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

I now understand someguy1's objection to 0.999... - 0.0999... = 0.9 therefore 0.999... = 0.9/(1 - 1/10).

Leibniz (if I recall what I read yesterday afternoon) was once provided the sum S = 1 + 1/2 + 1/4 + 1/8 + 1/16 + ....
and calculated since 2S = 2 + 1 + 1/2 + 1/4 + 1/8 + ... then 2 S = 2 + S therefore S = 2. (While Aristotle concluded it could be no number other than 2.)

But the procedure does not work for all geometric series as if Z = 1 + 2 + 4 + 8 + 16 + ... and 2Z = 2 + 4 + 8 + 16 + 32 + ... = Z - 1 leads to the absurd conclusion that 2 Z = Z - 1 and therefore Z = -1.

Thus Leibniz's mistake was in assuming the infinite geometric sum existed as a real number before doing operations on it. However, since the infinite geometric sum is a number if the absolute value of the ratio between successive terms is less than 1, when this condition is met (as it is in repeating decimal fractions) then Leibniz's trick is justified.

Since we should agree that 0.999... is real number and 0.999... = 0.9 + 0.09 + 0.009 + 0.0009 + 0.00009 + ... by definition, I submit we have not fallen into the same trap as Leibniz's guess before formalization of the ideas of limits and supremum.

Chinglu's last objection seems to be that no infinite sum can exist at all because doing the additions one-at-a-time would be a never-ending process. I agree, but do not accept that one-at-a-time is the only way to do addition. Lots of operations in set theory work even for infinite sets. Since all the terms being added are all positive, the limit of the partial sums as the number of terms grows without limit is the same number as the supremum of the set of all partial sums; the infinite sum (if it exists as a real number) can be no other number than this value. Further the digits of a decimal fraction themselves as bounded below by 0 and bounded above by 9. Thus every decimal fraction of the form $$a_0.a_1a_2a_3...$$ is a number bounded below by $$a_0$$ and bounded above by $$a_0 + 1$$.

Chinglu's quote-mining of his source in [post=3142405]post #1625[/post] is misleading, in that the value of the infinite series is actually introduced on that page.
An infinite series is an expression like this:

S = 1 + 1/2 + 1/4 + 1/8 + ...
...
It is clear what the pattern is: the n-th partial sum is

$$S_n = 2 - 1/2^n$$
When n gets larger and larger, Sn gets closer and closer to the number 2. When a sequence $$S_n$$ gets closer and closer and closer to a given number S, we say that S is the limit of the Sn's and we write
lim( $$S_n$$ ) = S.
So when you have an infinite series (a sum over all the terms of an infinite sequence), the value of that sum is defined to be the limit of the partial sums if such a sum exists.

For geometric series that limit exists if the ratio has an absolute value less than 1. And for the series formed from a sequence formed by the pair-wise multiplication of a bounded sequence and a sequence that would lead to a converging series already, that new series is also going to converge. 0 to 9 are the bounds on a decimal sequence of digits and 1/10 is clearly the geometric ratio involved, so all decimal fractions of infinite length are in fact infinite sums that form real numbers.

We cannot sum an infinite processs, and this is common sense.

Archimedes was doing it over 2000 years ago. Newton did it well, and systematized the procedure. Weirstrass et al formalized the procedure and put it on a logically sound basis; and Cantor, Zermelo, and the other set theorists of the late 19th and early 20th centuries showed that these infinite sums could be logically derived from a few simple axioms of set theory.

If you deny this, then the burden is on you to invalidate 2000 years of mathematical progress.

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But the procedure does not work for all geometric series as if Z = 1 + 2 + 4 + 8 + 16 + ... and 2Z = 2 + 4 + 8 + 16 + 32 + ... = Z - 1 leads to the absurd conclusion that 2 Z = Z - 1 and therefore Z = -1.
And yet, this conclusion, which is absurd in the Real Numbers is completely justified in the 2-adic numbers which has a different notion of "closeness" and "limit" than then real numbers. In 2-adic numbers, one is fully justified in writing $${...1111111}_2 = -1$$.

It is then obvious that any 2-adic integer is the limit, for 2-adic distance, of its sequence of partial sums. Consequently, every 2-adic integer is the limit of a sequence of positive integers.

