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

Undefined said:
it becomes important to distinguish which is the fundamental aspect and which is the trivial and non-relevant aspect when the string and actions upon same are being discussed in this context.
I suppose.

But let's look a bit more closely at my list model: if you have a number like 0.999 and multiply it by 10[sup]-2[/sup], the result is 0.00999, the single zero on the left of the decimal point must really be like a string of zeros extending infinitely to the right (which is inductively true since we can multiply by 10[sup]-2[/sup] indefinitely). But this is true for any number, there is this infinite string of zeros on the right. We just write numbers like 000.9 as 0.9, because it's: "0.009 multplied by 100".

In short, I agree that for convenience aspect, no problem using the non-fundamental non-action of 'shift the decimal point' to convey the result.
You seem keen to pursue this idea that one view is fundamental, the other isn't. This is not the case. Are numbers more fundamental than operations on them?
the distinction to be drawn between 'shifting decimal point' and 'shifting string' really does matter substantively if the discussion is not to confuse vague convenience aspects with specific fundamentality
A well-defined operation on a set is not a vague convenience. You really want to cling to this "there's a problem here" thing, don't you?

Hi, arfa. I'm really getting rushed now, so please forgive the brevity of my responses...

I suppose.
Yes, that is because of the context/perspective the discussion is in regarding what 'proofs' are validly based on actual operations, and what are invalidly based on formatting convenience after-the-fact 'operation' has concluded and a 'result' needs to be 'formatted' according the whatever you wish to use to convey the meaning of that result.

But let's look a bit more closely at my list model: if you have a number like 0.999 and multiply it by 10[sup]-2[/sup], the result is 0.00999, the single zero on the left of the decimal point must really be like a string of zeros extending infinitely to the right (which is inductively true since we can multiply by 10 indefinitely). But this is true for any number, there is this infinite string of zeros on the right. We just write numbers like 000.9 as 0.9, because it's: "0.009 multplied by 100".

Is that "zeros extending infinitely to the right" a typo? Did you mean to write "left" (not "right") there?

In any case, what you describe is not at issue with me. I only wish to make clear that it is the operation/numbers that make the string up and make its various 'behaviors' and 'properties' etc 'change/move' etc. And not the formatting conventions used after the operation/function is determined according to those functions/numbers involved in reality. That was all.

You seem keen to pursue this idea that one view is fundamental, the other isn't. This is not the case. Are numbers more fundamental than operations on them?

A well-defined operation on a set is not a vague convenience. You really want to cling to this "there's a problem here" thing, don't you?
Operations involve numbers/entities that have some 'reality' of effect to them IRRESPECTIVE of formatting conventions after the event/action. The question/aspect of which is the more/real fundamental comes into it only (as I already explained) when some try to use the decimal formatting per se as the fundamental aspect, rather than the operation involving the numbers/entities in the first place.

That was it on that particular aspect, arfa. No more than that. It is self-explanatory, and not really any cause for further argument unless you want to give formatting conventions more reality/effectiveness (as an operation) rather than the actual number/string manipulations which lead to new resulting strings/numbers which must THEN AFTER be formatted in whatever system of display/convey same that you think fit?

Gotta go. Really! Cheers until we speak again, arfa, everyone! Thanks again for all the interesting contributions to the discourse, arfa, everyone!

Is that "zeros extending infinitely to the right" a typo? Did you mean to write "left" (not "right") there?

Nope, "to the right" is correct. Your guess is wrong.

Yes. Thanks. I was only clarifying where I was coming from. Any cross-purpose misunderstandings are just that. I am fast running out of time for internet discussion/posting. I will soon have time to read-only (if I ever get out of here today! ) Thanks again for your patient engagement, Trippy. Good luck in your future discourse. I will read with interest anything else you have to contribute to sciforums discussions here and elsewhere. Apologies for any misunderstandings on my part!

Hi Trippy.

You sure put a lot of assumptions in my mouth when making your strawmen which you then go on to build your counter-arguments/assumptions on.
I put no assumptions in to your mouth. If I appear to have misunderstood you, I would suggest that the most likely explanation is that I have misunderstood you because you have not been clear.

