# 1 is 0.9999999999999............

10/9 <> 1r1. I don' know what you mean since for any n of the division algorithm, 10/9 = 1.1(n-1). That is all I can prove.
I thought you knew something about Maths - 1r1 (In this context at least) is 1 remainder 1.

Again, can you prove you produced a digit for all possible natural numbers?
I don't have to - I only have to prove that it's true for some numbers and for the general case. That's why we use induction.

I thought you knew something about Maths - 1r1 (In this context at least) is 1 remainder 1.

I don't have to - I only have to prove that it's true for some numbers and for the general case. That's why we use induction.

Are you saying 10/9 = 1R1 ? I thought you knew something about Maths.

And I provided a recursive algorithm that demonstrates your division does not produce an infinite number of digits.

Is it wrong?

Are you saying 10/9 = 1R1 ? I thought you knew something about Maths.

That's precisely what I am saying - this isn't any harder than what they teach in Gradeschool.

Here:
$$\hspace{53 pt}1 \\ \frac{10}{9}= 9 \overline{\big) 10}\\ \hspace{37 pt}-9 \\ \hspace{42pt}\overline{\hspace{7 pt} 1}$$
See that 1 on the bottom line? It never goes away, it will always be there, no matter how many times you perform the operation, because 9 will only ever go into ten once, and 10-9 will always be 1.

And so, 10/9 = 1 with 1 remainder (or 1r1, or $$1\frac{1}{9}$$)

And I provided a recursive algorithm that demonstrates your division does not produce an infinite number of digits.

Is it wrong?
Obviously your proof is wrong, because it fails to match reality.

You've failed to retain the context of the post I was replying to haven't you.

Also, inspite of all of your bluster and blither, you haven't answered the two very direct questions I asked you.

I have been taking the full context into consideration, Trippy, as included in my exchanges with you and Billy T and arfa brane, Q Q et al.

Pardon me, Trippy, but I took your two 'questions' in that post to be 'throwaway lines' rather than serious questions to be further addressed. I say, "further addressed" advisedy, because I already effectively addressed that sort of thing when I cautioned against trivial/facile 'constructions' and 'arguments' based on unitary self-selecting logics/definitions which don't throw any more light on the issues.

It's not in any way 'profound' that 1/1 = 1, nor that 9/9=1 etc. It ALSO has counter-productive effect when such facile unitary-obvious 'reasoning' and use of like/like 'constructions lead to such like/like situations like 0/0 where the 'reductio ad absurdum' of such possibly flawed approaches comes to a grinding "undefined" and/or "undetermined" halt where the axioms lose all sense at this logical boundary condition where the like/like 'construction' consistent with your argument says nothing but that it is 0/0.

Such facile self-selecting 'examples' offered as 'proofs' are based ultimately on facile constructions/assumptions which get us no further in actually answering the mathematical aspects/issues (the formating convention aspects/issues having already been agreed upon between me and Billy T et al).

So you see, Trippy, it seems a little unfair of you now to use that facetious questions 'tactic' and then say I did not address them; especially as the inherent points (and more) were already covered more than once by my posts in our exchange and in my exchanges with others.

Anyhow, as is the case lately, I have to get back to 'read-only' again for a few minutes and then log out again. Cheers and good luck in your discussions with others, Trippy!

Pardon me, Trippy, but I took your two 'questions' in that post to be 'throwaway lines' rather than serious questions to be further addressed. I say, "further addressed" advisedy, because I already effectively addressed that sort of thing when I cautioned against trivial/facile 'constructions' and 'arguments' based on unitary self-selecting logics/definitions which don't throw any more light on the issues.
I'm not terribly interested.

It's not in any way 'profound' that 1/1 = 1, nor that 9/9=1 etc. It ALSO has counter-productive effect when such facile unitary-obvious 'reasoning' and use of like/like 'constructions lead to such like/like situations like 0/0 where the 'reductio ad absurdum' of such possibly flawed approaches comes to a grinding "undefined" and/or "undetermined" halt where the axioms lose all sense at this logical boundary condition where the like/like 'construction' consistent with your argument says nothing but that it is 0/0.
Ignoratio elenchi, argumentum ad inorantiam.

I've pointed out to you, more than once, that the proof I posted relies on proving that 0.9(9) has the same properties as the multiplicative idenity. IE it proves that 9x0.9(9) = 9x1 and the only way this can be true is if 0.9(9) has the same properties as the multiplicative identity - IE 1.

If you don't like it, take apart step by step and address it with maths, otherwise, stop wasting my time, and everybody else time with this unmitigated bullshit you insist on posting.

