math question

[mesh]

.999... is also a geometric progression, albeit not finite so therefore it is a infinite series. Nevertheless, for shear entertainment:

note: 9/(10^0) is 9/1, which is 9. Therefore, I will assume that my solution is the solution to 0.999... = 1, but taken as 10 = 9.999... for ease of application to a geometric progression.

( 9 + 9/10 + 9/100 + 9/1000 + ... + 9/(10^n) ) where n >= 0;

9( 1 + 1/10 + 1/100 + ... + 1/(10^n) );

9( 1 + 1/10 + 1/100 + ... + (1^n)/(10^n) );

9( 1 + 1/10 + 1/100 + ... + (1/10)^n );

9( (1 - (1/10)^(n+1)) / (1 - (1/10)) )

see http://en.wikipedia.org/wiki/Geometric_progression for details on last step.

now we take the limit as n approaches infinity... because it does.

lim n->? 9( (1 - (1/10)^(n+1)) / (1 - (1/10)) ) =

= 9( (1 - 0) / (1 - 1/10) )
= 9( 1 / .9)
= 9.999...

therefore, geometric progressions have nothing to add to this discussion, good thing I stated early on that this was for shear entertainment.

[mesh]

Actually I will take back my last conclusion and assert that my last post illustrated that there is no rigorous way to prove to .999... is equal to one.

The previous posts induce that since

1/3 = .333...
2/3 = .666...

3/3 should therefore be interpreted as .999...

However, consider this. Since pythagorean times a rational number is one that can be expressed as an integer over an integer, 1/3 is rational as is 2/3. These numbers just happen to be equivalent to the real numbers .333... and .666... respectively. Asserting that 1/3 = .333... associates rational numbers with real numbers and therefore introduces approximation, not equality. Therefore, the inductive conclusion that 3/3 = .999... is not an equality but an approximation.

The point of asserting that .999... is equal to 1 is equivalent to asserting that our approximations are "good enough", rather than rigorous. In order to be rigorous, one must remain within the same domain.

[DCrazy]

http://en.wikipedia.org/wiki/Geometric_ ... ric_series

You need a sum in there ( lim_n->inf. Sum[ k = 0 -> n] (1/(10^n) ). Then it becomes an infinite sum, and Lothar's proof takes over.

It is incorrect to use the formula you did as it provides the n'th (infinity'th) term in the series, which is going to be a 9, NOT the sum of the terms or the point to which that sum converges.

[mesh]

Sorry to be trite, but I just don't see how I did not incorporate a summation. A ( 1 + 1/10 + ... + 1/(10^n) ) is a summation, an expanded Sum[ k = 0 -> n]. Am I missing something?.

Granted that I took the limit as n goes to infinity at a different point in the argument, but so does the Wikipedia page. And not just to appeal to authority, a geometric summation is equivalent to the shorthanded ( 1 - x^(n+1)) / ( 1 - x );

[SuperSheep]

K, here goes...

1/3 = 0.33333...

Now if we add 3 times we get...

3/3 = 0.99999...

but this IS equal to 1 because the way we long divide could be re-written like so...

3 divided by 3 equals 0 remainder 3.
30 divided by 3 equals 9 remainder 3.
rinse and repeat...

3 into 3 = 0.9R3 which is the same as 1.0.

This works regardless of how you choose to long divide. For instance 1/10...

1 divided by 10 = 0 R 1
10 divided by 10 = 9 R 1
etc...

1/10 = 0.0999999....

Thought of this last night and seems to make sense. Maybe some math geeks can point out if this is mathematically allowed.

[DCrazy]

mesh, you used the forumula for a geometric sum on finite bounds, when the article provides a formula for an infinite sum. Using that formula ( Sum[ k=1->inf. ] x^k = 1/(1-x) |x=10^-1 ) yields:

x=10^-1

9(10^0 + 10^-1 + 10^-2 + ... + 10^-inf.) =
9(x^0 + x^1 + x^2 + ... + x^inf.) = 9 * Sum[ n = 0 -> inf. ] (x^n)

= 9 * 1/(1-x) = 9 * 1/(1 - 10^-1) = 9 * 1/(1-(1/10)) = 9 * 1/(9/10) = 9 * 10/9 = 10.

[dissent]

I wonder if this is where we got the expression "dressed to the nines"??

I guess if you have enough nines, then you really are "the one".

