The limit = 1 but not .999...= 1 — EnPassant

the ... means the limit is taken. IE, .999... = 1

Try this: .999... is not a number because it has a indefinite extension. A number is an object and an object cannot have an indefinite extension. So discussing the status of .999... under the presumption that it is a number, is misleading oneself by starting with a false premise.

If a person wanted to say "1" they would say 1. If a person wanted to say ".999..." they would say .999.... The two symbols have a distinct meaning and it is not a case of using different symbols to refer to the same thing. The one refers to an object, the other something indefinite. Why try to argue that they each refer to the same thing? Since it is so obvious that the two symbols have a different meaning, it requires accepting that false premise just to start the discussion. Then as soon as you accept the false premise there is nothing to discuss anymore, because the falsity has been validated.

because that’s just what decimal notation means. And since the limit of the series of partial sums of that infinite series is 1, that means the total sum of that infinite series represented by 0.999... is also 1, so 0.999... = 1. — Pfhorrest

But it has to do with the way language is being used. 'The total sum' is not the same thing as 'equals'.

The total sum cannot be explicitly demonstrated. It is really a concept that cannot be substituted with 'equals'.

the ... means the limit is taken. IE, .999... = 1 — fdrake

Yes but 'limit' is not the same as 'equals'.

Yes but 'limit' is not the same as 'equals'. — EnPassant

0.999... is equal to the limit of the sequence {0.9, 0.99, 0.999, ...} is equal to 1.

Yes but 'limit' is not the same as 'equals'. — EnPassant

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

It's amazing how many people will respond without reading through and understanding @jorndoe's well written proof. Go educate yourself! Someone's made it super easy for you!

:up:

And 1 - 0.999... = 0.000... = 0 (sanity check)

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

True. But they don't mean .999... = 1. They mean the Limit of the infinite sum = 1. There's a difference.
The sum being 9/10 + 9/100 + 9/1000 + ...

.999... — EnPassant

That ... MEANS the thing on the left IS the limit. 0.999... IS the limit of the sequence {0.9,0.99,0.999,...}, which is 1.

Another argument, more or less following similar thinking, is whether a number could be found between 0.999... and 1.000... (like the mean).
If no such number can be found, then we might reasonably say they're one and the same. — jorndoe

But the problem is what does .999... mean? How many 9s are we talking about? An infinity of them, of course. But what is an infinity 'of' something?

That ... MEANS the thing on the left IS the limit. 0.999... IS the limit of the sequence {0.9,0.99,0.999,...} — fdrake

I don't think so. .999... describes the series. The limit of it is infinitely far away.

How many 9s are we talking about? — EnPassant

Well there's a 9 in place 1, a 9 in place 2, and 9 in place 3, and so on...

There's a 9 in place 20, a 9 in place 4 billion... apparently, there's a 9 in all places n where n is a number.

So, how many numbers are there?

What is $\frac 13$ in decimal?

Then you don't know what the symbols mean and should read the OP's article!

There are always people that disbelieve in it. It isn't so surprising, since it involves infinity, limits, convergence and monotonicity. What I find especially frustrating about it is that people can be shown a formal proof of it with sources for everything and still refuse to read it and ask exploratory questions.

What is 1/3 in decimal? — Michael

I have no idea!!! I suspect there is 'an infinity of 3s' but what does that mean? That's the crux of the biscuit.

So, how many numbers are there? — InPitzotl

I don't know because I don't know if 'how many' applies to infinity. At the beginning of the theory of limits mathematicians were careful to say that we should say a series 'tends towards' a limit. It is a conservative statement.

Then you don't know what the symbols mean and should read the OP's article! — fdrake

But does anybody know? Intuitively yes, we can see that the limit is 1. But limit is not the same as equals. The argument is subtle. What is being said is 'After an infinity of 9s'. That is what I am suspicious about. I'm not sure what 'an infinity of' means. Or if it is a coherent statement.

But does anybody know? — EnPassant

Yes, many people do. Those who've taken the time to study it.

Intuitively yes, we can see that the limit is 1.

0.999... IS the limit of the sequence {0.9,0.99,0.999,...}, which IS 1.

0.999... is not an element of the sequence {0.9,0.99,0.999,...}, it is the limit of that sequence, which is 1.