So you see, Trippy, it seems a little unfair of you now to use that facetious questions 'tactic' and then say I did not address them; especially as the inherent points (and more) were already covered more than once by my posts in our exchange and in my exchanges with others.
No undefined, it was perfectly reasonable, there was nothing facetious. I addressed specific questions directly to you and you ignored them. Your excuses for ignoring them are irrelevant, and at the moment, you're bordering on trolling.

chinglu said:
#1577 proves the Taylor series is not an infinite addition.
From wikipedia:
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

Since I do know what a Taylor series is, I don't need to google millions of hits that will say the same thing. Looks like chinglu has a lot of work to do correcting them all.

p.s. post #1577 is about the halting problem, I guess chinglu doesn't understand that either.

I'm not terribly interested.

Ignoratio elenchi, argumentum ad inorantiam.

I've pointed out to you, more than once, that the proof I posted relies on proving that 0.9(9) has the same properties as the multiplicative idenity. IE it proves that 9x0.9(9) = 9x1 and the only way this can be true is if 0.9(9) has the same properties as the multiplicative identity - IE 1.

If you don't like it, take apart step by step and address it with maths, otherwise, stop wasting my time, and everybody else time with this unmitigated bullshit you insist on posting.

No undefined, it was perfectly reasonable, there was nothing facetious. I addressed specific questions directly to you and you ignored them. Your excuses for ignoring them are irrelevant, and at the moment, you're bordering on trolling.

My bolding in that last sentence of yours....

Ok, such intimidatory words/tone from a 'mod' (who bases his opinion/threat on his own impression that I have made 'excuses' and have 'not addressed your questions' etc) must be answered without delay, despite my limited time resources at this juncture.

Trippy, I have NOT ignored NOR have I been irrelevant. I already pointed out where, in your following post as quoted....

Trippy said:
I trivially reformated nothing, I wrote out explicitly what 0.111(1) recurring is and extended it to the generic case to prove that the statements 9*0.111(1) = 0.999(9) is true.

Originally Posted by Undefined
...and then just as trivially used that facile RE-arranged format ONLY in further facile 'formatting based' treatments, so as to RE-INTRODUCE a '9' UNIT TIMES factor which merely effectively re-inserts the 9/9 unitary factor in order to further re-arrange the same re-formatted result into a UNITARY trivial reformatting of the argument based on the like/like construction you want to introduce circuitously.
Let me lead you through it by the hand, step by step, seeing as how you're obviously not comprehending it on your own.

Starting point (after proving it by stepping through the long division - I stated long ago that this should be done):

Expand the decimal representation of 1/9 to show the decimal powers (I mentioned long ago that people needed to keep in mind what the numbers actually represent):

Multiply both sides of the equation by 9

Expand the brackets

Simplify.

You make an obvious "statements 9*0.111(1) = 0.999(9) is true." claim that no-one is disputing because it is trivially true, simply because you multiply by 9 WHOLE UNITS, and NOT because of any fundamental FRACTIONAL aspect being elucidated as such about the fractional string itself.

You also (as I already pointed out previously) later in your above post go on that same trivial route of UNITARY dependence, by whole unit multiplication by 9 in order to recover your a-priori state of UNITARY state 'equivalence'. Not profound proof of anything excpet the circuitousness built in to the exercise as 'constructed'.

Why should that 'prove' anything? I addressed your points by asking that more than once. So please don't make unfair/intimidatory statements/accusations based on your own missing of the fact that I have been following/addressing your and others' contexts/arguments.

You at no stage prove via EXTERNAL MEANS (ie, not resorting to trivial manipulations using UNITARY multipliers and or unitary like/like constructions which again elucidate nothing about the fractional behaviour of the fractional string which makes it equal to 1). We have already covered the Limits route and that something more than that also is needed if we are going to convince anyone once and for all either way.

All your approaches have been done to death in earlier discussions elsewhere. The point I am making to you now is: Can you explain the MATHEMATICAL (not formatting convention/definition 'equality') of 1= .999... WITHOUT resorting to 1 from the starting state? Ie, without using unitary starting/ending points which are inherent in your approaches/proofs so far?

That is the only way to put an end to the differences of opinions. That's it. I make no other assessment either way, I merely ask for more ROBUST and independent 'proofs' approaches to the issue. My observations in context on your and others offered 'proofs' and 'approaches' is already in the record. I have no more to add at this juncture. Good luck with your other discussions, Trippy, everyone!

