# [1 paradox] Why 0.999... is not equal to 1?

Discussion in 'Physics & Math' started by transition, Mar 13, 2013.

1. ### transitionRegistered Member

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[1 paradox] Why 0.999... is not equal to 1?

Written in 2012

The current mathematic theory tells us, 1>0.9, 1>0.99, 1>0.999, ..., but at last it says 1=0.999..., a negation of itself (Proof 0.999... =1: 1/9=0.111..., 1/9x9=1, 0.111...x9=0.999..., so 1=0.999...). So it is totally a paradox, name it as 【1 paradox】. You see this is a mathematic problem at first, actually it is a philosophic problem. Then we can resolve it. Because math is a incomplete theory, only philosophy could be a complete one. The answer is that 0.999... is not equal to 1. Because of these reasons:

1. The infinite world and finite world.

We live in one world but made up of two parts: the infinite part and the finite part. But we develop our mathematic system based on the finite part, because we never entered into the infinite part. Your attention, God is in it.

0.999... is a number in the infinite world, but 1 is a number in the finite world. For example, 1 represents an apple. But then 0.999...? We don't know. That is to say, we can't use a number in the infinite world to plus a number in the finite world. For example, an apple plus an apple, we say it is 1+1=2, we get two apples, but if it is an apple plus a banana, we only can say we get two fruits. The key problem is we don't know what is 0.999..., we can get nothing. So we can't say 9+0.999...=9.999... or 10, etc.

We can use "infinite world" and "finite world" to resolve some of zeno's paradox, too.

2. lim0.999...=1, not 0.999...=1.

3.The indeterminate principle.

Because of the indeterminate principle, 1/9 is not equal to 0.111....

For example, cut an apple into nine equal parts, then every part of it is 1/9. But if you use different measure tools to measure the volume of every part, it is indeterminate. That is to say, you may find the volume could not exactly be 0.111..., but it would be 0.123, 0.1142, or 0.11425, etc.

Now we end a biggest mathematical crisis. But most important is this standpoint tells us, our world is only a sample from a sample space. When you realized this, and that the current probability theory is wrong, when you find the Meta-sample-space, you would be able to create a real AI-system. It will indicate that there must be one God-system in the system, which is the controller. Look our world, there must be one God, as for us, only some robots. Maybe we are in a God's game, WHO KNOWS?

(1)speedyshare.com/DQz9y/AiforSC.rar
(3)bayfiles.com/file/F5tD/B8M4Xh/AiforSC.rar

3. ### Prof.Laymantotally internally reflectedRegistered Senior Member

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Say you had 10 and divided it by 3. You would end up getting 3.333... going on forever. Then you multiply that number by 3, you would get 9.999... going on forever. By doing this operation you have broken mathmatical principals, the reason for this is because no matter what numbering system you use, all numbers in that system cannot be evenly divisable. So then you end up getting these numbers that run out into infinity. So if you started out with a 10 and then divided it by 3 and then multiplied it by three again, the answer should be 10.

It is like if you cut an apple into 3 peices, you put them back together you should get a whole apple. So then saying that 9.999... is equal to 10, then that would be the correct answer. If it was said to be equal to something less than 10 then you would have lost a peice of your apple from cutting it. So saying 9.999... is equal to 10 is just correcting for an error that comes up in division, that says that things could be divided out forever because they are not evenly divisable.

You can solve this problem by leaving numbers in fractions, so that way you never get answers that go on forever so then you know your final answer will be more accurate. This would be a good example of why that is a good idea.

5. ### rpennerFully WiredValued Senior Member

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0.999... = 1 is a non-intuitive result of how the Real numbers are defined. In software user interface design, any non-intuitive behavior can be flagged as a possible bug, but the bulk of software engineers (and all mathematicians) say the bug isn't in the code (or the theorems) but in the specification (or the axioms which define the real numbers). Before decimal notations, infinite continued fractions were used to define the reals, and they two suffered from "bugs" in that 1/(n+1) = 1/(n + 1/1), so that there were no unique representation guaranteed for any rational in that system either.

