no such thing as a circle. These items were designed as ideals, but the irrationality of the ratios demonstrates that this effort was a failure. — Metaphysician Undercover

And there you have it folks. MU is a genuine, triple-barreled whackdoodle.

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. — Andrey Stolyarov

Not "all" higher math is all about. Lots of topology topics, for example, don't revolve about infinities.

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 — Andrey Stolyarov

True, but frequently it concerns spaces of functions that are not so elaborate, like complex valued functions of the form

$S=\{{{f}_{s}}(t)={{s}^{2}}{{t}^{2}}+itSin(s+t):\text{ }s\in [0,1],\text{ }t:0\to 1\}$

, where the space extends over values of s (uncountably infinite, but one dimensional). One then considers a sigma-algebra of subsets of these functions and defines a kind of measure of these sets. Then, frequently, a functional, which takes a function to a real or complex number. For instance,

$\mathbb{F}[{{f}_{s}}]=Su{{p}_{t}}\left| {{f}_{s}}(t) \right|$

And so on. Just a simple example. Not everything in abstract math is infinite dimensional, though infinity comes into play.

It looks to go beyond this. — InPitzotl

Oh, indeed. Meta also cannot calculate an object's velocity, either, since for him the notion is absurd.

I pointed this out long ago. It is important to note that we are talking fractured ceramics here, not cranial Faraday cages. Hence the exercise becomes one of identifying and tracing the crack.

I'd point out that 2 + 2 = 4, but we've previously determined that you don't even believe that. — fishfry

Seems you have a short memory. What we previously determined is that I do not believe that 2+2 is the same thing as 4. Remember? You argued that 2+2 is identical to 4, ignoring the difference between equivalent and identical.

What do you mean "If that's the case, then"? There seems to be an implicit assumption that every thing should have exactly one name. — InPitzotl

The question was why does the same thing have two names. There was no implicit assumption that the same thing ought not have the same name, but an implicit assumption that if the same thing does have two distinct names, there is a reason for it having two distinct names.

I don't believe that ".999..." and "1" refer to the exact same thing. So I'm asking you, who apparently does believe this, why does mathematics, as a single unified discipline, have these two distinct symbols to refer to the exact same thing. If you could answer this for me, then you might help me to believe what you believe. Until then, I'll believe what seems very evident to me at this time, that these two have distinctly different meanings.

You might argue as others have, that it is a difference which does not make a difference. But in acknowledging that it is a difference you accept the fact that they do not refer to the exact same thing. So I warn you that this would be a self-refuting argument.

So 1/9 is a number, even for Meta, but 0.111... is not? And this despite their being equal?

Is this Meta's claim? — Banno

Did I say that I agree that 1/9 is a number? Check my definition of number, "particular quantity". How could 1/9 ever be construed as a particular quantity? A fraction is not a number.

And there you have it folks. MU is a genuine, triple-barreled whackdoodle. — tim wood

"Triple-barreled" now. Looks like I'm moving up in the world.

Did I say that I agree that 1/9 is a number? Check my definition of number, "particular quantity". How could 1/9 ever be construed as a particular quantity? A fraction is not a number. — Metaphysician Undercover

Gorgeous!

Divide the pie amongst the nine of us - but none for Meta, since a ninth is not a quantity of pie!