Undefined said:
You at no stage prove via EXTERNAL MEANS (ie, not resorting to trivial manipulations using UNITARY multipliers and or unitary like/like constructions which again elucidate nothing about the fractional behaviour of the fractional string which makes it equal to 1).
You should explain what "external means" means, do you mean a proof that doesn't use numbers, or just whole numbers?. Also, what is "fractional behaviour"?
We have already covered the Limits route and that something more than that also is needed if we are going to convince anyone once and for all either way.
You're implying there's a problem with 1 = 0.999..., and that no-one is convinced it's true. This is false.
"Anyone" would have to exclude people who understand why it is true, like people with degrees in mathematics or related subjects.

My bolding in that last sentence of yours....

Ok, such intimidatory words/tone from a 'mod' (who bases his opinion/threat on his own impression that I have made 'excuses' and have 'not addressed your questions' etc) must be answered without delay, despite my limited time resources at this juncture.
And once again you grasp the wrong end of the stick come out swinging.

Here's the part we're arguing about in context:
So you see, Trippy, it seems a little unfair of you now to use that facetious questions 'tactic' and then say I did not address them; especially as the inherent points (and more) were already covered more than once by my posts in our exchange and in my exchanges with others.
No undefined, it was perfectly reasonable, there was nothing facetious. I addressed specific questions directly to you and you ignored them. Your excuses for ignoring them are irrelevant, and at the moment, you're bordering on trolling.
Do you honestly think that my comment about your behaviour bordering on trolling has anything to do with your outright evasion of the direct questions I asked you?

Or do you think that it might have more to do with your personal noise/attack on me by calling me facetious?

Trippy, I have NOT ignored NOR have I been irrelevant. I already pointed out where, in your following post as quoted....

You make an obvious "statements 9*0.111(1) = 0.999(9) is true." claim that no-one is disputing because it is trivially true, simply because you multiply by 9 WHOLE UNITS, and NOT because of any fundamental FRACTIONAL aspect being elucidated as such about the fractional string itself.

And yet, here we are:
';lkjhg
You are right, but what fraction is .999999.... ?
9/9

Can I see your proof?

Please indicate why induction it not needed.

Or please indicate why it is needed.

Thanks

What's to prove?

Please excuse the formatting of this first step if the columns don't quite line up.

$$\hspace{37 pt}0.111\overline{1} \\ \frac{1}{9}= 9 \overline{\big) 1.0000...}\\ \hspace{37 pt}-9 \\ \hspace{42pt}\overline{\hspace{7 pt} 1}0 \\ \hspace{43 pt} -9 \\ \hspace{47 pt} \overline{\hspace{7}1}0 \\ \hspace{49 pt} -9 \\ \hspace{56 pt}\overline{\hspace{7}1}0 \\ \hspace{56 pt} -9 \\ \hspace{65 pt} \overline{\hspace{7}1...}$$

$$\frac{1}{9}= 0.111\overline{1}$$

$$\frac{1}{9}=1 \times 10^{-1} + 1 \times 10^{-2} + 1 \times 10^{-3} + 1 \times 10^{-4} + ... + 1 \times 10^{-(n-1)} + 1 \times 10^{-n} + 1 \times 10^{-(n+1)}+ ...$$

$$9 \times \frac{1}{9}= 9 \times (1 \times 10^{-1} + 1 \times 10^{-2} + 1 \times 10^{-3} + 1 \times 10^{-4} + ... + 1 \times 10^{-(n-1)} + 1 \times 10^{-n} + 1 \times 10^{-(n+1)}+ ...)$$

$$9 \times \frac{1}{9}= 9 \times 10^{-1} + 9 \times 10^{-2} + 9 \times 10^{-3} + 9 \times 10^{-4} + ... + 9 \times 10^{-(n-1)} + 9 \times 10^{-n} + 9 \times 10^{-(n+1)}+ ...$$

$$\frac{9}{9}= 0.999\overline{9}$$

Apparently you have failed retain the context of the conversation.

Chinglu didn't say what he wanted proven, but the context of the discussion was $$\frac{1}{9} \times 9 = 0. \overline{1} \times 9$$

So it was precisely what was being discussed/argued.

You also (as I already pointed out previously) later in your above post go on that same trivial route of UNITARY dependence, by whole unit multiplication by 9 in order to recover your a-priori state of UNITARY state 'equivalence'. Not profound proof of anything excpet the circuitousness built in to the exercise as 'constructed'.
No... ONce again... What I actually did was prove that 0.999(9) has the properties of the multiplicative identity, therefore 0.999(9) is trivially true.

Why should that 'prove' anything? I addressed your points by asking that more than once. So please don't make unfair/intimidatory statements/accusations based on your own missing of the fact that I have been following/addressing your and others' contexts/arguments.
Pull your horns in. My tolerance for this kind of invective personal noise/commentary is rapidly waning. If you don't want to get called out on trolling, don't troll around me. It's that simple.

