x = 0.999...
10x = 9.999... — Michael

Nice. But it still begs the question: what does it mean to say x = 0.999...?
It means an infinity of 9s but what can that mean when infinity is not a number?
You have to say x = lim 9/10 + 9/100 + 9/1000 + ...
And we are back to square 1. (Uh, I mean square 0.999...)

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

Because I know it is not any nameable number.

$1 = \sum_{i=1}^{\infty} 9 \times 10^{-i} = \lim_{n \rightarrow \infty} s(n) =0.999...$

fdrake — fdrake

Yes, but that is the limit which is different from equals. When you say 1 = you are saying 1 = the limit not simply 1 =

It should be written
lim $\sum$
not simply $\sum$

Yes, but that is the limit which is different from equals. When you say 1 = you are saying 1 = the limit not simply 1 = — EnPassant

What is the limit of the series {0.9,0.99,0.999,...}? Call this x.
What does the symbol "0.999..." represent? Call this y.

Is x=y ?

What is the limit of the series {0.9,0.99,0.999,...}? Call this x.
What does the symbol "0.999..." represent? Call this y.

Is x=y ? — fdrake

It depends on how you read these expressions. I'll grant you that 0.999... can be identical to the limit of the series if that's how you interpret it. But if you do you interpret it as a limit not as equals.
You can say 0.999...= 1 if by that you mean the limit of 0.999...

fdrake — EnPassant

What is the difference between
$\sum$ = x and
lim $\sum$ = x ?

There isn't. What you're looking for is the infinity symbol above the Sigma.

Need to figure out that sexy equation mode.

"How do you know it's infinity and not, say, an octillion?" -- InPitzotl
Because I know it is not any nameable number. — EnPassant

You answered only half of the question... the half about your knowing that it's not any nameable number. What about the other half... how do you know it's infinite?

ETA: Probably obsolete now... once you accept that it's infinite, we could then enumerate as statements the meaning of each finite decimal expansion, and agree that we have no such infinite statement; then we can define the infinite statement to mean the limit (after possibly a quibble that we're defining the finite expansion's meaning anyway).

You can say 0.999...= 1 if by that you mean the limit of 0.999... — EnPassant

That is exactly how it is meant. That is what 0.999... means.



The infinite sum notation $\sum_{n=1}^{\infty}$ just means $\lim_{n\rightarrow \infty} \sum_{i=1}^{n}$, that $\lim$ means; for a sequence $\forall \epsilon \gt0 \exists N : \forall n \gt N |s(n)-L| \lt \epsilon$, the value $L$ is the limit. $0.999...$ is the limit of the sequence of partial sums $s(n)$, which is equal to 1.



Wrote a guide here. It's essentially LateX if you're familiar with it, though without lots of the standard packages.

Wrote a guide here. It's essentially LateX if you're familiar with it, though without lots of the standard packages. — fdrake

$\lt3$

Just a comment about posting math material, symbols, equations, etc. I doubt if anyone here uses it, but MathType is very easy to use and is WYSIWYG rather than coding for each symbol. Under cut and paste preferences, choosing the Wikipedia option and pasting on this forum only requires changing <...> to [...], etc.

:up:

to .

There isn't. What you're looking for is the infinity symbol above the Sigma. — Kenosha Kid

I just left it out for brevity. I'm sure you know what i mean.

That is exactly how it is meant. That is what 0.999... means. — fdrake

Ok, I'll accept that. But what we are talking about here is subtle and the " = " sign in calculus can be misleading:

$\sum 1/x$ is a literal sum.

lim $\sum 1/x$ is not a sum. It is the limit towards which the sum (over the range) converges.

That is the difference.

Just a comment about posting math material, symbols, equations, etc. I doubt if anyone here uses it, but MathType is very easy to use and is WYSIWYG rather than coding for each symbol. — jgill

The first post on this math forum explains how to us the math tags.

https://thephilosophyforum.com/discussion/5224/mathjax-tutorial-typeset-logic-neatly-so-that-people-read-your-posts

Ok, I'll accept that. — EnPassant

Great. That means you accept $0.999...=1$!

Great. That means you accept
0.999... = 1 — fdrake

Only according to a strict interpretation of the ' = ' sign: 1 is not the sum. It is the limit of the sum. So 0.999... = 1 does not mean it is literally 1. It means 1 is the limit.

It means 1 is the limit. — EnPassant

Ok, I'll accept that. — EnPassant

You just accepted that 0.999... is the limit of {0.9,0.99,0.999,...}, and equal to 1.