The point was that it is the numbers/entities and the operations that affect the strings; be the strings finite of infinite. IE, that it is the numbers involved that make the string 'shift' across the decimal point, and NOT 'shifting the decimal point' which changes a 'frozen string'.
My post said nothing about shifting the decimal point other than to make the point that shifts in the decimal point and the additions of zeroes were simply because multiplying by powers of ten in a base ten numbering system performs operations on the decimal powers. This means that zeros which were there all along, but we agree not to write unless they are significant suddenly become significant.

That was the essential point I tried to make to Pete when he and others used the "shift the decimal point" as part of their 'explanation/proof' arguments. Ok?
I understood that and addressed that directly. I still assert that it is a nonsense that leads to nonsensical and contradictory results. I have attempted to illustrate these inconsistencies on many occasions, however, you have so far point blank refused to address them. You have only thus far directed glib dismissals toward them.

And further on that 'invoking infinity' in the Limits treatments:

Did you read where I pointed out that the Limits formulation/concept approach essentially invokes the INFINITESIMAL 'last step of effectiveness' concept/point/value/number or whatever is involved in that particular 'infinity' used?
I invoked no limits in anything I have said. I simply pointed out that there was an infinite number of zeroes on either side of '1' that we agree not to write because they are not significant zeroes. Multiplying the '1' by a power of ten changes that.

Do you understand where the last step of any infinity as used in the Limits formulae is effectively INVOKING an FINAL INFINITESIMAL of that infinity as a REAL CONCEPT which allows us to REASONABLY conclude the LIMIT IS REACHABLE (and hence something like 0.999...=1 IS ok to conclude from such reasonable invocation of the infinity/infinitesimal as per the LIMITS formulae?
There is no last step.

Which number in the sequence 0.(142857), the decimal representation of $$\frac{1}{7}$$ is the last number?

That makes even more sense when you consider that 'infinitesimal of effectiveness' is NOT '0'-nothing, but '0'-SOMETHING which is the last effective...
I'm sure this means something to you, but to me it reads as unintelligible nonsense.

...(non-nothing 3-d energy-space extent....ie, 'points' are NOT really "non-dimensional, as per previous posts/explanations) REAL step between one boundary condition/domain and another adjacent....and hence repeating decimals like the ones whose 'limit' is reached ARE REAL POSSIBLE STATES where the last effectiveness step IS a-non-nothing '0'.
So according to you I can not trisect a ten degree angle, however. reality says otherwise.

Further consideration that in PHYSICAL energy-space terms reality there is NO such thing as NON-existence; only balanced states, or transitioning states between SOMETHING and SOMETHING ELSE etc.
Again, I'm quite sure this means something significant to, but, I am unable to parse it in any meaningful way.

Zero states do actually exist in reality, it's just a question of whether they're absolute or relative.

That is where my perspective is coming from. No more than that. All the usual tit-for-tats based on the old stock-standard arguments and cross-purpose exchanges don't really interest me enough to want to pursue down the old 'two sides that never communicate' emotionally charged and status quo 'frozen' attitudes to any new perspectives which are not being read closely enough to distinguish the novel/reasonable new points/subtleties from the usual old dead-end 'pre-conceived' stances/notions which do nothing to elucidate those new points/subtleties I am interested in.
The only person failing to communicate at this point is you. You're still reacting to what you think I said in my post, rather than directly addressing it.

I put no assumptions in to your mouth. If I appear to have misunderstood you, I would suggest that the most likely explanation is that I have misunderstood you because you have not been clear.

My post said nothing about shifting the decimal point other than to make the point that shifts in the decimal point and the additions of zeroes were simply because multiplying by powers of ten in a base ten numbering system performs operations on the decimal powers. This means that zeros which were there all along, but we agree not to write unless they are significant suddenly become significant.

I understood that and addressed that directly. I still assert that it is a nonsense that leads to nonsensical and contradictory results. I have attempted to illustrate these inconsistencies on many occasions, however, you have so far point blank refused to address them. You have only thus far directed glib dismissals toward them.

I invoked no limits in anything I have said. I simply pointed out that there was an infinite number of zeroes on either side of '1' that we agree not to write because they are not significant zeroes. Multiplying the '1' by a power of ten changes that.

There is no last step.

Which number in the sequence 0.(142857), the decimal representation of $$\frac{1}{7}$$ is the last number?

I'm sure this means something to you, but to me it reads as unintelligible nonsense.

So according to you I can not trisect a ten degree angle, however. reality says otherwise.

Again, I'm quite sure this means something significant to, but, I am unable to parse it in any meaningful way.