The real question is "Must we accept non-intuitive results in mathematics." and the answer is a qualified yes. If you want to use mainstream math and logic, then you need to work with the results of Dedekind and of Cantor, or you aren't even working with the real numbers. You may choose to be an intuitionist or a finitist, but these assumptions you make would forbid you from considering 0.999... to be a number. So in both cases it is untrue that 0.999... < 1 because in one case 0.999... = 1 and in the other case 0.99... isn't a number. You can't mix-and-match here.

The implied question, "Can my post disestablish a mathematical truth." must be answered in the negative, since as I have pointed out many times before, there is no such thing as an appeal to authority in mathematics. And nothing backs your post but the soi-disant authority of your own assumptions and definitions.

If you believe 0.999... and 1 are real numbers, then 1 - 0.999... is a real number. I say it is zero, the only real number that has no multiplicative inverse. The opposite position is that it is a non-zero, positive, real number which does have a definite multiplicative inverse. Then 0.999... = 1 - 1/x, where x is a real number. If x is a positive real number, then $\log_{10}(x)$ is a real number, and $N = 1 + \lceil \log_{10}(x) \rceil$ is a positive integer and $y = 10^{- 1 - \lceil \log_{10}(x) \rceil}$ is a real number and y > x, so 1/y < 1/x, so -1/y > -1/x, so 1 - 1/y > 1 - 1/x = 0.999.... Read that carefully!
If 0.999... is less than 1, then there is at least one number $1 - 10^{-N}$ which is bigger than 0.999.... But 0.999... is obviously bigger than any number of the form 0.9..9 with a finite number of nines. So we must also conclude that 0.999... > 1 - 1/y. By triality, this is a contradiction. 1 - 1/y > 0.9r > 1 - 1/y is total poppycock and nonsense! Therefore the assumption that 1 - 0.999... is not zero must be false and then 1 - 0.999... = 0 or 1 = 0.999..., by reductio ad absurdum.

Other such proofs, include using the theorem that there is at least one real number between any two different real numbers. If 0.999... < 1 then there is a real number that is between those to. Imagine writing out 0.999... and change any one digit (to the right of the decimal point). The result is ALWAYS smaller than 0.999..., never larger. Imagine writing out 1.0000.... and change any one digit (to the right of the decimal point). The result is always larger than 1.0. Therefore there is no such real number between 0.999... and 1.0 and by the aforementioned theory of denseness* (part of the definition of real numbers that makes them distinct from the rationals), and reductio ad absurdum, 0.999... and 1 must be the same number.

Another proof says between any two different real numbers, there is at least one rational number. Therefore if 0.999... < 1, then 0.999... < p/q < q/q = 1, which says 0.999... < $1 - \frac{q-p}{q}$ but there is no such number. If 0.999... < $1 - \frac{q-p}{q}$ then write out the decimal expansion of $1 - \frac{q-p}{q}$. Sooner or later you come to a point where you have to write a digit different than 9, and then you have proven $1 - \frac{q-p}{q}$ < 0.999..., which contradicts your assumption that 0.999... < 1. Thus 0.999... = 1.

Other people have shown that saying 0.999... < 1 is equivalent to saying that Achilles (who runs 10 times as fast as the tortoise) can never catch up if the tortoise is given even a small head start (One of Zeno's paradoxes) which is contrary to all intuition and experience. So 0.999... = 1 is intuitive in geometry and physics, it's just not a pretty statement to people who are hung up on symbols rather than the concepts behind the symbols.

So if you insist 0.999... < 1, let's take Hamming (see quote below) up on his challenge.

If 0.999.... specifies a number different than 1, how do you do math with that difference?

Does 9 + 0.999... = 10 × 0.999... ?
Does ( 0.999... )² = 1 − 2 ( 1 − 0.999... ) + ( 1 − 0.999...)² = ( 0.999... ) ?
Does 0.999... − 0.9 = (1/10) × 0.999... ?
How does 0.999... in base 10 compare to 0.111... in base 2?