You at no stage prove via EXTERNAL MEANS (ie, not resorting to trivial manipulations using UNITARY multipliers and or unitary like/like constructions which again elucidate nothing about the fractional behaviour of the fractional string which makes it equal to 1). We have already covered the Limits route and that something more than that also is needed if we are going to convince anyone once and for all either way.
What does this even mean?

Why should I be interested in convincing you of anything? At this point I would be happy to stop you from confusing others.

All your approaches have been done to death in earlier discussions elsewhere. The point I am making to you now is: Can you explain the MATHEMATICAL (not formatting convention/definition 'equality') of 1= .999... WITHOUT resorting to 1 from the starting state? Ie, without using unitary starting/ending points which are inherent in your approaches/proofs so far?
I didn't start off with 1, I started off with the rational number $$\frac{1}{9}$$. You understand the difference, right?

That is the only way to put an end to the differences of opinions. That's it. I make no other assessment either way, I merely ask for more ROBUST and independent 'proofs' approaches to the issue. My observations in context on your and others offered 'proofs' and 'approaches' is already in the record. I have no more to add at this juncture. Good luck with your other discussions, Trippy, everyone!
Or you could spend an afternoon with google, wikipedia, or a good textbook to actually learn about those things you're objecting to.

I mean, if your objection is that you don't like the fact that mathmeticians choose to leave $$\frac{0}{0}$$ undefined.

Did it ever occur to you that the reason why these proofs seem so trivially to you is because the statement is trivially true? That maybe if you want to come up with a new form of maths, you're starting in the wrong place?

You should explain what "external means" means, do you mean a proof that doesn't use numbers, or just whole numbers?. Also, what is "fractional behaviour"?
You're implying there's a problem with 1 = 0.999..., and that no-one is convinced it's true. This is false.
"Anyone" would have to exclude people who understand why it is true, like people with degrees in mathematics or related subjects.

No, as I have already often stated/made clear, I mean that any arguments purporting to 'prove' the mathematical equivalency (as distinct from formatting convention equivalency) should try to avoid using constructions which may have in-built circuitously self-selecting treatments which only revert to the initial inbuilt 'pre-conclusions' results. That is why I already and often asked for proof arguments which start from the fractional string state and independently from there on (ie, without introducing a UNITARY operator or other constructions such as like/like trivial expressions like 1/1, 9/9, 8/8 etc which ARE effectively already unitary....and which also lead to 0/0 "undefined" things because the axioms are not fully capable at that stage). Like I said, trivial and unitary manipulations merely massage the string into what you want, by simply MAKING the expresion a UNITARY one from the outset of the 'exercise'. That is what I mean; and what I ask should be avoided if the issue is to be really proven properly without such obvious UNITARY 'set-ups'.

And once again you grasp the wrong end of the stick come out swinging.

Here's the part we're arguing about in context:

Do you honestly think that my comment about your behaviour bordering on trolling has anything to do with your outright evasion of the direct questions I asked you?

Or do you think that it might have more to do with your personal noise/attack on me by calling me facetious?

And yet, here we are:
';lkjhg

Apparently you have failed retain the context of the conversation.

Chinglu didn't say what he wanted proven, but the context of the discussion was $$\frac{1}{9} \times 9 = 0. \overline{1} \times 9$$

So it was precisely what was being discussed/argued.

No... ONce again... What I actually did was prove that 0.999(9) has the properties of the multiplicative identity, therefore 0.999(9) is trivially true.

Pull your horns in. My tolerance for this kind of invective personal noise/commentary is rapidly waning. If you don't want to get called out on trolling, don't troll around me. It's that simple.

What does this even mean?

Why should I be interested in convincing you of anything? At this point I would be happy to stop you from confusing others.

I didn't start off with 1, I started off with the rational number $$\frac{1}{9}$$. You understand the difference, right?

Or you could spend an afternoon with google, wikipedia, or a good textbook to actually learn about those things you're objecting to.

I mean, if your objection is that you don't like the fact that mathmeticians choose to leave $$\frac{0}{0}$$ undefined.

Did it ever occur to you that the reason why these proofs seem so trivially to you is because the statement is trivially true? That maybe if you want to come up with a new form of maths, you're starting in the wrong place?

Trippy, what is trivially true is trivially true. No argument.

I also pointed out that though you started with 1/9, you then introduced a multiplier of 9, thus creating the unitary inevitability because you convert a fractional string to a UNITARY-plus-fractional string which MAKES the string 'cross the decimal point' such that you are no longer dealing with the same fractional string...in short, you FORCED the original string by your unitary+ multiplier. That's what I was trying to point out.

