# M for Maths

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1090602.  Thu Aug 21, 2014 9:30 am

 Zziggy wrote: (personally I'm not 100% certain I agree with the assertion that "Well explained maths can be understood by children, almost regardless of the difficulty of the intermediate calculations").

Yes maybe that was a tad hasty - it was more a turn of phrase to try and counter the view among non-mathematicians that maths is this mysterious,impenetrable, incomprehensible subject.

1090758.  Fri Aug 22, 2014 6:17 am

 ConorOberstIsGo wrote: Sorry but I trained as a maths teacher so would like to point out that while the idea of 0.9(recurring) being equal to 1 may not have been explained well, it is most definitely true in a non-tautological fashion. If anyone is still unconvinced after a cursory youtube search of this, then I am happy to take them through it. More importantly, while you may understand several types of infinity, there are actually about 20 different names given to different types of infinity. You might say that two or three are qualitatively the same but it's definitely not two different types plus all of Cantor's.

I would say that the only relevant distinction of infinities for the purposes of this discussion of how to show that 0.999... = 1 to a non-mathematician are that between potential and complete (actual) infinities. However, I am tangentially interested in these other infinities that are not related to Cantor's work. Can you describe a couple of the more tractable ones to me, plz?

I took a look at youtube & didn't see anything that held water unless actual infinity had already been assumed. This is the crux of the issue. I think unless this (and probably limits) are properly explained then the non-mathematician is being short-changed by the various "explanations" I have seen on youtube. They are tautologies.

If the new student is (most likely) thinking of potential infinity in their mind, then 0.999... is a process and cannot be manipulated arithmetically. You cannot then logically multiply it by 10 or subtract it from itself.

If completed infinity is explained and understood and accepted*, then 0.999... has already reached its limit and equals 1.0 and there is no need for the smoke'n'mirrors algebra I've seen on youtube**. There's even a guy on there showing that 0.999 ... does not necessarily equal 0.999... using set theory and it shows the nonsense that can be thrown up by this approach:

Decimal numbers arrived in Europe long before the development of calculus. We can assume that accountants of the time were trying to write the familiar & precise 1/3 fraction as close as they could in the new format, e.g. 33.3% or better still 33.33% or let's just write it as 0.333... and pronounce that as: "naught point three three three recurring". That suggests that there's a string of 3s being forever added onto by some hidden maths 'demon'. It was a very close approximation to 1/3, not exactly equal to it.

The very word "recurring" suggests a process that is ongoing. So, it's unsurprising that a non-mathematician has in their head that the decimal string is something being continually added onto - "recurred" would be a less misleading word for the modern use of completed 0.333...

This is the issue.

*If, like the inestimable Gauss and Poincaré, they don't accept completed infinity, then the notion of limits has to be explained and used. Maths in this arena is more an art than a science.

** I'd be v interested to see a youtube demo using 1/7 = 0.142857... hehehe! :-D

1090819.  Fri Aug 22, 2014 12:56 pm

Could you explain exactly where you believe the tautology to be in these proofs please? I'm really struggling to understand your problems.

 gruff5 wrote: 0.999... is a process and cannot be manipulated arithmetically. You cannot then logically multiply it by 10 or subtract it from itself.

I don't see why this works as a complaint against the proof I offered earlier, which didn't involve doing anything to 0.9... other than rewriting it.

Wikipedia says that
 Quote: Potential infinity is something that is never complete: more and more elements can be always added, but never infinitely many.

I don't believe this applies to 0.9... - that is not what is happening here. I'm also not convinced a non-mathematician would think so. Sorry but I am just not convinced the lack of training in mathematics would render someone incapable of understanding 'a neverending sequence of nines'.

 1090822.  Fri Aug 22, 2014 1:21 pm There is something in your point that decimal numbers are not very well suited for expressing exact fractions, but this - in my eyes - just makes the concept stronger to be honest. In the same way, if we were using binary notation, 1 = 0.111...