This is not considered to be your blog, your vanity publisher or your peer-reviewed mathematics journal, so why would you think we would want to see your pre-composed post?

A paradox in mathematics is when single group of axioms lead to theorems that contradict each other. What you have here is a theorem (see below computer-checked proof) contradicting your intuition, which is at best a stumbling block for your progress in mathematics and not an apparent problem with the axioms.

Still not a problem with the axioms. It also looks like you are distorting the content of Gödel's incompleteness theorems which do not say what you think they say. Nor is there a demonstration that philosophy has any outright strengths above and beyond those of its subdicipline, mathematics.
Just because 0.999... is neverending doesn't imply it contains the totality of all things. It doesn't even have an 8, let alone God.

But mathematics has since 1871 been perfectly comfortable with the infinite world. Have you heard of Cantor?

Your argument applies equally well to 2 and -1. You can't have 2 identical apples, you can only have 1 apple and another 1 apple, which may be quite similar or not. The concept of 2 is a mathematical abstraction which relates well to commerce in the case 2 similar apples each retail for identical prices. You can't have -1 apples.

Apparently, since 2012 you never completed this thought.

A real sequence is a particular ordered collection of real numbers, like {1, 2, 3, 4, 5, .... } or { 1, 1/2, 1/4, 1/8, 1/16, ... } or { 1, 2, -3, 2, 1 } and may be finite or neverending, in which case we are talking about an infinite sequence.
A infinite sequence may have a limiting value, in which case for any positive margin of error, x, there is a point in the sequence N, such that all values past N are closer to the limiting value than x.
Example: The limiting value of { 1, 1/2, 1/4, 1/8, 1/16, ... } is zero, because if x = 1/16, N = 5. Indeed, we can write $N(x) = \sup \left{ 0, \lfloor - \log_2 x \rfloor \right}$

A sequence may be converted into a sequence of partial sums of all terms encounters up to that point.
Thus {1, 2, 3, 4, 5, .... } generates a sequence of partial sums: { 1, 3, 6, 10, 15, ... }
Thus { 1, 1/2, 1/4, 1/8, 1/16, ... } generates a sequence of partial sums: { 1, 3/2, 7/4, 15/8, 31/16, ... }
Thus { 1, 2, -3, 2, 1 } generates a sequence of partial sums: { 1, 3, 0, 2, 3 }, etc.
If a sequence generates a finite sequence of partial sums, the last value of the sequence of partial sums is the sum of all the elements of original sequence.
If an infinite sequence generates a sequence of partial sums that has a limiting value, that value is defined as the sum of all the elements of the original sequence.
Another name for the sum of all the elements of a sequence (when it exists) is a series.

If the sequence is {1,2,3,4,5} (a finite sequence) then the series is 1+2+3+4+5 or 15
If the sequence is {1,1/2,1/4,1/8,1/16,...} then the series is the limit as n goes to infinity of the sum of the first n elements of the sequence, or 2.

The infinite series is the limit of truncated series. Or if you like, the partial sums of the first infinite sequence generates another infinite sequence whose limiting value is the value of the series.

The infinite sequence { 9/10, 9/100, 9/1000, 9/10000, ... } has a series 9/10 + 9/100 + 9/1000 + 9/10000 ... which is naturally written in close analogy with finite decimals as 0.9999.... which is equal to 1. Not only is this basic to the arithmetic of real numbers (and their decimal representations) but is a famous point demonstrated on this other forum.

Demonstration with computer-checked math proof: http://us.metamath.org/mpegif/0.999....html

Since the definition of an infinite sum (series) already uses the concept of the limit of sequence of partial sums, 0.999... = 0.9 + 0.09 + 0.009 + 0.0009 + ... = $\sum_{k \in \mathbb{N}} \frac{9}{10^k}$ = $\lim_{n\to\infty} \left[ \sum_{k \in \left{ 1 \dots n \right} } \frac{9}{10^k} \right]$= 1. And since things represented by symbols to the left and right of an equals sign are the same thing, the symbols to the left and right are just two different names for the same thing, and thus 0.999... = 1 is a fundamental truth about the names given to real numbers.