[mesh]

Ah, I think I see what you are saying. It is not that I used the formulas of the Geometric sum incorrectly... for example

lim n->inf. ( 1 - (x^(n+1)) ) / (1 - x ) is equal to, only if x is a fraction and by happenings that I do not quite understand, x is always a fraction, the equation thus becomes:
( 1 - 0 ) / ( 1 - x ).

Which is stated in the Wikipedia page, and which is also used in the post in question.

So, pointing the blame at the summation confused me. Sorry, I like refutations to hit me in the face like a shot of fully charged fusion.

What is to blame in my post, is that I was not consistent. And wow holy almighty, it's really facinating. Let me elaborate. (yes, this may be ad-hoc, but i really want to address the issue of converting rational fractions into real numbers ). And yes, I may just be that late bloomer on the bus...

My error:

=9( (1-0) / (1-1/10) )
=9( 1/.9 )
=9( 1.111... ).

While I remained within the domain of rational numbers, here at the end I jumped. This jump is so... intuitive its almost unnoticeable. And my formula was blamed. HAH! javascript:emoticon(':P')

I AM A BELIEVER!! THERE IS A GOD!! an integer of one at least.

3/3 is still one though... nice try guys but I am not buying that one.

[Dedman]

Geeks! The lot a ya

[snoopy]

3/1 = .33333333333333333333333333333333333333333333
3333333333333333333333333333333333333333333333333333
33333333333333333333333333333333333333333333333333333
ad infinitum.

Hate to break it to you, but 3/1 = 3. I know, it's incredibly painful to have your bubble burst

[DCrazy]

Mesh: Wow. Nothing like taking a hard U-turn at the last step of the problem.

[Foil]

There is no proof that demonstrates 0.999... = 1 ...

ROFL. That seriously pegged my WTF-meter. Never thought I'd see anyone talk that way about math, except maybe those LaRouche supporters.

Oh, man... I had the exact same reaction.

For those who don't already know, my educational background is a Masters in (Pure/Theoretical) Mathematics.

I'm going to say it just once:

Lothar, Drakona, The American Mathematical Society, and everyone else who presented rigorous proofs here, are absolutely correct. Based on every well-defined definition of the set of real numbers and infinite sums and series, we can say with rigorous, absolute certainty that "0.999..." is the same as, and is exactly equal to 1.


Some notes:


TopGun is right that at first it seems counter-intuitive. But work your way through the proofs (be careful about it, to make sure you are sure of each step) if you need convincing. Once you finally see it, you'll understand completely, and think, "Wow.. how did I ever think they were not equal?"

Or better yet, go ask any collegiate professor of Mathematics (or even Statistics or Physics for that matter). Sometimes a one-on-one explanation of a proof can be more easily grasped than seeing a written proof.


Look carefully... there is a flaw (sometimes quite hard to find) in every attempt I've ever seen at proving "0.999... is not equal to 1".

Here are some things to look for:
- Jumps/breaks in logic based on "intuitive" concepts. If it doesn't follow directly from a definition or exact logic, it can't be used in a truly rigorous proof.
- Poor/incorrect definitions, or poor logic. If the pieces of the argument aren't well-defined, your conclusion is pretty much meaningless.

For example, I've heard people say, "because there's a difference between 0.9 and 1, and there's a difference between 0.99 and 1, (and so on), there must be a difference between 0.999... and 1." Huh? Where's the logic? They're using a set of numbers, none of which are 0.999..., and saying that it implies something about a number they never even used anywhere in their argument!

Or for example, it was argued in this thread that "you can't equate real numbers (like 0.999...) and rational numbers (like 1/3, 2/3, 3/3, etc.)". But go back and look at the definitions... the set of reals includes the set of rationals, so you've got an exact correspondence by definition!



Anyway, sorry for my fervency here. It's just that it's my field of expertise (as well as Lothar's, Drakona's, and others'), and it's frustrating to see flawed information and arguments flying around about it.

The links that have already been posted to all the various proofs of "0.999... = 1" are probably the best resource on the matter, so I won't go into any more technical detail, but feel free to PM me if you have any detailed questions.

[Lothar]

my last post illustrated that there is no rigorous way to prove to .999... is equal to one.

Funny... my last post contained the rigorous proof you claim can't exist.

.999... is exactly equal to one, because of the way repeating decimals are defined.

Asserting that 1/3 = .333... associates rational numbers with real numbers and therefore introduces approximation, not equality.

Uh... there's no approximation involved here. The rationals are a subset of the reals. You don't have to approximate 1/3 to make it real; it already is.