 1090824.  Fri Aug 22, 2014 1:33 pm Okay I think I understand what you are getting at; that most non-maths folk think that 0.3recurring has to end somewhere and so cannot be equal to 1/3, right? But I think everyone's favourite topic of division(!) is useful here, especially the conversion of fractions to decimals that you allude to. Oddly I look at a fraction as an incomplete process because it literally describes a division (1/3 means 1 divided by 3) that has not yet been resolved. But I suppose that it is very much like a surd in that is left unresolved for brevity (and accuracy, rather than rounding or truncating). But 1/9 = 0.1recurring 0.1recurring x 9 = 0.9recurring This is probably the youtube proof you saw? along with. 0.9recurring x 10 = 9.9recurring 9.9recurring - 0.9recurring = 9 if 9 x 0.9recurring = 9 then it is also true that 0.9recurring x 1 = 1 I know this is precisely what you were talking about but how is this a tautology? What about 10/2 = 5.0recurring How many 2's go into 10? 5 do and your remainder is 0 okay so your next step in gcse maths division would be now deal with the remainder by multiplying it by 10 and add your answer to the previous one decimal place to the right. 2's into 00 go 0 times so add a zero and repeat ad infinitum.

 1090825.  Fri Aug 22, 2014 1:36 pm Here is the end of a conversation I just held with the most non-mathematicianest person I know: [her]: I don't understand how 1 can equal 0.999 etc ... because won't there always be the 0.00...01 left over? [me]: no [me]: because it's an infinite number of 9s [me]: 1 - an infinite number of 9s = an infinite number of zeroes. You'll never ever reach that 1. [her]: oh ... hang on ... *thinks* [her]: yes, I can see that ... if you get to 0.00...01 there has to be an end number for it to be the remainder of. If its infinite, by definition, there cannot be an end. [me]: exactly! :D So I guess non-mathematicians can get it :)

1090934.  Sat Aug 23, 2014 9:01 am

 Zziggy wrote: Sorry but I am just not convinced the lack of training in mathematics would render someone incapable of understanding 'a neverending sequence of nines'.

0.999... with potential infinity is a 'a neverending sequence of nines'

 Zziggy wrote: ... So I guess non-mathematicians can get it :)

From the way you described your conversation with your friend you were using the potential infinite (which is NOT the same as coming to an end and stopping!) and arithmetic with such a sequence of decimals is intractable, as I'll illustrate below.

 Zziggy wrote: 1 - an infinite number of 9s = an infinite number of zeroes.

Such a subtraction with the potential infinite would give you an undefined non-zero infinitesimal. Yes, someone may agree with your proposition, but that doesn't necessarily mean the foundational issues have been understood.

has 20x more upticks than downticks, so it seems people think the matter has been properly explained to them in the vid. It hasn't. It's tautological because if it is assuming actual infinity for 0.999... then that expression should properly be written as 1.0 and the rest of the "explanation" is redundant and circular. If it doesn't assume actual infinity, then it's downright wrong.

Quote from the wiki link to actual infinity I provided earlier:

 wiki wrote: During the Renaissance and by early modern times the voices in favor of actual infinity were rather rare.... The majority agreed with the well-known quote of Gauss [not accepting actual infinity] ... The drastic change was initialized by Bolzano and Cantor in the 19th century.

Even after Cantor there has remained a significant minority of scholarly mathematicians who have not accepted actual infinity. These have included such luminaries as Poincare, Kronecker, Weyl, Brouwer, Wittgenstein.

These scholastic mathematicians and philosophers spent their professional lives on such matters. If they had a natural inclination towards potential infinity, then that is what we should expect to be the normal naive perspective for the non-mathematician when looking upon 0.999... Actual/completed infinity is a highly abstract concept and cannot be grasped by the human imagination. The concept of infinite space is difficult enough to grasp, for example, and is thought of as something that goes on and on, without end - this is akin to potential infinity.

We are talking about explaining to people who are studying maths at high school or took maths no further than that, so we should certainly not expect them to already know about Cantor, actual infinity etc Here is an interesting link to a lecturer teaching maths at university level who every year spends time on 0.999... and potential vs actual infinity with his maths students.

http://mathoverflow.net/questions/178284/can-one-live-without-actual-infinity

With the non-mathematician's natural mindset of potential infinity, you cannot then arithmetically manipulate 'his' potential unending sequence of 9s - it's not tractable. Think of it like buckets of water. If the buckets are completed (full) then they can of course be added, subtracted, multiplied. 1 bucket + 1 bucket = 2 buckets etc etc But if the bucket is being filled (it is a potentially full bucket), how can you sensibly add filling buckets together and get a defined result? You can't. Same thing with arithmetic manipulation of uncompleted decimals. You can only arithmetically manipulate the limits (the volume of space up to the rim of the filling bucket, if you will).