Zero states do actually exist in reality, it's just a question of whether they're absolute or relative.

The only person failing to communicate at this point is you. You're still reacting to what you think I said in my post, rather than directly addressing it.

Ok, Trippy. Sorry for whatever offense I may have given to you. My misunderstandings and my perspectives are already well covered, and I don't have time to go over them again. If my posts to date in context (here and in the other associated thread) have not clarified what my take is on the various aspects raised (and cross-purpose exchanges etc), then it's all my fault and I take all the blame for it. I have no more to say for now. Thanks again for your time and trouble, Trippy. Good luck and enjoy your discussions.

Undefined said:
Is that "zeros extending infinitely to the right" a typo? Did you mean to write "left" (not "right") there?
Yes, sorry, I should heve said they extend to the left, from the most significant digit.

But one should be careful about what happens to any string of digits, since zeros extend to the right of any finite decimal fraction too. Since the discussion is about repeating decimals, there are no zeros on the right of the string unless division by some power of 10 means they "shift" across the decimal point and become part of what I've been calling the "tail" of the list, in which case they "spontaneously" become a potentially infinite string of zeros on the right of the decimal point. However, there will always be a nozero digit on the right of this string of zeros.

I think it's easier to just consider the left or right of the decimal point itself and the multiplication or shifting function acting on it, and zeros treated specially.

Last edited:
Frog( 1 ) NOT = Dog( 0.9999...)--zero( 0 ) NOT = 0.111....

rr6....Yeah RP, this the key point--- called mapping ---where Column X with n becomes a in Column Y with a etc....again this at first sight and with minimal knowledge of what will appear to most humans, as complex branch mathematics, that, is changing X-n( frog ) into Y-a( dog ) i.e. to me all of this still leads to what is obvious to most humans who would view the original statements again;
You have failed to give us a relatively simple, rational logical and understandable to the average human, within this one email message with ingecti, surjectives etc, o whatever alien terms....thx again, but no banana for you RP.

rational finite( 1 ) does not equal irrational infinite 0.999 .....

non-counting( 0 ) does not equal irrational infinite 0.111....

Space = macro-micro infinite
..micro = >IN<--subdividing space INward--multiplication-by-division

0 = non-counting number

0 is representative of macro-micro infinite non-occupied space beyond our finite occupied space

0 = empty column on abacus.

0.111....does not equal zero( 0 )

0.999.....does not equal one( 1 )

Inifinity is irrational concept except in relationship to infinite space.

Space
..1a) non-occupied and macro-micro infinite, embraces;
..1b) occupied space...

Mathematics represent values associated with;

1) finite occupied space,

2) infinite non-occupied space,

3) imaginary concepts of mind, that may or may not directly correlate to either of the latter #2 and #1 above.

Rational does not equal irrational.

Irrational values are expressed using numerical symbols similar to rational values being expressed using numerical symbols.

The irrationally infinite, macro-micro infinite non-occuped space, that embraces our finite Universe of occupied space may be the only exception to our mathematical rules, as that is the only time we can correlate to an infinite value existence.

Who values the irrelevant non-occupied space( 0 ) that exists beyond our finite occupied space( 1 ) Universe?

The non-occupied space exists purely as a infinite reference set of zero representated value/non-point(s) having being of no immediate value to humans, other than as the most cosmic refersence set( infinite ) for our finite Universe.

Fuller believed in the possibility of micro-infinite subdivision of our finite occupied space Universe i.e. eternal increase of quanta via micro-infinite multiplicaiton-by-division of our finite Universe of occupied space.

I dont believe so. The graviton is the cosmic minimal limit. Does the graviton have variable limits? Didn't L. Smolin suggest such on the ultra-micro scale and on a ultra-macro cosmological scale of Universe that expands to varying radial values at differrent times of expansion-contraction?

r6

Yes, sorry, I should heve said they extend to the left, from the most significant digit.

You shouldn't have apologized, you were right , multiplication by $$10^{-2}$$ means division by $$10^2$$ so the zeroes extended indeed from the decimal point to the right (and not to the left, as Undefined confused you). You were right all along. He was wrong.

...
Rest assured, Undefined, your reports have not gone unnoticed. Neither have your posts.
Thank you! I was beginning to think that the report avenue was futile, since the trolls have continued so blatantly for so long without any remedy. Cheers!