The only way in which you get an approximation is if you truncate the decimal expansion. So, for example, if I said 1/3 = .333 (without the 3 dots at the end) that would be an approximation. But 1/3=.333... is not an approximation, because the ... means the 3's go out to infinity, and you can prove the equivalence of the infinitely long decimal expansion to the fraction quite easily using the method from my previous post.

[Stryker]

All this just begs the question....

Does .899... = .9?

[jakee308]

3/1 = .33333333333333333333333333333333333333333333
3333333333333333333333333333333333333333333333333333
33333333333333333333333333333333333333333333333333333
ad infinitum.

Hate to break it to you, but 3/1 = 3. I know, it's incredibly painful to have your bubble burst

you know that's a typo. sheesh what a bunch of cheap shot artists. and they get upset if I do that. hah! talk about double standard.

edit: oh yeah, forgot, as i said .9999... isn't equal to 1. even the proofs that have been cited do not state equality they claim equivalency. now in theoretical math this is useful. however i want someone to divide a 10" x 10" piece of cake into 3 equal parts. can't be done. you know it, i know it. better yet buy a 1 oz. gold bar. divide it into 3 equal pieces and let me pick. .9999... is the equivalent to 1? ok, fine but it ain't equal by definition only 1 = 1. btw people believed for many years that the earth was the center of the universe. they evolved logical proofs for this claim. THEY WERE WRONG!!! i'm not claiming that this is the same, i'm just saying dogmatic adherence to a claim due to "logical proof" don't prove s***. closed doors = closed minds.

[DCrazy]

Lothar, after a couple of misfires it's been sorted out. He did in fact prove that .999... = 1.

jakee, it's quite obvious you have no clue what you are talking about. "The proofs that have been cited do not claim equality but equivalency"? Now you're grasping for straws, and this one does not make sense.

It is just as possible to separate a 10" x 10" piece of cake into 3 equal parts as it is possible to cut a 3" x 3" piece of cake into 3 equal parts.

Multiple proofs have been given that by definition, .999... = 1.

"Closed doors = closed minds" is a quite accurate way to describe your current situation. Not ours.

[Lothar]

as i said .9999... isn't equal to 1

And you've had 3 people with masters degrees in (some form of) math PROVE that .999... is in fact exactly equal to 1, directly from the definitions.

the proofs that have been cited do not state equality they claim equivalency

In this context, equivalence and equality are equivalent concepts. If you read my proof carefully, you can see that my conclusion was that .999... = 1.

But, of course, it's kind of pointless to prove something mathematically to people who don't understand the mathematics involved. Do you know what a limit is? Do you know what completeness is? Have you ever seen an epsilon-delta proof? If not, then it's kind of ridiculous to say anything about my proof except that you didn't understand it.

i want someone to divide a 10" x 10" piece of cake into 3 equal parts.

Trivial. All you need to do is trisect one edge. That can be done regardless of length. There's nothing magical about 10" that makes this any harder than 3" or 3.52" or any other length.

.9999... is the equivalent to 1? ok, fine but it ain't equal by definition only 1 = 1.

So are you saying 4/4 isn't equal to 1, since only 1=1?

.999... is equal to 1 because of the way .999... is defined. If you don't understand the definition (if you don't know what a cauchy sequence is, for example) then just say "sorry, I don't understand the definition" and stop pretending.

By definition, .999... is equal to 1.

dogmatic adherence to a claim due to "logical proof" don't prove s***.

Hahahaha... dude, that's really pathetic.

I adhere to the claim that .999... = 1 because I understand the definition of what .999... means. The fact that it's equal to 1 follows directly from the definition.

And, if we leave the real numbers for some other field (like, say, the hyperreals) I won't adhere to the claim, because within the hyperreals .999... doesn't equal 1 based on the definitions within the hyperreals.

There's nothing "dogmatic" about it. It's just understanding the definitions. It's knowing what ".999..." means, how the real numbers work, etc.

[mesh]

Um, a non-math major proved it too. Thank you. ( after some miss-fires )... damn neurons *clunk*. And, i know that you posted solutions, however I need to move stuff with my own mind for it to sink in. Thank you for participating. =D

Lothar, is sandbagging the best way to argue against closed mindedness?

[Vindicator]

Wow, WTF @ jakee. At this point I think he's just trying to keep us riled up.