Moreover, it can be shown that every negative integer is a 2-adic integer and that every ratio a/b of integers a and b with b odd is a 2-adic integer.

Needless to say, the set of 2-adic integers is not an identifiable subset of the set of real numbers.

Hi origin. Reading through as usual, and just have time to comment on this from you to rr6 et al...
...the 3rd grade, where fractions were taught. So 1/3 + 1/3 + 1/3 does not equal 1?

...

IIRC, the 1/3 is merely noting a concept of some division function YET to be carried out? Any connection you make about its equivalence to 1 when writing down 1/3 + 1/3 + 1/3 is merely reflecting the assumption that when each of the three terms are actually 'done and dusted' you will have three 'equal part results' to add up in fact rather than just philosophical expectation of it being so in reality. The sticking point is that 1/3 does not actually mean anything unless you attach that philosophical 'overlay' to the notation. In itself, the notation implies a function/operation yet to be done, and not an actual number 'result' per se. Likewise, as far as I can judge by all the discussions so far, the .999... notation also denotes a function of sorts which is assumed to be 'already done' in philosophical terms, but not yet achieved in reality? That seems to be the cross-purpose misunderstandings which drive the two 'sides' in these discussions. The philosophical and the mathematical are all mixed in together until one actually comes up with a way of 'proving' the assumptions by routes which do not inherently depend on such a-priori setup 'hidden' philosophical 'definitions/assumptions' which start from the presumption that 1/3 is already 'done and dusted' and that .999... is 'already there as a definite number', and so on.

Can't stay and talk. Just wanted to remind/advise why it might be best not to be so dogmatic/insulting when depending on 1/3 for the basis of your 'retorts' to rr6 et al.

Back to read-only and log out! Cheers all.

0.333.. + 0.333... + 0.333...= infinite value 0.99.... ergo eternal process/procedur

Archimedes was doing it over 2000 years ago. Newton did it well, and systematized the procedure. Weirstrass et al formalized the procedure and put it on a logically sound basis; and Cantor, Zermelo, and the other set theorists of the late 19th and early 20th centuries showed that these infinite sums could be logically derived from a few simple axioms of set theory.If you deny this, then the burden is on you to invalidate 2000 years of mathematical progress.

SG, just because a flawed/invalid process, is repeated for 2000 year, s does not make it valid.

0.333.. + 0.333... + 0.333... = 0.999... as best as I can tell. Common since not repeated invalid, mathematical flaw

You and others need to answer my specific comments as stated ex see the wiki article I posted and Origin and RP both have posted stuff very similar to that.
And here I have to repeat again, that, in the "infinite series sequence" they last thing they add in( inlcuded ) toi their given formula/equations to left side of the equal sign is is the dots "..." and then to the right side of the equal sign the 3 dots "..." those dots ".." have mysterious vanished/excluded from the final formula.

Seems suspcious to me and I asked RP abput that and have yet to see him address my comments specificallyl, in those regards.

Next I have to see if MK has addressed my comments specifically in those regards in her last reply to me.

Origin has never address my comments specifically as directed at him, because he lacks even the simplest rigor to explain any thing posts.

0.333.. + 0.333... + 0.333... = infinite value 0.999... ergo and eternally existent processs/procedure. The only way to sum the parts of any infinite subdivision is;

1) STOP the process/procedure--- that is what a calculator does,

2) and then round to next higher number for practical purposes.

Origin is clueless on these common sense issues because his ego blocks his acknowledgement of truth.

r6

r6

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Undefined said:
In itself, the notation implies a function/operation yet to be done, and not an actual number 'result' per se.
Well, no; a + b = c implies several things.