You just accepted that 0.999... is the limit of {0.9,0.99,0.999,...}, and equal to 1 — fdrake

Yes of course. I have not said it is not the limit. I said 0.999... is not equal to 1 if we are talking about a literal sum. If we are talking about a limit, yes, the limit is 1. I keep saying a sum and a limit are not the same thing.

0.999... is the limit of {0.9,0.99,0.999,...} — fdrake

https://en.wikipedia.org/wiki/Limit_(mathematics)#:~:text=In%20mathematics%2C%20a%20limit%20is,)%20%22approaches%22%20some%20value.&text=The%20concept%20of%20a%20limit,direct%20limit%20in%20category%20theory.

Try this: .999... is not a number because it has a indefinite extension. — Metaphysician Undercover

.999... is obviously not a number. It is a numeral. 1, 2, 3, ..., are obviously not numbers. They are numerals. It's a difference that makes a difference. I'm surprised you need to have that pointed out to you.

$0.999...$ is the limit of the sequence of partial sums $s(n)$ — fdrake

Yes, 'partial' sums. That means the sums are finite. Calculus does not speak about literal infinite sums. It speaks about finite sums approaching a limit. As more terms are added indefinitely, the limit is approached more closely. That is what calculus is saying.

What I am saying is that a literal infinite sum is probably an incoherent concept.

Calculus does not speak about literal infinite sums. It speaks about finite sums approaching a limit. — EnPassant

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

.999... is obviously not a number. It is a numeral. 1, 2, 3, ..., are obviously not numbers. They are numerals. It's a difference that makes a difference. I'm surprised you need to have that pointed out to you. — tim wood

Whether or not .999...qualifies as a numeral is a matter of interpretation. What I meant, as you seem to have difficulty in understanding, is that it does not signify a number. Whether or not .999... is a numeral is a semantic issue involving the meaning or definition of "numeral" being applied, and is irrelevant to the point that I was making, that .999... does not signify a number. If you'd like to discuss this point, please do not just create a distraction like that.

Whether or not .999...qualifies as a numeral is a matter of interpretation. — Metaphysician Undercover

How so? What else would it be?
As a numeral, it's nothing in itself but a sign of something. But a sign of what? Well, the people who define these things have told us. Except that some people don't care and jump into the middle of the thing and with no sense at all say approximately that everyone else is wrong and they alone are correct - forgetting that the sign is nothing but a matter of definition, being in itself nothing. So go back to the OP, read it, understand it, and then tell us what is wrong with it - but that you cannot do; that you will not do. As to telling us that the definition is wrong, that's simply delusional.

As a numeral, it's nothing in itself but a sign of something. But a sign of what? Well, the people who define these things have told us. — tim wood

A numeral is a special type of sign. To know whether .999... qualifies as a numeral would require a definition which dictates the criteria for being a numeral.

But as I said, that issue is just a distraction. 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. Whether or not these could still be numerals, which do not represent numbers, is a matter of one's interpretation of "numeral", and this is not relevant.

The rest of your post doesn't seem to make any sense.

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. — Metaphysician Undercover

Hm. So present the object 1; that object to which "1" refers. Then your point will be made.

But you cannot. Numbers are not objects.

"1" does not refer to anything.

Whether or not .999...qualifies as a numeral is a matter of interpretation. What I meant, as you seem to have difficulty in understanding, is that it does not signify a number — Metaphysician Undercover

.999... is obviously not a number. It is a numeral. — tim wood

:rofl:

, indeed, calling infinitesimals

Ghosts of departed Quantities — Berkeley

goes along with (contemporary) calculus. They do occasionally come up as matters of convenience or tradition (e.g. in notation), but not of necessity.

Another possible source of confusion could be the Archimedean properties^[23][24][25]: neither ∞ nor infinitesimals^[26] are real numbers^[27][28].

Ghosts couldn't comprise a "boundary" between 0.999... and 1.


_{23, 24, 25, 26, 27, 28}

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

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

I suggest that we look at this issue from the standpoint of expressions. Just like someone might say, "I love you to death" and then express the very same thing as, "I love you more than anything else in this world", the quantity 1 can be expressed as 0.999...

In other words, the quantity 1 = the quantity 0.999...

The only difference is in the expression.

The difficulty with accepting the truth of 1 = 0.999...is in no small part due to the fact that the one's place in 0.999... is vacant while that in 1 is occupied by the digit 1. Quite obviously there's a discrepancy here - how can 1.000... be equal to 0.999...?

But seeing it that way is to ignore the infinite 9's that follow the decimal point in 0.999... Infinity should never be ignored is the main lesson here.