At any rate, leave it to an engineering student to put it in the simplest form

[roid]

it seems the only thing seperating 0.9999... from 1 is an infinitely small gap (could this gap be defined as 1 divided by infinity? coz of the dividing by 0 thing, i'm not sure if dividing by infinity is even allowed in mathematics).
It seems that just as an infinitely small sphere is defined as a "point", anything infinitely small is equal to 0.

So the gap between 0.9999... and 1 is equal to 0. So there is no gap, it's seperated by a "point".

All this just begs the question....

Does .899... = .9?

yep, i can't see why not.

[Jeff250]

At least Mobius knew when he was beat.

[Jeff250]

it seems the only thing seperating 0.9999... from 1 is an infinitely small gap (could this gap be defined as 1 divided by infinity?

No gap. You're still trying to think of it as an infinite sum in your mind (as .999... increasingly stretching out forever). ".999..." symbolically is just another way of writing "1" or "3/3" or "e^(2pi*i)" or "unity" or "one." .999... is just a particular decimal notation of a value, of whatever "1," "3/3," and all the other aforementioned symbols represent. This is my understanding.

[Top Gun]

On a somewhat-related side note, I've seen a lot of people equating the expression e^(2pi*i) with 1. I was completely puzzled as to why this was true, until some Google searching reminded me of Euler's formula. It's amazing how quickly I manage to forget formulae/concepts after the class they're used in is over.

[fyrephlie]

i've said it before and i will say it again:

MATH IS DUMB AND YOU NEVER USE IT IN REAL LIFE


























[/sarcasm]

[dissent]

This reminds me of a joke I heard once.

A Mathematician, a Physicist, and an Engineer all go to the same job interview....

Hey Dedman. Good 0.9999999999999...!!

[fyrephlie]

This reminds me of a joke I heard once.

A Mathematician, a Physicist, and an Engineer all go to the same job interview....

Hey Dedman. Good 0.9999999999999...!!

ROFLOL
OMG I actually clicked back before I got it.

[World War Woodi]

Ya all are purty smart!

This math problem reminds me of the thought problem of the falling stone, stopped in time at halfway, then stopped at the next halfway point and so on.

So does it ever reach the ground?

I wonder if the whole of the distance to the ground equals one, then once it reaches .9999.... the distance to the ground, is it there?

I suppose since it is always divided by half it would never come to .999...

Hmmm school, I shoulda listened ......

[fyrephlie]

Ya all are purty smart!

This math problem reminds me of the thought problem of the falling stone, stopped in time at halfway, then stopped at the next halfway point and so on.

So does it ever reach the ground?

I wonder if the whole of the distance to the ground equals one, then once it reaches .9999.... the distance to the ground, is it there?

I suppose since it is always divided by half it would never come to .999...

Hmmm school, I shoulda listened ......

i think you are discussing the philosophical principles of .999~ not equalling 1...

[mesh]

What? from one perspective it seems that a civilizations accomplishments are measured by the depth of their math. Sometimes I have this perspective.

While it seems that science lays down the framework for mathematical intuition, the simplification process of physics which delivers it to the masses is not possible without some understanding of math. Example: gravity on earth being reduced to the approximate acceleration of 9.8 meters per second towards the center of the earth.

[Jeff250]

Ya all are purty smart!

This math problem reminds me of the thought problem of the falling stone, stopped in time at halfway, then stopped at the next halfway point and so on.

So does it ever reach the ground?

I wonder if the whole of the distance to the ground equals one, then once it reaches .9999.... the distance to the ground, is it there?

I suppose since it is always divided by half it would never come to .999...

Hmmm school, I shoulda listened ......

That's one of Zeno's paradoxes. I think that it is solved by considering that when the space being considered becomes infinitesimal, so does the time required to pass it. Anyone who has worked with Calculus is already familiar with this sort of approach to modeling.

[Foil]

...
This math problem reminds me of the thought problem of the falling stone, stopped in time at halfway, then stopped at the next halfway point and so on.

So does it ever reach the ground?
...

That's one of Zeno's paradoxes.
...

You're exactly right!

In fact, here's a way to use a variation of that paradox to illustrate that 0.999... does in fact equal 1 (parts copied and edited from http://www.mathacademy.com/pr/prime/art ... /index.asp ):

Argument 1
Let's consider the distance of exactly 1 mile (or any unit of length for that matter, since units are arbitrarily defined).