If a,b and c are numbers then it says the c is the sum of a and b. We interpret this as meaning "if we don't have c, then add a and b together"; it does imply an operation, but not that the operation is required. Maybe you do know what c is, then a and b might have several solutions. The "result" is arbitrary, the notation only says the left side equals the right side "under" the operation.
So, it doesn't say you have to add a and b. Instead you might subtract b from c, for instance.
Likewise, as far as I can judge by all the discussions so far, the .999... notation also denotes a function of sorts which is assumed to be 'already done'
That's one way to think about it, or you could just say it's a number.
The philosophical and the mathematical are all mixed in together until one actually comes up with a way of 'proving' the assumptions by routes which do not inherently depend on such a-priori setup 'hidden' philosophical 'definitions/assumptions' which start from the presumption that 1/3 is already 'done and dusted' and that .999... is 'already there as a definite number', and so on.
But you have to start somewhere. You have to be able to represent one third, or some other fraction, in an unambiguous way.
Otherwise you get caught up in questions like: "where do numbers come from, do they really exist?", and so on, which although perhaps philosophically interesting, doesn't get far in a proof. You have to assume that they do exist, and that you can write them symbolically and manipulate the symbols according to well-defined rules.

Infinite 0.999..STOP Process = 0.9 > Round To Value 1-- Practical Purposes-Simple

Monimonika..I'm not talking about picturing what 0.999...Don't become a jerk like Undefined.

Huh? I have no idea what your talking about MK. Sorry but I dont have my original comments here so not sure where your coming from, but guess it is not addressing the real issues here that have clearly laid out in many posts in many replies.

Confusing as your post is, I am going with that second paragraph above to be your true thoughts about the matter of summing sequences (i.e. you think they don't work). I'm not talking about that, so drop this irrelevant tangent. My point is much simpler.

I rather doubt Ive ever stated any regarding your stated above "summing sequences". If you have a point that is relevant to the main issues I have pointed out and asked your camp to reply too, is what I'm looking for from you in this reply to me.

What the heck does "9 parts would sum to value 1" have anything to do with what's in my post? Stop adding irrelevant stuff.

I'm process new ways of understanding this. If you find it irrelevant skip. It is important to me. I

Starting with value one and subdividing into 9 or any number of parts is not irrelevant and certainly not inrelevant to the main point of this thread and that is in regards to and finite subivision of a value 1 whole. Whole line segment specifically is what had finally came to my mind, for my clarity and yours.

If you want to ignore micro-infinite subdivision of a value 1, line segment, fine. I'm still working through how to visualize this in my own mind as there seems to be two approachs;

1) start with a whole we label as 1,

2) attempting to sum infinite set tha is less than one and some mysterious way, would sum to value 1.

Now lets go and then lets me see if you actually address any of my other comments/questions specifically as stated.

Look at that part you quoted again. Where is there anything about getting closer to the value one (1)? You seem to have missed that there is no "1" being used there. You're confusing yourself with irrelevancies of your own creation. Focus on what I actually typed, okay?

Not irrelevant to my processing the info in mind to better understand and be able to express to others with greater clarity. If your find it irrelevant to you move on. II think your getting on your high horse for no good reason, and Ive yet to come across your addressing my comments/questions as specifically stated to you and many others here.

Where is the beef of your comments? I have found much relevant beef in your comments so far.

And finally I can get to my point (which you missed due to your irrelevant musings)! I did not mistype anything. I only just did exactly what you do when the left side of the inequality is "1" instead of "0.999...". In fact, I had presented it to you quite clearly:

I see I asked you questions but forgot the "?" sign, and you certainly did not address my comments / questions.

Again, most of what your posted is meangless and adds nothing to clarifying anything, that I can see ergo my questioning you as what is the point of your posting 0.999... > 0.999... is meaningless to me and I even stated you appear to have mstyped or whatever, tho it appears to be irrelevant comment by you.

this means that you made the leap from:
1.0 > 0.999(finite "n" of 9s)99

Huh? What is your point MK. We have already made this clear previously.

Yeah, 1.0 is greater than 0.999 and 1.0 is greater than 0.999....

to 1.0 > 0.999...

It is not a leap. A leap implys a process/procedure and you jumped all over my case above when I was processing a new of for me to envision what is happening i.e. micro-infinite subdivision of value 1 line segment. You need to get off you high horse and address my comments as specifically stated or move along.

Most of what you have posted appears irrelevant, not to the point of the issue where dealing with and certainly have not addressed my question regarding why theon that wiki page, in the"infinite series" part the dots "..." to left of "=" sign are mysteriously disregarded to the right of the same equal( = ) sign.