And about that 0/0. No, Trippy, it's NOT as you stated it; mathematicians DON'T CHOOSE to leave it "undefined", they are FORCED to leave it so because the AXIOMS don't cover it. Hence the problem which I point to with using such like/like 'constructions where the AXIOMS dealing with such things are NOT complete, and lead to FORCED "undefined" brick walls! Do you see what I am trying to say?

Anyhow, Trippy, if you admit to being human, and admit that it is possible that YOU are misunderstanding where I am coming from on this, then you must admit that your current 'assessment' that I am the one 'trolling/evading' may be a misjudgement on your part, based on your own misunderstandings? If you ARE human, and allow that sometimes it is you not me who is missing the point, then please stop invoking the cloud of intimidatory/punitive reaction just because you have mod status and I don't. Please.

Thanks anyway for your trouble. I can see that if I haven't by now got across to you what I wanted to get across, then it's possible we are BOTH misunderstanding each other, and I would like to leave it at that and not bring trolling and combative tone etc into it.

Disengaging amicably. Cheers!

I also pointed out that though you started with 1/9, you then introduced a multiplier of 9
Look.
In order to make 0.111(1) into 0.999(9) I have to multiply it by nine correct?

Elementary school maths say that if I multiply one side of an equation by a value, I must multiply the other side of the equation by nine.

thus creating the unitary inevitability because you convert a fractional string to a UNITARY-plus-fractional string which MAKES the string 'cross the decimal point' such that you are no longer dealing with the same fractional string...in short, you FORCED the original string by your unitary+ multiplier.
This doesn't even mean anything. As I just explained to you we multiply both sides by 9 because $${\frac{0.\overline{1}}{0.\overline{9}}=9$$.

Get it?

The fact that $$\frac{1}{9} \times 9$$ happens to equal 1 is not the point of the exercise.

That's what I was trying to point out.
I understand what you were trying to point out, I happen to think that what you were trying to point out is wrong and wrong headed.

Do you understand? Just because I disagree with you doesn't mean I haven't understood you.

And about that 0/0. No, Trippy, it's NOT as you stated it; mathematicians DON'T CHOOSE to leave it "undefined", they are FORCED to leave it so because the AXIOMS don't cover it. Hence the problem which I point to with using such like/like 'constructions where the AXIOMS dealing with such things are NOT complete, and lead to FORCED "undefined" brick walls! Do you see what I am trying to say?
Well, that's your opinion anyway.

Anyhow, Trippy, if you admit to being human, and admit that it is possible that YOU are misunderstanding where I am coming from on this,
I have yet to show where I have misunderstood you, or how.

...then you must admit that your current 'assessment' that I am the one 'trolling/evading' may be a misjudgement on your part, based on your own misunderstandings?
As I have explained to you once already, the comment about you trolling was in relation to your insulting me by calling me facetious, nothing else. Now, this is to be the end of the personal noise. Get it?

If you ARE human, and allow that sometimes it is you not me who is missing the point, then please stop invoking the cloud of intimidatory/punitive reaction just because you have mod status and I don't. Please.
Stop insulting people and I won't have to take action as a mod. It's that simple. Now, let this be an end to it. Address the questions and the posts and stop dragging your personal issues into the discussion.

Undefined said:
That is why I already and often asked for proof arguments which start from the fractional string state and independently from there on (ie, without introducing a UNITARY operator or other constructions such as like/like trivial expressions like 1/1, 9/9, 8/8 etc which ARE effectively already unitary
But if you want to prove 0.999... or 0.(9)* is equal to 1, how do you avoid such trivial expressions, assuming 1 = 1/1 = 2/2 = ... is "trivial"?

What about starting with 0.(9)* = 0.(3)* + 0.(3)* + 0.(3)* which contains none of what you refer to as "trivial expressions".

When is it ok to introduce a 1? or a 9/9 in the proof?

But if you want to prove 0.999... or 0.(9)* is equal to 1, how do you avoid such trivial expressions, assuming 1 = 1/1 = 2/2 = ... is "trivial"?

What about starting with 0.(9)* = 0.(3)* + 0.(3)* + 0.(3)* which contains none of what you refer to as "trivial expressions".

When is it ok to introduce a 1? or a 9/9 in the proof?

Good point.

But if you want to prove 0.999... or 0.(9)* is equal to 1, how do you avoid such trivial expressions, assuming 1 = 1/1 = 2/2 = ... is "trivial"?

What about starting with 0.(9)* = 0.(3)* + 0.(3)* + 0.(3)* which contains none of what you refer to as "trivial expressions".

When is it ok to introduce a 1? or a 9/9 in the proof?