First, of course, I must cover 90% of the distance (0.9 miles).
Then, I must cover 90% of the remaining distance (0.99 miles traveled).
Then, I must cover 90% of the remaining distance (0.999 miles traveled).
Then I must cover 90% of the remaining distance (0.9999 miles traveled).
... and so on forever. The consequence is that I can never travel 1 mile, because I always have another 10% of the remaining difference left!

Guess I shouldn't try to go home, then, since I can never get there... right?

Argument 2
That's essentially the same argument which is used in saying 0.999... is not equal to 1: "They can't be equal, because you never get to the end of the sequence 0.9, 0.99, 0.999, and there's always a difference of 0.000...1!"

So you can never get from 0 to 1 (because you always have to go 90% of the way first)... right?


So, if you haven't already grasped it, here's the kicker:

If you take the stance that 0.999... is not equal to 1, you must also accept Zeno's paradox (it's the same argument), and conclude that we can never get anywhere!

But of course, we all know that we can get from place to place. So both of those arguments must be wrong.


BTW, for those taking notes, that's not a formal proof by any means - but it's a good illustration that may help make sense of the matter to a few folks.

Oh, and the method of argument used there is called Reductio Ad Absurdium (Latin). Basically, in this case it goes something like, "If you assume (A) is true, and it leads directly to a conclusion which we know is wrong, then (A) must not be true!"

[Foil]

On a somewhat-related side note, I've seen a lot of people equating the expression e^(2pi*i) with 1. I was completely puzzled as to why this was true, until some Google searching reminded me of Euler's formula. It's amazing how quickly I manage to forget formulae/concepts after the class they're used in is over.

One of my favorite professor's favorite formulas is e^(i*pi) + 1 = 0 , which is difficult to grasp conceptually (mostly because functions on the complex plane are very hard to visualize), but is easily shown by Euler's formula.

The cool thing about e^(i*pi) + 1 = 0? It uses every basic mathematical unit (e, i, pi, 1, 0), and every basic mathematical operation (addition, multiplication, exponentiation, and equality)!

(Yep, this establishes it... I'm still a math geek, even years after college.)

[roid]

this Zeno's paradox was what was throwing me before (didn't know it had a name though, thx).

how i ended up working through it was when i realised that i was wrongfully imagining the universe as a giant computer simulation. In my mind i knew that as the amount of decimal places increased in a calculation - it would take more CPU cycles to calculate - so an infinite number of decimal places would take an infinite amount of time to calculate - so the universe (that was merely a computer simulation) would theoretically "stall" when trying to render it's own physics.
heh, that kinda thought pattern is one of the problems of growing up with computers i think.

it seems the only thing seperating 0.9999... from 1 is an infinitely small gap (could this gap be defined as 1 divided by infinity?

No gap. You're still trying to think of it as an infinite sum in your mind (as .999... increasingly stretching out forever). ".999..." symbolically is just another way of writing "1" or "3/3" or "e^(2pi*i)" or "unity" or "one." .999... is just a particular decimal notation of a value, of whatever "1," "3/3," and all the other aforementioned symbols represent. This is my understanding.

Yeah. Infinitely small gap = No gap.

[fyrephlie]

heh, that kinda thought pattern is one of the problems of growing up with computers i think.

i have the same problem all the time.

ever almost say 'lol' to someone in the 'real world'? even if you didn't laugh?

[World War Woodi]

...
This math problem reminds me of the thought problem of the falling stone, stopped in time at halfway, then stopped at the next halfway point and so on.

So does it ever reach the ground?
...

That's one of Zeno's paradoxes.
...

You're exactly right!

In fact, here's a way to use a variation of that paradox to illustrate that 0.999... does in fact equal 1 (parts copied and edited from http://www.mathacademy.com/pr/prime/art ... /index.asp ):

Argument 1
Let's consider the distance of exactly 1 mile (or any unit of length for that matter, since units are arbitrarily defined).

First, of course, I must cover 90% of the distance (0.9 miles).
Then, I must cover 90% of the remaining distance (0.99 miles traveled).
Then, I must cover 90% of the remaining distance (0.999 miles traveled).
Then I must cover 90% of the remaining distance (0.9999 miles traveled).
... and so on forever. The consequence is that I can never travel 1 mile, because I always have another 10% of the remaining difference left!

Guess I shouldn't try to go home, then, since I can never get there... right?

Argument 2
That's essentially the same argument which is used in saying 0.999... is not equal to 1: "They can't be equal, because you never get to the end of the sequence 0.9, 0.99, 0.999, and there's always a difference of 0.000...1!"

So you can never get from 0 to 1 (because you always have to go 90% of the way first)... right?