You yourself have added the dot( ... ) representation of infinite to the right side of the inequality when you used this argument about 0.999... never reaching the value of 1 as being relevant to 0.999... not equaling 1.

Huh? You appear to be having a mental breakdown for no apparrent reason MK. This thread places the dots "..." there. All of my posts are in regards to the 0.999... not 0.999. Your the one who through for some unknown, and confusing process, introduced 0.9 and all the other 0.999's. That may have what led me to initial subdivsion of a finite, value 1 line segment and then onto micro-infinite subdivision of line segment--- i.e. 0.999... ---you start jumping on my case about it for no apparrent good reason ??? Duhh?

My point was to show that a contradiction would occur, but you've shown me that you are a hypocrite who uses double-standards.

Huh? MK, you appear to be having a mental breakdown for no apparrent reason. Now read my lips/text MK, and if you want to address my comments as stated then please do so, without having a mental breakdown. Please.

If you want ifnite value 0.999... to equal finite value 1, I will repeat this again, for umpteenth time we will have to do two things;

1) STOP an eternally existent, micro-infinite, subdividing process/procedure i.e. 0.999... becomes 0.9,

2) round finite value 0.9 to finite value 1.0 for all practical purposes.

It is that is rational, logical, common sense and clearly explained unlike what wiki, you and others in your camp, have offerred. RP has made some attempts but he, like you--- and others here in same camp ---has not addressed my given comments above as stated, nor my questions regarding why the dots ".." are mysteriously not included in the final transition of the given formula/equation, in the "linfinite series" section of the wiki page.

All of you have been silent on that so far, to best of my knowledge.

Sorry you to see you have become so mentally bent out of shape. Reminds me of some of the behavioural slide of the some of trolls around Sci-forum, when they dont get their way.

r6

Mind/Intellect > Numbers > Value-1 Universe > value-1 Quantum bit value-1 > 5 vertice

arfa brane .."You have to be able to represent one third, or some other fraction, in an unambiguous way."

On cal. I divide 1 by 3 ergo 1 / 3 = 0.333 and the 0.333 limited to spaces of the cal.

So then wirthout punching 'C' without clearing out key, I times( * ) limited value 0.333 and it gives value 1. I think this is what Origin keeps going on about. Tho he has not so apparrently cannot explain what he keeps reposting. :bugeye:

So then, I tried this. I typed in 0.333 till all of cals. spaced were full, then typed + and repeated procedure twice more i.e. 0.333 spaces filled on my cal. sums( + ) to value 0.999{n} spaces fill on cal.

So how the cal. handles the calculation is differrent-- times gives 1 and addition gives 0.999{n} depending on how we do the procedure.

I asked many post(s) ago and a in differrent posts, why 9, 7, 6 and 3 give those filled spaces value-- inferred infinite value ---and why 10, 8, 5, 4 and 2 give the less spaces occupied finite resultant.

Nobody addressed those questions that I'm aware of plus some other more recent ones. Seems suspcious to me that some questions are not answered and that may be, because, there exists no answers for those specific questions ergo the smart ones avoid them for fear of stickin their foot in mouth. Smart! I dunno.

Otherwise you get caught up in questions like: "where do numbers come from, do they really exist?",

Mind/intellect ergo, our finite Universe of occupied space = value 1.

A minimal quantum info bit = value 1.

and so on, which although perhaps philosophically interesting, doesn't get far in a proof.

Can we prove that energy( physical ) cannot be created nor destroyed ergo can we prove that our finite occupied space Universe, was truly a creation of energy( physical )?

You have to assume that they do exist, and that you can write them symbolically and manipulate the symbols according to well-defined rules

My MS calculator does not seem to understand the "rules" as it multiplys the inferred infinite value 0.333 an gives a resultant finite value 1, however, the same calculator does not give same resultant, when adding inferred infinite value 0.333 three times.

Mind/Intellect makes the rules and those rules on occasion correlate to our finite occupied space Universe and its associated cosmic laws/principles/rules.

r6

rpenner said:
Leibniz (if I recall what I read yesterday afternoon) was once provided the sum S = 1 + 1/2 + 1/4 + 1/8 + 1/16 + ....
and calculated since 2S = 2 + 1 + 1/2 + 1/4 + 1/8 + ... then 2 S = 2 + S therefore S = 2. (While Aristotle concluded it could be no number other than 2.)