You can, however, do all the arithmetic you like with limits, they are just ordinary numbers. But if the notion of a limit has been understood and accepted, then there's no need for those tautogical algebraic youtube "proofs". A commentator on youtube summed it up nicely with regard to being explicit about limits:

"Numerals for example, are a representation for a number. The numeral "1" represents the number one. One thing to note is that 0.999... is not a number. It is a process involving a limit which results in the number one. The [...] are interpreted as "limit" and not just "repeating" which seems to be the main confusion. "Please research this properly, you clearly don't understand what infinity means in mathematics." The [...] don't mean only infinity, but also limit"

The original inexact 0.333... (naught point 3 recurring) notation & pronunciation was developed centuries ago as an approximation to 1/3, before limits had arrived and with potential infinity as its basis. I'm beginning to think it is quite inappropriate and confusing for actual infinity to be applied to this ancient decimal expression.

1090936.  Sat Aug 23, 2014 9:10 am

 ConorOberstIsGo wrote: Okay I think I understand what you are getting at; that most non-maths folk think that 0.3recurring has to end somewhere and so cannot be equal to 1/3, right?

No, I doubt many folk think that it ends somewhere. Potential infinity doesn't mean it ends, just that it's an ongoing process. Like how we might think of infinite space that just goes on and on without end. It's natural for all of us, maths and non- maths, to think of infinity like that in the potential way and that's the way we'll generally think of 0.3recurring. If you think about actual/completed infinity too hard, you'll probably go nuts!

 ConorOberstIsGo wrote: Oddly I look at a fraction as an incomplete process because it literally describes a division (1/3 means 1 divided by 3) that has not yet been resolved.

Not odd at all, you're right. As long as the fraction is in its lowest possible terms,ie 1/2 rather than 2/4 for example, that's as much as you can ever do. If you write that same fraction as decimal 0.5, all you're really doing is re-writing the fraction of 1/2 as 5/10 (5 tenths). Which is even more incomplete! Which looks more resolved - 1/7 or 0.1428571428571... ?! There's nothing complete about 0.333.... (3/10 + 3/100 +3/1000 and on and on ). Even more so with pi and e - best off leaving them as undecimalised symbols as much as you can. You only really find decimals useful if you want to do fine measurement in the real world and that's never going to be exact or complete either. Decimals are just a messy version of fractions, really (combos of 10ths, 100ths, 1000ths etc). I hated decimals when they were introduced in primary school - ugly, ungainly beasts compared to elegant fractions!

 ConorOberstIsGo wrote: 1/9 = 0.1recurring 0.1recurring x 9 = 0.9recurring This is probably the youtube proof you saw?

tommyk outlined that as a "proof" earlier, which I responded to.

 ConorOberstIsGo wrote: 0.9recurring x 10 = 9.9recurring 9.9recurring - 0.9recurring = 9 if 9 x 0.9recurring = 9 then it is also true that 0.9recurring x 1 = 1 I know this is precisely what you were talking about but how is this a tautology?

This is the "proof" in the youtube link I've posted in my response to Ziggy (post just above) & respond to above, also.

 ConorOberstIsGo wrote: What about 10/2 = 5.0recurring How many 2's go into 10? 5 do and your remainder is 0 okay so your next step in gcse maths division would be now deal with the remainder by multiplying it by 10 and add your answer to the previous one decimal place to the right. 2's into 00 go 0 times so add a zero and repeat ad infinitum.

I don't know what this is about?
10/2 = 5 as far as I know

PS I'd still be interested in hearing about the non-Cantorian infinities - I couldn't find anything on wikipedia about them.

 1090943.  Sat Aug 23, 2014 9:49 am Um, as someone who has no schooling past high school, I've just accepted 0.999... as being equal to 1 and 0.333... being equal to 1/3. I don't necessary understand it, and most of the explanations here looks to me to be quibbling over very minute differences. (Or in other words, I haven't the foggiest idea what you're on about.) I don't know if that settles anything, but I have no problem accepting it.