So, if you haven't already grasped it, here's the kicker:

If you take the stance that 0.999... is not equal to 1, you must also accept Zeno's paradox (it's the same argument), and conclude that we can never get anywhere!

But of course, we all know that we can get from place to place. So both of those arguments must be wrong.


BTW, for those taking notes, that's not a formal proof by any means - but it's a good illustration that may help make sense of the matter to a few folks.

Oh, and the method of argument used there is called Reductio Ad Absurdium (Latin). Basically, in this case it goes something like, "If you assume (A) is true, and it leads directly to a conclusion which we know is wrong, then (A) must not be true!"

Very nice indeed, that was my point completly, the stone does hit the ground. No matter how complicated you make the problem, the facts are there.
.999... is equal to 1 because the difference {or gap} is insignificant.

[Foil]

...here's the kicker:

If you take the stance that 0.999... is not equal to 1, you must also accept Zeno's paradox (it's the same argument), and conclude that we can never get anywhere!
...

Very nice indeed, that was my point completly, the stone does hit the ground. No matter how complicated you make the problem, the facts are there.
.999... is equal to 1 because the difference {or gap} is insignificant.

Technically, there is no gap at all. The reason we tend to perceive a gap is that we usually try to get to "0.999..." by thinking of the sequence "0.9, 0.99, 0.999, etc.", and there's a small gap at each step.

Of course, since "0.999..." is technically defined as the infinite limit of the sequence (see any Calculus I textbook if you want to an introduction to infinite limits), one can never get there that way!


Here's another analogy, which might help illustrate it for people:

Consider the always-decreasing sequence "0.1, 0.01, 0.001, 0.0001...".

At every step of the sequence, you're looking at a positive number, right? So does that mean that the infinite limit of the sequence is a positive number?

Of course not! The limit of the sequence is 0 (if you assume it's not zero, it leads to some pretty ridiculous conclusions), which is certainly not positive.

Well, the sequence "0.1, 0.01, 0.001, 0.0001..." is just the difference at each step between 1 and "0.9, 0.99, 0.999...". And since the limit of this difference is zero, then they must be the same!

[Lothar]

it seems the only thing seperating 0.9999... from 1 is an infinitely small gap (could this gap be defined as 1 divided by infinity?

No gap.

Yeah. Infinitely small gap = No gap.

Nope. Infinitely small gap is different from no gap.

The sequence .9, .99, .999, ... will have a gap that shrinks infinitely small... but the number .999... has no gap.

In some number systems, that's really significant. See, for example, the hyperreals, where the number epsilon lives in the infinitely small gaps.

[fyrephlie]

this is just me, not having my masters in any form of math... but the whole argument depends on what you are looking to get out of it. for all practical purposes .999~ = 1, but without limits there is in fact a gap. (as you stated above).

it could be said, i think, that they are not equal given certian mathematic situations, but none that exist in the real world of real mathematics that we use.

i myself hear the statement .999~ = 1 and think... it can't because .999~ will never reach 1, EVER!, its as close to 1 you can get without being one. i think that just comes from the way i think tho. i accept infinity whole heartedly.

i guess i just see both sides of the fence on this one. even if one side is 'wrong' as hell.

[roid="quote"]i'm not sure if dividing by infinity is even allowed in mathematics[/roid]

sure it is, it's just like a-milliondy-one times infinity!

*p.s. i saw what i did there... when i hit preview it made me laugh so hard i just had to leave it!*

[Foil]

Actually, by definition, "0.999~" is the limit of the sequence.

So to say, "without limits there is in fact a gap" doesn't work, because without limits you don't even have a value for "0.999~"!

Lothar made a distinction between the sequence (0.9, 0.99, 0.999, etc.), and the limit (0.999~).

In other words:

Between 1 and the sequence "0.9, 0.99, 0.999, etc.", there is a gap at each step, which gets "infinitely small" (there is a technical definition for this term in regards to infinite sequences).

Between 1 and the value (or limit) "0.999~", there is not any gap.

This is one of the sticky points for most people; "0.999~" is actually not in the sequence "0.9, 0.99, etc." at all! It's the mathematical limit, but it's not equal to any number in the sequence.

It's very similar to the analogy I made before, like 0 not ever being in the sequence "0.1, 0.01, 0.001, etc.", although the sequence clearly has 0 as its limit.

...
sure it is, it's just like a-milliondy-one times infinity!
...

Oh, boy... back to square one!