But the procedure does not work for all geometric series as if Z = 1 + 2 + 4 + 8 + 16 + ... and 2Z = 2 + 4 + 8 + 16 + 32 + ... = Z - 1 leads to the absurd conclusion that 2 Z = Z - 1 and therefore Z = -1.

Thus Leibniz's mistake was in assuming the infinite geometric sum existed as a real number before doing operations on it. However, since the infinite geometric sum is a number if the absolute value of the ratio between successive terms is less than 1, when this condition is met (as it is in repeating decimal fractions) then Leibniz's trick is justified.

Just to be explicit here: the important point in these kinds of sums is whether the infinite series converges or not.

The series S = 1 + 1/2 + 1/4 + 1/8 + ... converges.
The series S = 1 + 2 + 4 + 8 + ... doesn't.

There are a number of tests available to see if a given series converges or not.

I now understand someguy1's objection to 0.999... - 0.0999... = 0.9 therefore 0.999... = 0.9/(1 - 1/10).

My objection is that term-by-term multiplication by a constant of an infinite series is not justified until it is proven from first principles. It does happen to eventually be true; but by the time you've proved it, you've already used principles far more powerful than the mere fact that .999... = 1. So it's more of a heuristic or plausibility argument than it is a formal proof. However, as a formal proof, it's not wrong ... as long as you state "I invoke the theorem that says I can multiply a constant times a convergent infinite series term-by-term, and the result is the constant times the limit."

But if the person you're presenting the proof to were capable of understanding that, they would already be mathematically sophisticated enought to be able to produce their own proof that .999... = 1; and they would not be in need of a fake one.

You're right about the Leibniz example. The theorem says that you can multiply a convergent infinite series by a constant; and the resulting series also converges, to the value of the constant times the limit of the original series.

It's a premise of the theorem that the series has to be convergent in the first place.

Yes. Once it has been established S = 1 + 1/2 + 1/4 + 1/8 + ... converges in the domain of real numbers then then Leibniz's trick for determining the value of that convergence is justified and correct.

... There are a number of tests available to see if a given series converges or not.
Can you illustrated one applied to S = 9/10 + 9/100 +9/1000 ... ?
I'm not doubting, just wanting know of one.

Billy T,

Ok. First, we recognise that

$$S=\left(\frac{9}{10}+\frac{9}{100}+\frac{9}{1000} + \dots\right) = \sum_{n=1}^\infty \frac{9}{10^n} =\sum_{n=1}^\infty a_n$$

The ratio test says that an infinite series of the given form converges if

$$\lim_{n\rightarrow \infty}\left|\frac{a_{n+1}}{a_n}\right| < 1$$

In this case

$$\lim_{n\rightarrow \infty}\left|\frac{9}{10^{n+1}}/\frac{9}{10^n}\right| = \lim_{n\rightarrow \infty}\left|\frac{1}{10}\right| = \frac{1}{10}$$

so the given series converges according to the ratio test.

This test doesn't tell us what it converges to, but it does tell us it converges to a finite value.

Billy T,

Ok. First, we recognise that

$$S=\left(\frac{9}{10}+\frac{9}{100}+\frac{9}{1000} + \dots\right) = \sum_{n=1}^\infty \frac{9}{10^n} =\sum_{n=1}^\infty a_n$$

The ratio test says that an infinite series of the given form converges if

$$\lim_{n\rightarrow \infty}\left|\frac{a_{n+1}}{a_n}\right| < 1$$

In this case

$$\lim_{n\rightarrow \infty}\left|\frac{9}{10^{n+1}}/\frac{9}{10^n}\right| = \lim_{n\rightarrow \infty}\left|\frac{1}{10}\right| = \frac{1}{10}$$

so the given series converges according to the ratio test.

This test doesn't tell us what it converges to, but it does tell us it converges to a finite value.
Thanks, I was looking for something other than the |a| < 1 requirement of the geometric series.

Hi origin. Reading through as usual, and just have time to comment on this from you to rr6 et al...