ℕ is the set of (all) natural numbers (beginning at 1) and is an infinite set. Thus the notation Σk ∈ ℕ [ d_k ] is a sum over an infinite number of (ordered) terms which is precise and permissible in set theory and therefore many branches of mathematics. Nothing sloppy about it. To emphasize how not sloppy it is, I linked to a rigorous of the claim " Σk ∈ ℕ (9 / (10↑k)) = 1 " that has been mechanically verified all the way back to set theory and the postulates of logic.

I also have several college textbooks which precisely talk about infinite products and sums. Any similar level material on the topic of the real numbers (the basis of analysis) covers that 0.999... is just another name for 1. The concept of Dedekind cuts explains that any real number can have multiple names and that its the nature of the real numbers that there are no natural ways to exclude such double-namings.

You are free to disagree with the "mathematical authority" of textbooks, Wikipedia, math professors and people that genuinely know what they are talking about, but that disagreement comes with a price:

from American Math Monthly, vol 105 no 7

So if you disagree with 1 = 0.999... then you are not talking about the same real numbers as the rest of the world has been using since at least 1871. This means that you aren't allowed to manipulate them until you explain what rules you are using.

Likewise, you may deny that infinite sums (or even infinite sets) exist as a class of mathematical concepts, but your authoritative claims amount to nothing. Indeed, since the value of an infinite sum is the limit of sequence of partial sums (where such a limit exists) this sidetrack has been nothing but your baseless claim to the authority to make a pedantic notational quibble.

https://www.dpmms.cam.ac.uk/~wtg10/decimals.html A Cambridge mathematician writes about the real numbers

That's a problem with the precision of your cuts that contradicts your assumption that you cut the apple into nine equal parts. You have flip-flopped on your assumptions and run your ship of reasoning aground on a obstacle of your own creation. It doesn't contradict the abstraction that if you cut an object into nine equal pieces they would be nine equal pieces..

The division algorithm for dividing 1 by 9 goes like this : 1 is less than 9. Output 0. Shift the decimal place so 1 becomes 10. 10 is not less than 9 times 1. Output 1. Subtract 9 times 1 from 10 giving 1. Shift the decimal place so 1 becomes 10. 10 is not less than 9 times 1. Output 1. Subtract 9 times 1 from 10 giving 1. Shift the decimal place so 1 becomes 10. 10 is not less than 9 times 1. Output 1. Subtract 9 times 1 from 10 giving 1. Shift the decimal place so 1 becomes 10. ....

The output of 1's is never-ending. This is an infinite-loop and why 1/9 is equal to 0.111....

Giving up on man-made philosophical and mathematical problems and appealing to knowledge held only by God is no way to progress in philosophy, mathematics or discussion on an Internet forum.

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8. ### DinosaurRational SkepticValued Senior Member

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There was/is a Thread with many Posts relating to this issue. Many (most?) of the posts to that thread were poorly worded and/or clearly erroneous due to posters not having prerequisite formal education (or perhaps they forgot what they learned in some pertinent course or book relating to the subject).

In the better books relating to limits, the problem is not viewed using radix notation. It is viewed using the limit of a geometric series. There is good reason to avoid using any form of radix notation.

.FFFFFF . . . . is always greater than .999999 . . . . . & less than 1.0000000 . . . .

Using radix notation, you cannot prove that the above two series both have the same limit, which is one. In view of the above inequalities, the two series cannot have the same sum. Any valid proof to the contrary (using radix notation) must be viewed as a paradox.

Using geometric series you can prove that the two series have the same limit, which is surely a valid notion.

The correct method of dealing with the above deals with the sum of a general geometric series: a + ar + ar[sup]2[/sup] + ar[sup]3[/sup] + . . . . . . + ar[sup]n[/sup], where r < 1

For r = 1/10 & a = 9/10, the series represents decimal notation: .999999 . . .