Before I log out again I will take this opportunity to answer the following from Tach since it is on-topic...

arfa brane to Undefined said:
Yes, sorry, I should heve said they extend to the left, from the most significant digit.

You shouldn't have apologized, you were right , multiplication by $$10^{-2}$$ means division by $$10^2$$ so the zeroes extended indeed from the decimal point to the right (and not to the left, as Undefined confused you). You were right all along. He was wrong.

You missed his contextual "extending infinitely", Tach. Here...
...zeros extending infinitely to the right (which is inductively true since we can multiply by 10[sup]-2[/sup] indefinitely).

The only infinitely extending string of zeros that affects the value of the string starting from the first significant digit is the zeros extending infinitely to the left from the leading significant digit, as arfa just correctly clarified he meant to say. So Tach, your "corrections" and your "speaking for others" because of you again missing crucial context is neither here nor there in the context in which my conversation with arfa was being conducted. Thanks.

You missed his contextual "extending infinitely", Tach. Here...

I didn't miss anything, you were wrong and you persist in being wrong. And it has become very clear that you weren't asking any question, you were making the same incorrect statement you are repeating right now. If one multiplies with an ever increasing n as in $$10^{-n}$$ , this means inserting an ever increasing number of zeroes (n of them) to the right of the decimal point.

The only infinitely extending string of zeros that affects the value of the string starting from the first significant digit is the zeros extending infinitely to the left from the leading significant digit, as arfa just correctly clarified he meant to say.

Nope, to the right. Mate.

Let's have a go at making things a bit clearer.

Since 0 = (0)* is true on the left of the first nonzero digit which itself is to the left of the decimal point, and not true on the right of the decimal point, we have two things to worry about when dividing a repeating decimal by 10[sup]n[/sup], where n is not zero or negative.

But if the decimal has period one, so we don't have to worry about a repeating pattern of digits, then (d)* can represent the string's suffix. So shifting the decimal point to the left, ("into" (0)*), means you append n zeros to the right of the decimal point, and the suffix does change, but not its length. But the head of the list structure (all the digits to the left of the decimal point) is always (0)* for a repeating decimal unless you shift the decimal point to the right, equivalently multiply by 10s.

So "to the left" or "to the right" needs to be in reference to some part of the structure of the string of digits. Arguing about left or right of what probably isn't all that much help.
But don't let me stop you.

I didn't miss anything, you were wrong and you persist in being wrong. And it has become very clear that you weren't asking any question, you were making the same incorrect statement you are repeating right now.

Please stop making unfounded accusations and strawmen. I ASKED what he meant, not told him. He answered accordingly. You are being dishonest there. Please retract that strawman/accusation or be reported for dishonest 'framing' of others' words/intents. Thankyou.

If one multiplies with an ever increasing n as in $$10^{-n}$$ , this means inserting an ever increasing number of zeroes (n of them) to the right of the decimal point.

But unless the division DOES go on infinitely, there is a finite value for the string being divided, with the only meaningful zeros always falling in front of the leading significant digit of the string at all stages of division. So the ONLY meaningful infinite string of ZEROS is to the left. The zeros to the right are always 'notional', and never actually contribute to the value of the digit string at any stage. That is why I ASKED arfa whether he meant to imply INFINITE zero string to the left/right, and for clarification of his point in using that statement, so I could better read and understand what he was trying to get at.

Nope, to the right. Mate.
You again intruded your usual venomous tone into the polite conversation, without letting him and me finalize the clarifying exchange that would lead to my more proper reading of his point/explanation. Please stop this bad habit, Tach, and let people sort out questions of clarification between themselves, without putting your rude, arrogant and unwanted confounding oar in all the time. Thanks.

But unless the division DOES go on infinitely, there is a finite value for the string being divided, with the only meaningful zeros always falling in front of the leading significant digit of the string at all stages of division.

The meaningful zeroes are added to the right of the decimal point and even LESS "in front of the leading significant digit". There is no such thing as "meaningful zeroes in front of the decimal point" and even LESS "in front of the leading significant digit".

Let's have a go at making things a bit clearer.

Since 0 = (0)* is true on the left of the first nonzero digit which itself is to the left of the decimal point, and not true on the right of the decimal point, we have two things to worry about when dividing a repeating decimal by 10[sup]n[/sup], where n is not zero or negative.