IIRC, the 1/3 is merely noting a concept of some division function YET to be carried out? Any connection you make about its equivalence to 1 when writing down 1/3 + 1/3 + 1/3 is merely reflecting the assumption that when each of the three terms are actually 'done and dusted' you will have three 'equal part results' to add up in fact rather than just philosophical expectation of it being so in reality. The sticking point is that 1/3 does not actually mean anything unless you attach that philosophical 'overlay' to the notation. In itself, the notation implies a function/operation yet to be done, and not an actual number 'result' per se. Likewise, as far as I can judge by all the discussions so far, the .999... notation also denotes a function of sorts which is assumed to be 'already done' in philosophical terms, but not yet achieved in reality? That seems to be the cross-purpose misunderstandings which drive the two 'sides' in these discussions. The philosophical and the mathematical are all mixed in together until one actually comes up with a way of 'proving' the assumptions by routes which do not inherently depend on such a-priori setup 'hidden' philosophical 'definitions/assumptions' which start from the presumption that 1/3 is already 'done and dusted' and that .999... is 'already there as a definite number', and so on.

Can't stay and talk. Just wanted to remind/advise why it might be best not to be so dogmatic/insulting when depending on 1/3 for the basis of your 'retorts' to rr6 et al.

Back to read-only and log out! Cheers all.

Just read your post, mate! It is completely inane/nonsensical! I can't believe you cannot even handle fractions/addition/arithmetic! Got to go and work with people who have an education!!

Billy T,
See:

Two other tests that demonstrate 0.999... is a finite real number are:

Theorem 4.3.5 (Cauchy’s Convergence Criterion for Series) A series $$\sum a_n$$ converges if and only if for every $$\epsilon \gt 0$$ there is an integer N such that
$$| a_n + a_{n+1} + \dots + a_m | \, \lt \epsilon \quad \quad \quad \textrm{if} \; m \geq n \geq N$$​
This works for 0.999... as $$N(\epsilon) = \textrm{max} \left( 0, \; 1 - \textrm{floor} \left( \log_{\tiny 10} \epsilon \right) \right)$$ gives a function such that $$0 \leq \left| \sum_{k=n}^m \frac{9}{10^k} \right|\leq \sum_{k=n}^m \left| \frac{9}{10^k} \right| \leq \frac{10}{10^n} - \frac{1}{10^m} \lt \frac{10}{10^n} \leq \frac{1}{10^{N-1}} = 10^{\textrm{min} \left( 1, \; \textrm{floor} \left( \log_{\tiny 10} \epsilon \right) \right)} = \textrm{min} \left( 10,\; 10^{ \textrm{floor} \, \log_{\tiny 10} \epsilon} \right) \leq \epsilon$$ and so the series converges to a finite value.

Theorem 4.3.10 (The Integral Test) Let
$$c_n = f(n), \quad \quad \quad n \geq k$$​
where f is positive; nonincreasing; and locally integrable on $$\left[ k , \, \infty \right)$$: Then
$$\sum c_n \lt \infty$$​
if and only if
$$\int_k^{\infty} f(x) dx \; \lt \; \infty$$​

Since for $$F(x) = - \frac{9}{\ln 10} 10^{-x}$$ we have $$f(x) = \frac{d F}{d x} = 9 \times 10^{-x}$$ it follows that $$\int_k^{\infty} f(x) dx = \lim_{x\to\infty} F(x) - F(k) = - F(k) = \frac{9}{\ln 10} 10^{-k} \lt \infty$$ and so the series converges to a finite value.

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Again, I should have made clearer what I meant. Namely, to invoke the unitary states when 'setting up' an exercise/proof is what I meant. That is different to the unitary state being the 'result' of that exercise, as distinct to being the circuitous inevitability of the exercise which had the unitary state built-in and the rest of the manipulations merely followed self-selecting logics from that setup unitary state.

In short, I am looking for a proof that achieves the unitary equivalence as the 'result' and not the 'starting' condition as inbuilt and inevitable via circuitous routes 'back to' that initially invoked/inbuilt unitary state. I hope that is clearer as to the difference between starting/predetermining with the unitary state and ending with the unitary state via a route independent of any staring unitary state conditions inherent from the get-go?

Cheers.
Here's why your objection is absurd.

What if I define 1 based off 1/3?