I have read books which state that an infinite series is equal to the limit.

To the best of my knowledge, the better books dealing with limits use verbiage similar to the following:
The above dodges the issue of the series sum being equal to its limit.

Those books might make a statement similar to the following:
I do not remember reading such a statement when taking pertinent courses. I am certain that verbiage similar to the first of the above quotes appears in all good books dealing with limits.

9. ### DinosaurRational SkepticValued Senior Member

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BTW: While I was composing & carefully proof reading my Post, several Posts were made to this Thread.

The Post by R. Penner might be valid but is very verbose & digresses from the basic topic of the Thread. I found it time consuming to read & study well enough to verify that it was valid in all of its statements. I suspect that there are some dubious statements relating to the basic Thread topic.

10. ### Prof.Laymantotally internally reflectedRegistered Senior Member

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You could just ask yourself what is the limit of 0.999..., it gets increasingly closer to 1, so then the answer would be 1. Then $\lim_{x \to 0.999...} = 1$. There are no real numbers that are like 0.000...1, you couldn't have a number with an infinite number of zeros and then a one after it. An infinite number cannot be reached, so then you could never get to the end to put a 1 on the end of it. Since this number does not exist, then the difference between these numbers do not exist. You would never get a division problem where there is an infinite number of repetitions and then a different number on the end of it. No two numbers could be divided by each other in order to get 0.000...1. So the answer as to what real value the number is, is actually the limit, 1.

But then you could say, well 0.999... gets closer and closer to one and never actually reaches it. I think this idea is only a concept of mathmatics, and it doesn't actually apply to our reality. If you tried to cut something up an infinite number of times, would you never actually get to the point where you only make a cut that is the same decimal number over and over again? 0.999... would have to be left out of the infinite division process. But like I say I don't think it applies to reality, in reality they would end up cutting until they reached the Planck Length. After they decided to make a cut shorter than $10^{-34}$ cm , they would no longer be able to make a cut smaller than that without some source of infinite energy. They would try to cut, and then be like oh, we cut it smaller but then we can only measure $10^{-34}$ cm. So once it came that close there would be no way to measure if it actually reached it or not, it would be as close as it could get. It would then have to make a jump of $10^{-34}$ cm in order to even be able to measure that it had changed a distance, so then it would be there.

Look at Zeno, he kept walking around in a white robe, how did he do that when he had to walk over an infinite number of distances in order to take a step. He could take a step. So he did not cross an infinite number of locations. Walking is possible, even in contrary to mathmatics. You just have to take it at least $10^{-34}$ cm at a time, rounded up of course.

11. ### eramSciengineerValued Senior Member

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And Socrates says:

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13. ### Russ_WattersNot a Trump supporter...Valued Senior Member

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I don't think I've ever found it un-intuitive. The fact that there is no number between .999... and 1 seems pretty straightforward to me.

14. ### ash64449Registered Senior Member

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This proves that some infinity is equal to finite!!! 9.999.... continuing is equal to 10.. let me take like this- 9.9999.... is closer to 10. So when we go a little bit farther away,we will see that 9.9999 occupies the same position of 10!!!

15. ### KitemanSARegistered Senior Member

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It does = 1. Quit wasting our time.

16. ### GeoffPCaput gerat lupinumValued Senior Member

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Infinitesimal density functions. 0.999 is not equal to 1.

17. ### wellwisherBannedBanned

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Instead of treating math similar to the sacred documents of a religion, we need to realize math is man-made. Some will find logical problems with various assumptions, such as division by fractions violates energy conservation. However, others will circle the wagons to protect the dogmas even when there is logical inconsistency with laws of nature.

Math is a tool, like a cart, which needs theory or a horse to pull it. If you place the cart before the horse, or let math think for you, since it is manmade, it will funnel and perpetuate man-made bias until it looks self evident. Man-made may not lead to natural results.