It's not that 1/1 is a problem per se. The axioms are ok with that (until the like/like of 0/0 rears its ugly head and the axioms fail....hence my reservations where the arguments/proofs depend on such like/like constructions as crucial to the 'proof' argument.

What I am trying to say is that, just as the LIMITS approach avoids the need for any such 1/1. 9/9, etc trivial and un-elucidating 'devices/exercises, I am looking for an equally 'non-trivial' way of getting to the same point but not using the unitary/trivial 'devices' mentioned. A way of starting from the fractional string itself and making logical actions/assumptions bout it such that the mathematical 'transition' from fractional infinite string to 'unitary' is achieved without the use of trivial 1/1, 9/9 etc manipulations which only introduce what I observe as a bias to the argument/treatment which gives the result you want but is not in the same 'triviality independent' limits method. That' all. I am looking for such a way so that no arguments can be 'faulted' for any reason....especially for the reason that using axiomatic trivialities such as 1/1 is not a 'complete' approach since the axioms break down when 0/0 comes along. That's it. Good luck.

Look.
In order to make 0.111(1) into 0.999(9) I have to multiply it by nine correct?

Elementary school maths say that if I multiply one side of an equation by a value, I must multiply the other side of the equation by nine.

This doesn't even mean anything. As I just explained to you we multiply both sides by 9 because $${\frac{0.\overline{1}}{0.\overline{9}}=9$$.

Get it?

The fact that $$\frac{1}{9} \times 9$$ happens to equal 1 is not the point of the exercise.

I understand what you were trying to point out, I happen to think that what you were trying to point out is wrong and wrong headed.

Do you understand? Just because I disagree with you doesn't mean I haven't understood you.

Well, that's your opinion anyway.

I have yet to show where I have misunderstood you, or how.

As I have explained to you once already, the comment about you trolling was in relation to your insulting me by calling me facetious, nothing else. Now, this is to be the end of the personal noise. Get it?

Stop insulting people and I won't have to take action as a mod. It's that simple. Now, let this be an end to it. Address the questions and the posts and stop dragging your personal issues into the discussion.

Ok, Trippy. I get you. Please read my post to arfa brane above for some more clarification as to where I am coming from and trying to get to.

As to the 0/0 "undefined" being a choice by mathematicians, I again point out that it is not my 'opinion' that the axioms force the "undefined" tag. Mathematicians have no choice in the matter UNLESS the relevant axioms are (as I am involved in doing) modified/enhanced in such a way that such things as 0/0 are no longer "undefined" by them. That's another aspect which I feel is crucial to making the mathematics 'complete' so as to handle more unusual states and be more capable of reality modeling than at present incompleteness demonstrates.

And just to be clear, I didn't mean to imply you were facetious as such, I just thought your 'throwaway lines' were a 'facetious questions' tactic. My mistake. My apologies for any offense, Trippy.

Again I can't stay. Cheers and 'read you round', Trippy, arfa brane, everyone!

Macro-finite = 1---Micro-infinite = 0.999.... > >...oo...Eternal Subdivision........

Quantum Quack.."Can I suggest that the number one is only a balloon full of decimal places?"

Sure you can, i.e. BT's line segment has vale of 1 an you and he are infering micro-infinite subdivision of th value 1, line segment.

You know .....the number one is a boundary for 0.999...

Yeah, i.e. we begin with a given, macro-finite value of 1.

imagines a balloon full of o.999...

It is and error to use words like "full", large or small in association with infinite concepts.

"the surface of the balloon is the number one."

BT'sl ine segment

so therefore 0.999... = 1 balloon

Line segment value 1, inherently has two associated aspects, ending and beginning ergo a finite.

If we want to subdivide the line segment into two line segment parts, we initial the first 0.1 on the line segment, thereby dividing the line segment into two line segments via subidivision of line segement by introducing 0.1 i.e;

terminal beginning of value>1---0.1--------1<value teminal ending

Who says that the number one is finite any way?

QQ, you need to be able to distingush between marco and finite, and in this specific case, you need to distinguish between macro-finite and mirco-infinite subidvison of your given balloon, my given polygon(s) and Billy T's line segment.

Macro-finite = our finite Universe of occupied space.

Micro-finite = gravitatonal quantum limits, that may be variable yet still finite.

Macro-infinite = non-occupied space, beyond our finite occupied space Universe

Mathematics are used to exemplify aspects of our finite occupied space Universe ergo for any all alledged practical purposes.

Mathematics is also used to exemplify that, which appears to be beyond the aspects of our finite occupied space Universe ergo mathematically illusionary, mental masturbation, that has not practical--- ergo contrived ---purpose.