 1091027.  Sun Aug 24, 2014 6:23 am I fear I'm boring people, but no matter ... <- a dot-dot-dot joke ;-D CD, it's not a matter of whether 0.999... is equivalent to 1.0 in modern mathematics - it is, by definition. You may be happy that's all you need to know, but plenty of other people appear to want to know why. There is no need to "prove" it is so and most of the attempts to do so (eg on Youtube) are highly misleading or plain wrong. It's simply a definition & Ziggy gave an outline of the definition in one of his earlier posts. In modern mathematics 0.999... is not a number, it's a symbol that represents a geometric series. Here's another series: 1/2 + 1/4 +1/8 + 1/16 + ... Like 0.999... this series also converges to 1.0 Those three dots are part of the geometric series symbol and don't just represent an endless continuation of the terms, but also incorporate the essential notion of a limit. The series converges to the limit of 1.0 and that is its result. If the limit is not taken as the value of the series process you're going to get into deep water; infinitely deep water! The omission of explanation of limits is the cause of all the confusion you see on Youtube comments & elsewhere. More info on series & limits can be found in wikipedia. That might be all you want to know & that's fine. You can stop reading now. ----------------------------------------------------------------------------------- Still here? I'll describe where I think people have got mixed up with so-called "proofs". I take issue with the various attempts to "prove" this definition of 0.999... = 1.0 using digit manipulation & I'll try and explain why. Wikipedia in its article on "0.999..." describes these 'proofs' as "less rigorous", I'd be blunter than that and say they're tautological, pointless and misleading. I'll explain why. One of the most viewed Youtube vids supposedly proving 0.999... = 1.0 with digit manipulation is the one I linked to above by 'singingbanana'/'Numberphile' who has a PhD in maths. It's a tautology because he needs 0.999... to be a finite and completed quantity to do his arithmetic in the first place, so he's relying on his proposition to demonstrate his result - doh!! He's implicitly using the limit (as described above) without saying so. If he's not using limits, when he takes 0.999... away from 0.999... and announces 0.0 he's effectively doing ∞ - ∞ = 0 You might think that you can always take something away from itself and get zero, but it's not a valid thing to do with infinity. Infinity is un-finite, unfinished (the clue is in the name) and is not a number and arithmetic cannot be done on it like that. You might as well say "salary - salary = 0" or "infinite space - infinite space = no space". He even confuses himself when in the 2 minute vid he does the sum 1.0 - 0.999... and says it's infinitesimally small, realises after the recording he's contradicted himself and posts a correction. You can stop reading now. --------------------------------------------------------------------------------- Really? Still here?? Mathematicians today generally can treat 0.999... as a number (one trailing an infinity of repeating 9s) that equals 1.0 without needing to use limits, but they are using actual/completed infinity in their interpretation. Gauss, Poincare and other eminent and leading mathematicians of the past did not accept actual infinity as a valid part of mathematics and the majority of non-mathematicians will really struggle to understand the distinction between potential infinity (which is the 'familiar infinity' & the one used above) and the actual infinity popularised by Cantor in the 19th Century. Maths degree students at university spend some time on the difference between the two kinds of infinities & its relationship to 0.999... and you shouldn't expect to grasp it easily.﻿Last edited by gruff5 on Wed Aug 27, 2014 4:00 am; edited 1 time in total

 1091060.  Sun Aug 24, 2014 10:58 am I absolutely support people wanting to understand it and not just take it on blind faith. I'm still convinced a number of my friends at school were bad at math because they just did what the text book told them mechanically without actually understanding the process behind it. As for 0.999... = 1, I just can't be bothered to understand it, I think. It's not relevant to anything I do, but I am aware it is out there. I also still haven't the foggiest what you're on about, but my concentration is whack now anyhow.

1091289.  Tue Aug 26, 2014 4:39 am

I think our differences of opinion are as follows (apologies if I put words in your mouth, these are only my understandings of your arguments):

ETA: 0.
 gruff5 wrote: CD, it's not a matter of whether 0.999... is equivalent to 1.0 in modern mathematics - it is, by definition.

This is incorrect. 0.999... is not defined to be equal to 1.

1. You seem to consider 0.999... as a manifestation of 'potential infinity'. Having read up on the distinction, I do not. To me, it is one number, which happens to have an infinite number of digits. Now, of course I am biased, but I imagine this is how most people naturally conceive of 0.999... . (For reference, this is the resource which I found most helpful and am therefore basing this on.)

2. Actually, even considering 0.999... as an infinite series rather than a number, I don't see why you can't manipulate it arithmetically. Maybe if it were a series of random numbers you couldn't, but part of the whole point of 0.999... is precisely that it isn't random.
 gruff5 wrote: If he's not using limits, when he takes 0.999... away from 0.999... and announces 0.0 he's effectively doing ∞ - ∞ = 0

I truly don't understand the logic here. As far as I can see he is simply doing 9 - 9, infinitely many times.