As an analogy, a hammer is a tool, which needs a human to swing it. If the human leads, he may find other ways to use the hammer. We can use it for a lever, for a door stop, as a throwing weapon, etc. If we force the humans, to let this man-made tool lead, and define the hammer as a tool used to drive nails, all the other options are precluded. All the rest are taboo.

If we treat the tool (hammer) like a divine instrument, instead of man-made we may try to force only one way. But if we treat it as a tool, then there is always the need for additional R&D to make the hammer more versatile.

For example, statistics is a useful math tool. There is a push to let this sacred cart lead the horse; universe is restricted to chaos. One is not allowed to use this tool and other ways which do not lead to chaos. There is a way to do this. For example, if take a six-sided dice we can calculate the odds. If I load the dice with a weight, thereby change the tool for other uses, the odds change. With a heavy enough weight one side always comes up. If we add weight, to change the use of the dice tool, the universe loses chaos. This is taboo.

If we go back to 10/3 =3.333333 while 3.33333 X 3 =9.9999999, where did the extra go? Again there is a problem with mass and energy conservation since we lost something, albeit small. Man-made is about doing that which is not natural, which can include things like sky scrapers which can nevertheless be useful to humans.

18. ### Prof.Laymantotally internally reflectedRegistered Senior Member

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Thank you, I actually came up with it once waiting for Diablo 3 to come out. Everyone on the Diablo 3 forums, went on and on about it for pages. I finally felt like I actually got somewhere with this one.

It is as simple as saying that $\frac {a}{b} = c$ then $a = bc$. Then if you have a division problem that creates an infinite series then "b" times "c" does not equal "a". So I think it is really better advice than we think when they say it is better to keep values in fractions, because if you divide them out then it could no longer be valid with these equations. It wouldn't just be a rounding issue.

I think the real problem here is just long division. Say you had 10 apples, you could ever prove that if you cut them forever into smaller and smaller peices, that you would never end up in a situation where you could only cut it one decimal place smaller? What other numbers would be left out of the infinite division of the number line? An infinite number of them? An infinite number of them for each number? An infinite number of them for each series of repeating numbers? How could the number line still be infinitely divisiable after taking away infinities upon infinities of numbers away from the total amount of numbers that you could infinitely divide into?

To me it really seems like long division that creates an infinite series would be wrong by an infinitely small amount. If you left the problem in fractions, you would actually just get 10.

19. ### AlphaNumericFully ionizedRegistered Senior Member

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Except that fractions cannot completely describe numbers in a sufficient way. 0.9r is defined as a limit of a series, in this case a limit of rational numbers (rationals being numbers expressible as a fraction) and it is itself rational since 1 is rational. However it is possible to construct series of rational numbers whose limit is not rational, just consider the decimal expansion of an irrational number such as $\pi$ being the limit of 3, 3.1, 3.14, 3.141, 3.1415, .... . This is why we cannot restrict ourselves to rationals when considering concepts with limits, such as calculus, the rationals are not big enough.

20. ### originHeading towards oblivionValued Senior Member

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Math is not religion because math is logical and religion is not. Religion is man-made too.

Huh, what extra?