But if the decimal has period one, so we don't have to worry about a repeating pattern of digits, then (d)* can represent the string's suffix. So shifting the decimal point to the left, ("into" (0)*), means you append n zeros to the right of the decimal point, and the suffix does change, but not its length. But the head of the list structure (all the digits to the left of the decimal point) is always (0)* for a repeating decimal unless you shift the decimal point to the right, equivalently multiply by 10s.

So "to the left" or "to the right" needs to be in reference to some part of the structure of the string of digits. Arguing about left or right of what probably isn't all that much help.

Thanks for that further clarification, arfa. Yes, that is why I asked if you meant to write right ot left, so that I could better read the thrust of what you were trying to explain by referencing 'infinite zeros' ( just wanted to get straight exactly which 'direction' you meant, so I could avoid misreading you).

Anyhow, I think I get what you are saying. I can only say in return what I said to Tach above. Unless the dividsion does go on and on infinitely, the significant value digit strings are finite. Hence any zeros on the right of that string are not 'in play' unless the divison operation 'terminates'......which it can't if you keep conceptually including all the 'null effect' remaining zeros to the right of the string.

That was all I was trying to say. Only the infinite zeros on the left of the string leading significant digit are 'there' at all times in the fractional string 'generation' process. All zeros on the right are 'assumed' and insignificant 'forever' immediately from the last significant digit in the fractional string being generated.

Ie, the leading zeros are 'real' and the trailing zeros are 'notional' UNLESS and UNTIL the fractional generation division operation goes to infinity and COMPLETES 'somehow'....in which case only THEN can some last/final 'infinitesimal of effectiveness 0' be said to 'exist' on the right of that string....such that that represents the final step into whatever 'limiting value' the operation reaches (if you divide by ten infinitely, does the ever-reducing value/string reach a 'limit' of 'final zero'?)

I have to go again, now. I just came in to check a PM and answer these two posts. Cheers and keep up the polite conversation, mate. I will read with interest even if I haven't time start/engage in any new discussions for a while. Thanks again for your polite contributions!

The meaningful zeroes are added to the right of the decimal point. There is no such thing as "meaningful zeroes in front of the decimal point".

Can't you read? I said "in front of the LEADING SIGNIFICANT DIGIT". Please STOP with your misunderstandings-based insinuations/framings. If you can't read the context, what is the good of you making posts and opinions at all? Please stop trolling your 'version', Tach, and let the people who are discussing IN CONTEXT clarify their mutual exchanges for themselves, without you ballsing up everything as usual until the discussion is beyond repair and the mods have to step in to tell you to quit it, as usual. Thanks.

Can't you read? I said "in front of the LEADING SIGNIFICANT DIGIT".

There is no "zeroes in front of the LEADING SIGNIFICANT DIGIT". This is why it is "THE LEADING SIGNIFICANT DIGIT". Fail. Again.

The meaningful zeroes are added to the right of the decimal point and even LESS "in front of the leading significant digit". There is no such thing as "meaningful zeroes in front of the decimal point" and even LESS "in front of the leading significant digit".

NOT for FRACTIONS. The value of the string begins with the position of the leading significant digit where it falls to the right of the decimal point in a decimal fraction string.

For a decimal FRACTIONAL string, the leading zeros/places have significance too, whereas the trailing zeros/places are 'null' and 'notional' until actually 'occupied' by a trailing significant digit of the value string.

Get it? It is the mirror image of what happens from the left side of the decimal point positions in strings of unit or greater value strings.

There is no "zeroes in front of the LEADING SIGNIFICANT DIGIT". This is why it is "THE LEADING SIGNIFICANT DIGIT". Fail. Again.

Please read my above post (and my previous posts) in context. I explained where you are mistaken. Now please stop.

Maybe we should also have:

0 = 0.0 = (0)*.(0)*.

0.d = 0.d0 = (0)*.d(0)*, where d is in {1,2,3,4,5,6,7,8,9}, or wlog is a finite string of them.

But 0.(d)* = (0)*.(d)* and I don't know if (0)*.(d)*(0)* makes sense, it isn't something I would class as constructible.

Maybe we should also have:

0 = 0.0 = (0)*.(0)*.

0.d = 0.d0 = (0)*.d(0)*, where d is in {1,2,3,4,5,6,7,8,9}, or wlog is a finite string of them.

But 0.(d)* = (0)*.(d)* and I don't know if (0)*.(d)*(0)* makes sense, it isn't something I would class as constructible.

But "we" don't. For good reason.