What if I take a meter ruler and define it as being a standard unit, and 1 is three of these standard units. And 1/2 is 1.5 of them.

That's part of the problem here, your thinking is too linear and unicentric.

Just read your post, mate! It is completely inane/nonsensical! I can't believe you cannot even handle fractions/addition/arithmetic! Got to go and work with people who have an education!!

So, you have completed the mathematics and the physics theories? Good. Please show your work.

What's that? You didn't read properly so you missed the important subtle implications of the points being raised/discussed, and instead of making an intelligent remark you resort to one-liners and sarcasm? How very scientific of you.

If you can't contribute positively and fairly to the discourse in this thread, then maybe you had better leave the discussion to those who understand what's going on, origin.

If you really have any time and intellect to spare on subtle things like this, then re-read my posts, including my latest response to Trippy below regarding the proof requirement to be non-trivial and independent of 'convenient' self-selecting constructions based on unitary setup circuitous contrivances that prove nothing and elucidate nothing except what they were set up to 'prove' trivially. If after that you still don't understand, then good luck is all I can say to you. Cheers.

Here's why your objection is absurd.

What if I define 1 based off 1/3?

What if I take a meter ruler and define it as being a standard unit, and 1 is three of these standard units. And 1/2 is 1.5 of them.

That's part of the problem here, your thinking is too linear and unicentric.

Why do you again resort to the same kind of trivial 'constructions' I cautioned against many times already?

You now merely 'defining' things as you want them in order to construct a multi-unitary 'unitary' (ie, your example above of 3 ruler units to 'make up' 1 unit so you can of course 'trivially undo' back to 3 ruler units etc) is just the sort of circuitous in-built self-referencing logic and unitary setup 'definitions' I am asking we should try hard to avoid at all costs.

Starting from 'known' FINITE units/parts and then trivially 'dividing' them back to the same FINITE parts is not really doing anything except constructing and deconstructing FINITE UNITARY things.

They in no way treat FRACTIONAL cases per se. Such things are not what rigorous independent proofs are made of, as I already pointed out.

Thanks anyway for your consideration and trouble in responding, Trippy! Cheers.

PS: With specific reference to the point I made that the axioms as currently obtaining can only lead to "undefined" for the 0/0 construction? Can you explain precisely where/why you think the mathematicians have a 'choice' in the matter if they apply said axioms as is? Back again tomorrow if I can. Thanks.

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Why do you again resort to the same kind of trivial 'constructions' I cautioned against many times already?

You now merely 'defining' things as you want them in order to construct a multi-unitary 'unitary' (ie, your example above of 3 ruler units to 'make up' 1 unit so you can of course 'trivially undo' back to 3 ruler units etc) is just the sort of circuitous in-built self-referencing logic and unitary setup 'definitions' I am asking we should try hard to avoid at all costs.

Starting from 'known' FINITE units/parts and then trivially 'dividing' them back to the same FINITE parts is not really doing anything except constructing and deconstructing FINITE UNITARY things.

They in no way treat FRACTIONAL cases per se. Such things are not what rigorous independent proofs are made of, as I already pointed out.
I haven't divided anything, that's the point.

I start with something I will give a symbold, a. a is a number defined such that:
0 = { }
a = {0}
Then I say that:
1 = {a, a, a}
And that:
2 = {a, a, a, 1, 1, 1}
And that:
3 = {a, a, a, 1, 1, 1, 2, 2, 2}

There's no division involved.

PS: With specific reference to the point I made that the axioms as currently obtaining can only lead to "undefined" for the 0/0 construction? Can you explain precisely where/why you think the mathematicians have a 'choice' in the matter if they apply said axioms as is? Back again tomorrow if I can. Thanks.
Really? You still can't figure this one out?

Let me ask you a question that gets back to the root of what division is: If I have no pies, and I share them out equally among no people, how much pie does each person get? That's right, none! An answer which is in and of itself consistent with the rules of multiplication and division.

I think where it starts to ge a little messy (and I'm sure Rpenner will likely chip in with a few examples here) is when you start considering functions like 1/x where the limit as you approach zero is ±∞ rather than zero with x=0 representing a discontinuity. And so, they choose to leave it undefined rather than define it as zero.

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