0.999... IS the limit of the sequence {0.9,0.99,0.999,...}, which IS 1. — fdrake

.999... could not be the limit. To write the limit you'd have to have .999999999999999999999999999999999999999999999999999 - an infinity of 9s. And we can't write that, whatever it means.

.999... is a symbol for 'an infinity of' 9s. But what does that mean?

n infinity of 9s. And we can't write that, whatever it means. — EnPassant

Which is exactly why you write 0.999"...". It is the limit.

Which is exactly why you write 0.999... — fdrake

Yes, I understand what you are saying. But if infinity is not a number how can you have an infinity "of"?

I have no idea!!! I suspect there is 'an infinity of 3s' but what does that mean? That's the crux of the biscuit. — EnPassant

It's $0.\overline{3}$.

But if infinity is not a number how can you have an infinity "of"? — EnPassant

In one respect it is shorthand.

The sequence {0.9,0.99,0.999,..} is the sequence of partial sums $s(n)=\sum_{i=1}^{n}9 \times 10^{-i}$. IE {0.9, 0.99,0.999...}={s(1),s(2),s(3),...}. 0.999... is the limit of that sequence. It is equal to the limit of that sequence. Which is 1. 0.999... is equal to1.

In another respect, and you will not like this even more because the math is more advanced, the cardinality of the set of sequence elements {0.9,0.99,0.999,...} is aleph-null, the smallest infinity. That transfinite number is not a real number.

If you don't understand these issues, you should read through @jorndoe's document. If you have any questions regarding its content, ask in thread and I will try and address them for you.

What is 1/3 in decimal? — Michael

Infinitesimals have never really been understood rigorously. Have you heard of Berkeley's "Ghosts of departed quantities"? Below 'Fluxions' means infinitesimals.

Ghosts of departed quantities
Towards the end of The Analyst, Berkeley addresses possible justifications for the foundations of calculus that mathematicians may put forward. In response to the idea fluxions could be defined using ultimate ratios of vanishing quantities (Boyer 1991), Berkeley wrote:

It must, indeed, be acknowledged, that [Newton] used Fluxions, like the Scaffold of a building, as things to be laid aside or got rid of, as soon as finite Lines were found proportional to them. But then these finite Exponents are found by the help of Fluxions. Whatever therefore is got by such Exponents and Proportions is to be ascribed to Fluxions: which must therefore be previously understood. And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?[6]

Edwards describes this as the most memorable point of the book (Edwards 1994). Katz and Sherry argue that the expression was intended to address both infinitesimals and Newton's theory of fluxions. (Katz & Sherry 2012)

Today the phrase "ghosts of departed quantities" is also used when discussing Berkeley's attacks on other possible foundations of Calculus. In particular it is used when discussing infinitesimals (Arkeryd 2005), but it is also used when discussing differentials (Leader 1986), and adequality (Kleiner & Movshovitz-Hadar 1994).

Infinitesimals have never really been understood rigorously. — EnPassant

It has nothing to do with infinitesimals.

If you don't understand these issues, you should read through jorndoe's document. If you have any questions regarding its content, ask in thread and I will try and address them for you. — fdrake

I have read through it. These are mathematical expressions and as such they are symbols. They represent infinity. But mathematicians were aware of these issues when formulating the calculus and they cautioned against saying 'equals'. They said we should say 'Tends towards the limit'

That only shows you didn't understand it. @jorndoe defines what a limit is in it! It doesn't even need infinity in the definition.

Tends towards the limit' — EnPassant

The sequence elements tend towards the limit. The limit is not a sequence element. 0.999... is the limit. It is equal to 1.

$\begin{align*}x&=0.999...\\10x&=9.999...\\10x&=9+0.999...\\10x&=9+x\\9x&=9\\x&=1\end{align*}$

The sequence elements tend towards the limit. The limit is not a sequence element. 0.999... is the limit. It is equal to 1. — fdrake

Note that in the article cited in the op they don't write

$\sum_{i=1}^{\infty}$ etc = x

They write

lim $\sum_{i=1}^{\infty}$ etc = x

These are two different concepts.

I don't know because I don't know if 'how many' applies to infinity. — EnPassant

How do you know it's infinity and not, say, an octillion?

It is a conservative statement. — EnPassant

So if you want to be conservative, just say what you're talking about... "the number of counting numbers".