3. I think you are saying you accept a proof as long as it incorporates limits?

 gruff5 wrote: Maths degree students at university spend some time on the difference between the two kinds of infinities & its relationship to 0.999... and you shouldn't expect to grasp it easily.

They must have skipped it for the 5 years I've currently been studying maths at university then since I'd never heard of it before ...

1091296.  Tue Aug 26, 2014 4:50 am

gruff5 wrote:
 Zziggy wrote: 1 - an infinite number of 9s = an infinite number of zeroes.

Such a subtraction with the potential infinite would give you an undefined non-zero infinitesimal. Yes, someone may agree with your proposition, but that doesn't necessarily mean the foundational issues have been understood.

Non-zero infinitesimals do not exist in the real number line though.

1091499.  Wed Aug 27, 2014 5:42 am

 Zziggy wrote: This is incorrect. 0.999... is not defined to be equal to 1.

I suppose it depends on how you define the word "definition"! The (apparent) maths guys on the forums were using the word "definition" in this context. What is being said is that 0.999...=0 is a consequence of the axioms of the modern mathematical system. It is there by the definitions of the system and is not something that is there to be 'proved'.

 Zziggy wrote: You seem to consider 0.999... as a manifestation of 'potential infinity'.

I don't consider 0.999... as anything. It's a human-constructed symbol to me (like a mark on an Ordnance Survey map), not a number, and this symbol with those three funny dots needs to be defined. For Isaac Newton (no slouch at mathematics) the symbol had a difference of a non-zero infinitesimal from 1.0 For Gauss, the symbol was equivalent to 1.0 by virtue of viewing it as the limit of a geometric series. In modern mathematics it is a symbol of a number with a completed infinity of 9s digits and this makes it equal to 1.0.

 Zziggy wrote: Now, of course I am biased, but I imagine this is how most people naturally conceive of 0.999... . (For reference, this is the resource which I found most helpful and am therefore basing this on.)

That first sentence is a bit ambiguous. Do you mean that most people naturally conceive of 0.999... using the potential infinite? If so, yes, I totally agree with that. That reference you provided is the best summary I've seen of potential vs completed infinity. I particularly like these two sentences within that:

 Schechter wrote: Some of these mathematicians may be impatient with the few students who still have difficulty with completed infinities. But their impatience is not justified; they are forgetting what difficulty the mathematical community had in reaching its present perspective.

Schechter there is referring to the "few students" doing maths at university. The vast, vast majority of people will not have done a maths degree at university and will be thinking of infinity as it might be considered to exist in the real world - that is potential infinity. Such people may even have studied & understood calculus and using Newtonian mathematics the statement is not valid. There was a poll on a physics forum about 0.999...=0 and it was only 56% who concurred with it (and I think only those physicists who had insight into the matter would have been reading that thread in the 1st place!).

 Zziggy wrote: Actually, even considering 0.999... as an infinite series rather than a number, I don't see why you can't manipulate it arithmetically.

You mean series using the potential* infinite here, yes? Well, if your infinite geometric series converges to a limit (as it does in UK high school maths for A-level students at ages 16-18), then yes, you can manipulate it arithmetically. I don't think people who took maths only as far as GCSE would know about limits, so for them (as with Isaac Newton) 0.999... would be infinitesimally different from 1.0 You can't do arithmetic with non-zero infinitesimals, that's equally nonsensical as doing arithmetic with potential/incomplete infinity.

∞ + ∞ = 2∞ ?????

 Zziggy wrote: I truly don't understand the logic here. As far as I can see he is simply doing 9 - 9, infinitely many times.

I think you (and singingbanana) find it extremely hard to step outside the completed infinity mindset, Ziggy. This is not surprising as you've been doing maths at uni for 5 years (I didn't do maths at uni). To understand the problems people have with the issue, you need to put yourself into the potential infinity mindset - this is what the majority of the viewers of the video will be in.

In this mindset, that "doing 9 - 9, infinitely many times" doesn't ever finish! Do you see? I think your non-mathematician friend was still in the potential infinity mindset. She said something like: "you never get to the 0.000....1" but there isn't a 0.000....1 there at all in completed infinity.