Complex Mental Masturbation > Simple Mental Masturbation > finite vs infinite

Billy TMath is not done by "common sense" nor by popular vote, nor by opinion, BUT BY PROOFS.

I agree, however, we cannot discount common sense when it is needed and we all need and use it on occassion.

On this occasion common sense tells me, that your performing mathematically illusionary, mental masturbation i.e. I understand rounding higher, and in this specific case finite 1.0 = 0.999... your need to do two operations, to convert/map 0.999 to 1.0;

1) we stop infinite--- infinite recursion? ----ergo 0.999... becomes a finite 0.999,

2) we 0.999 to 1.0.

This is elementary or probably junior high school level of understanding, is my guess. i dunno

Several different proofs that 1 =0.999... have been presented.

Aledged "proofs", that I certainly do not understand so cannot verify that they are a proof and there are some who dispute those proofs with there math.

You nor anyone else has, and never-- nada/zip/zero ---will be able to, present a rational, logical, common sense and relatively simple explanatory guide to explain any of those alledged "proofs" for the 99% of humanity that will never ever deal with such complicated mathematics.

Some are even simpler than mine,

YOurs is NOT simple dude. Yeah Origins is simple but illogical and certainly not a proof, just as I belive none of them are truly "proof" only mathematically illusionary, mental masturbation.

but not as rigorous as require multiplying an infinitely long decimal expression to conclude that 1 and 0.999... are identical values,

If you think those two are identical, then;

1) you need glasses,

2) you do not understand the differrence between the concept of infinite and finite and those are incommensurate except in the situation of starting with a give macro-fiinite, value 1 line-segment, and then conceptually/abstractly initiating a micro-infinite( ... ) subivision of value 1 line-segment. In so doing, teh first 0.1 conceptually/abstractly divides the line segment into seperate line parts.

both rational and finite as they are the same , identical, number, only two different names for that same number.

For practical purposes we state that, for absolute truth we do not.

My proof, without need of multiplying an infinitely long decimal string, nor any limiting procedure that some valid proofs do use, and every step is logically derived from stated definitions, so it is quite rigorous.

No infinitely long decimal strings exist in our macro finite occupied space Universe. What we do to express infinite abstractions is "..." or "oo" etc....ergo mental masturbation.

In contrast you give ONLY your opinion and no supporting proof;

You give alledged "proof", as so others here claim to do, and tho I understand that there are many pathways-- see Feynman ---to the same resultant, if you want to continue to deny, that even those in your camp, have issues with each others mathematical approach, then continue to deny. The "proof" of such is in the many posts here, unless they go back and edit it out.

I don't need a "proof" as my eyes with appropriate glasses can see that 1.0 is not identical to 0.999....and common sense, what 99% of humanity uses when neccessary, tells me that a finite value 1 will never ever be equal to an infinite value 0.999..., except for practical/contrive purposes.

further more you can not find any error in the proof I gave and I even number the steps for you to tell what step you did not think was valid.
SUMMARY: YOU HAVE ZERO UNDERSTANTING OF MATH (and very poor comprehension of English)

Is that the best rational, logical common sense and explantory guide to your, Origins or anyone elses alledged "proof"? Ha dude, how many in your camp actually can follow RPenners and Chinglus mathamatics?

How many humans who have more math knowledge than me, can follow those maths/

And your also stating false nonsense in the latter above, because, your ego blocks you to simple absolute truths that any human only needs a working set of eyes along with some common sense to know that your blowing ego blocking hot air.

I will admit you now show a slight improvement in you written text: I. e. no longer write "infinite value 0.9999..." but still think that in same old false claim (only you ignorantly assert)* "that finite 1 can not equal infinite 0.999..." That is of course false as 0.999... is not infinite in value, only has an infinitely long decimal expression that has the value of very finite 1. (Much like 1/3 =0.3333.... has an infinitely long decimal expression for finite, rational fraction 1/3.)

* Still persisting in the ignorant claim that the value must be infinite if the decimal expression of it is infinitely long.

Huh? What are going on about now. Infinite is infinite dude. Your playing grammar games.

You don't even realize / understand that 1/4 = 0.25 is also infinitely long decimal expression given more correctly (no assumption about less significant decimal place / locations being zero needed as that is explicitly so stated.) as 1/4 = 0.25000000000000000000 ...

C,mon dude, your playing more mental masturbation games here. 0.999....is infinite approach to finite 1.0 and I believe I saw that posted early on in this thread.

Your given infinite value 0.2500....is inherently rounded off to a finite value 0.25. The cal. does this also. Get yourself and try the experiment for youself BT.

All of the spaces do not fill in the cal. and there are not zeros after the 5. Your playing more mental masturbation games again.