That video & all these digit-manipulation 'proofs' attempt to show that 0.999... is a finite and completed number (1.0), and in doing so rely on 0.999... being a finite and completed number to make the arithmetic work - don't you see the problem with that?

 Zziggy wrote: I think you are saying you accept a proof as long as it incorporates limits?

Perhaps I don't express myself clearly in my writing, but I hope you can see now that my view is much wider than that. I was emphasising limits because I think they're more tractable to explain than completed infinity and limits are part of the A-level curriculum. I personally am more interested in maths as it relates to the real world, so as a matter of taste, i don't care for completed infinity. This doesn't mean I don't understand it or its place in modern mathematics. I really don't care for set theory, either, as a matter of taste.

 Zziggy wrote: They must have skipped it for the 5 years I've currently been studying maths at university then since I'd never heard of it before ...

I was taking extreme liberties there. I was using a quote of a maths academic who did spend time every year with his students on this as part of the foundations of mathematics. The issue throws up problems so consistently for students that it is used as a way to investigate methods to improve maths education.

*we could really do with completely different words for potential infinity and complete infinity. They are such different concepts that when natural brevity arises and the single word "infinity" is used, I don't know what is being talked about in this context. In a weird way, completed infinity is more finite because it is complete and finished, so I reckon the "new imposter" is the one that will need to think up a new word for itself!

PS I've edited the last para of my previous post following someone pointing out it was an inaccurate reflection of the stance of modern mathematicians.

1091531.  Wed Aug 27, 2014 8:35 am

gruff5 wrote:
 Zziggy wrote: This is incorrect. 0.999... is not defined to be equal to 1.

I suppose it depends on how you define the word "definition"! The (apparent) maths guys on the forums were using the word "definition" in this context. What is being said is that 0.999...=0 is a consequence of the axioms of the modern mathematical system. It is there by the definitions of the system and is not something that is there to be 'proved'.

But you could say that about anything in maths. The whole point of maths is that you start with axioms and make the rest out of the axioms. To say that something is 'defined' and 'doesn't need proving' simply because it follows from axioms is ... wrong. Sorry, I'm struggling to think of a nicer way of putting it ... maybe I've simply misunderstood you here?

gruff5 wrote:
 Zziggy wrote: Now, of course I am biased, but I imagine this is how most people naturally conceive of 0.999... . (For reference, this is the resource which I found most helpful and am therefore basing this on.)

That first sentence is a bit ambiguous. Do you mean that most people naturally conceive of 0.999... using the potential infinite?

No it isn't, and no I don't! What I said was 'I think you are using the potential infinity; I do not, and I think most people don't'. That's some creative reading to think that is ambiguous and probably agreeing with you! :P

 gruff5 wrote: Schechter there is referring to the "few students" doing maths at university. ... There was a poll on a physics forum about 0.999...=0 and it was only 56% who concurred with it (and I think only those physicists who had insight into the matter would have been reading that thread in the 1st place!).

Actually I read that as referring to the few maths students who didn't understand it. Different interpretations I guess.
Bit disappointed that 56% of physicists agree that 0.999..=0, but what can you do hey, if they were clever they'd be mathematicians :P (you also wrote 0 instead of 1 in your first paragraph)

 gruff5 wrote: You mean series using the potential* infinite here, yes? ... In this mindset, that "doing 9 - 9, infinitely many times" doesn't ever finish! Do you see?

(Just edited down for space)
If you are considering an infinite series, you can manipulate it term by term, why not? Even if the limit is undefined. For instance, summing the natural numbers n is unbounded. I see no reason why you can't sum 10n. The limit is also unbounded. However, 0.9 recurring is not infinity. Yes "doing 9 - 9 infinitely many times doesn't ever finish", but any and every place it does happen will be the same, i.e. 0.

 gruff5 wrote: I think your non-mathematician friend was still in the potential infinity mindset. She said something like: "you never get to the 0.000....1" but there isn't a 0.000....1 there at all in completed infinity.

Exactly. It is an informal proof by contradiction.

 gruff5 wrote: That video & all these digit-manipulation 'proofs' attempt to show that 0.999... is a finite and completed number (1.0), and in doing so rely on 0.999... being a finite and completed number to make the arithmetic work - don't you see the problem with that?

Nope - I don't even see the correct in that :P

Look, I don't think we are going to agree on this. Shall we just mutually accept that 0.999... = 1, that many people accept this even if they don't understand it, and leave the rest as undefined?

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