Come back and talk to me, when you can offer us a rational, logical, common sense and relatively simple, explanatory guide to your alledged proof, or especially Origins two lines of formula--- what a joke his is ---or RP's complex references to "mapping" mathematics.

1.0 is labeled A in column XX, and we move it over to column S and relabled/reidentify A as G and that is the "proof" that 1.0 = 0.999..."yeah RIGHT!!!!!" says Bill Cosby in his Noah's voice to a mathematical God.

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God of Mathematical "mapping" > Simply NOT Simple----

1.0 is labeled A in column XX, and we move it over to column S and relabled/reidentify A as G and that is the "proof" that 1.0 = 0.999..."yeah RIGHT!!!!!" says Bill Cosby in his Noah's voice tomathematical "mapping" God.

RPenner:
.."mapping" God...

Chinglu:
..God of an alternate mathematical viewpoint....

Billy T:
..base 10..line segment...1/1 = 0.999....

Origin:
.....1/3 = 0.333333 1/9 = 0.999....yeah, RIGHT!!!!.....

Arfa-brane:
..ask Ronald Mac Donald about infinite value vs infinite decimal vs infinite number etc.....

Someguy:
..???....

Undefined:
..circuitous trivia is not a proof?...

QQuack:
..does balloon have 0.999....

R6:
...finite 1.0 NOT equal to infinite 0.999...

Ok, Trippy. I get you. Please read my post to arfa brane above for some more clarification as to where I am coming from and trying to get to. ...
I think he refers to post 1532, as I asked him for that link about 4 pages back and he gave 1532. His (originally Alpha Baine's) reference does indicate 1 = 0.999... is not well proven, but then continues to note that if that is not true, then there must be a number N such that 1> N > 0.999... and then remarks that "none has ever been found."
But I note, as is well known: "Absence of evidence for X is not proof of absence of X." where usually this is applied with X = God.

While I think I have proven* 1 = 0.999... (without use of the limit process or any other process one could question like addition or multiplication applied to infinitely long decimal strings) I do see some lack of rigor (or need to assume or by definition, define) that long convergent partial sums of ever smaller value are equal to their limit as number of terms becomes infinite. I have no problem with making such a definition. - I. e. defining the value of an infinitely long sum of progressively smaller terms which is limited as that limit. In fact I prefer that definition be done and accepted by all as then all questions about the validity of infinitely extended use of induction does not arise.

Likewise I have no problem with defining 0/0 = 1 and prefer to do that too as then the rule that a/a = 1 has no need for a footnote stating "except when a = 0. With the use of these (and other?) definitions, I think several proofs that 1 =0.999... have been given, but only my proof stands valid if one does not want to accept these definitions or the assumptions that processes known to be valid for finite strings, like induction, etc. remain valid even if applied an infinite number of times to infinite strings, which of course they can not actually be.

My proof starts by defining the line segment of unity length and names the ends 0 & 1. Then adds a copied of its self to make one twice as long with the new name "2" given to the end not still called 0. Based on this "joining" I defined 1 + 1 = 2 but to avoid people thinking I am doing conventional addition with the plus sign, it would have been less clear, but better to avoid that confusion if I had said: 1 & 1 = 2. Then I do more "joinings" of unit length line segments, one at a time, to arrive with meaning for 3, 4, 5, 6, 7, 8 and 9 well defined and the "& operation" also well defined except for problem discussed in next paragraph.

I then note that in base 10 symbols, there is no symbol for what 9 & 1 equals, so begin my discussion of the "base, place, and decimal point" notation system, and its properties, especially the effect on values of moving the decimal point n spaces to the right in a less than unity value decimal like 0.abcde etc. where these letters are chosen from the ten member set {0,1,2,3,4,5,6,7,8,9} etc. to make the proof without use of limits, induction, any multiplication or addition, and only indication (no algorithm) of division for rational integer, a & b, by the notation a/b. I do use a limited version of subtraction. I. e. make unit step value reduction from a starting point (not necessarily an integer) one after another if needed on the endless "number line" and require that y - y = 0 for all y, especially less than unity decimal fractions as that is the only use of y - y = 0.

All the subtractions I do have ONLY zero (y-y=0) or positive results which are values of the form (10^n -1) greater than the RD they are concerned with. I do assume without proof that doing same things to both sides of an equation stating equality, does not destroy that equality but results in a new valid equation stating equality between the two sides.

Not actually part of requirements of my proof of my derived methodology for finding the rational fraction equal in value decimal expression's proof, is the conventional "reduction of fractions." I.e. in one of my examples, I apply my procedure to 0.123123123123... and fine it is equal to 123/999 but reduce that result to 41/ 333.