21. ### AlphaNumericFully ionizedRegistered Senior Member

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Mathematics is not about reality, that is how physics makes use of mathematics. Mathematics is a conceptual framework built from logic. Axioms are asserted to be true and then mathematics is about deducing the implications of those axioms. It's about saying "Statements A, B and C are taken to be true. D follows from them.". Whether or know A,B,C,D have any kind of physical manifestation is irrelevant, all that matters is whether D indeed follows from A,B,C. For example consider A : All men are Italian. B : Fred is a man. Therefore C : Fred is Italian. C indeed follows from A and B, mathematically this is unquestionable. Obviously not men are Italian but that is a statement about whether or not the conceptual axioms are physically valid. C follows from A and B regardless of whether A and B are true in reality. A and B being true in reality would mean C is true in reality but that is a separate thing. Obviously in this example I used notions associated to reality, like Italian and male, but it was for illustration purposes only. In the case of 0.9r we have mathematical conceptual definitions for things which are used to construct the concept of 0.9r. It can then be shown from the same conceptual definitions that the concept known as 1 is equal (which is also conceptually defined in mathematics) to the concept 0.9r, within that framework. Saying things like "But you cannot infinitely divide objects into smaller and small bits, you reach atoms!" or the like is irrelevant since such complaints are grounded in physical reality, not conceptual frameworks. Given the axioms which lead to 0.9r then 0.9r=1, that is logically certain. Saying "Oh but maths isn't sacred" ignores that we define the rules of the game. Now you could say "Okay, I disagree with the axioms used!". Fine but then you cannot construct the same concepts of 0.9r and 1. Sure, you can construct new versions of them within whatever axiomatically defined conceptual construct you wish to consider but within the axioms used to construct the Reals in usual arithmetic 0.9r=1

Too often laypersons think there is only one mathematical framework and even worse they think it is somehow contingent upon reality. Both points of view are wrong. The latter I just explained. For the former it depends upon the axioms you wish to use, people can and do consider different axiomatically defined mathematical systems where each system is in contradiction to the others (but the axioms within a particular system are not contradictory to one another). For example, the usual axiomatic construction of arithmetic is done using the ZF axioms. People then build up the implications of these axioms, showing things like they can lead to 1+1=2 or 0*1 = 0, that sort of thing. But sooner or later the logic leads to a statement which can neither be proven true nor false. Worse, it doesn't matter which one you decide, either decision doesn't contradict all the results you've already deduced. These are 'undecidable statements' and every axiomatic system powerful enough to do arithmetic has them (see Godel). One such question pertains to the Axiom of Choice. Now many mathematicians don't like the axiom, preferring to avoid having to assume it as an axiom (ie a proof of some statement which uses the AoC is considered less elegant than one which doesn't have to use it, which in turn is less elegant than a proof which doesn't need to even consider the axiom at all!) but someone who says "The AoC is true!" is as consistent with ZF constructs as someone who says "The AoC is false!". The resultant mathematical deductions in both of those points of view, the former being known as ZFC theory, are as valid as one another but are not simultaneously valid.

This is different from physics where we object an external objective reality we are trying to describe. It is possible to construct a multitude of mathematically sound models for physical phenomena but only one will be right, the rest will be in error. Doesn't mean they are mathematically wrong but instead means the physical phenomena we associate to the mathematical objects in the construct do not behave in a way described by the mathematical relationships within said construct. To give a less convoluted example the Galilean transforms of Newtonian mechanics didn't stop being mathematically sound the day we discovered Lorentz transforms of Special Relativity are physically more accurate.

But the mathematical procedure of division doesn't become mathematically invalid just because the use of that mathematical concept in modelling reality doesn't lead to a viable physical description of reality. In most physical models energy conservation follows from a time translation invariance in a Hamiltionian, ie $H(t) = H(t+T)$ for all t,T. Given some H it is simple to work out whether it possesses such a property by crunching some algebra. If it does then it means that it might be able to describe some physical system, since if $H(t) \neq H(t+T)$ for all t,T then it means it doesn't include the structure we associate to energy conservation. Suppose someone indeed says "I claim H describes this system", works out some predictions, does the relevant experiments and finds the predictions are wrong. What does that mean? Does it mean the maths is wrong? No, if H(t) = H(t+T) then H(t) = H(t+T)! What it means is that the conceptual mathematical structures which follow from H(t) do not correctly capture the physical structures observed in the relevant physical phenomena.

0.9r=1 is true as a mathematical construct. There is no 'circling of the wagons' when people defend that result, they are standing by a perfectly valid logical implication deduced from a set of axioms. Does 0.9r=1 follow from the axioms used to define arithmetic, ie the ZF axioms? Yes. Arguing against that is like arguing against (-1)*(-1) = +1, they follow irrefutably and unavoidably from the axioms used to define them. The only thing you can do if you don't like it is to go back to the axioms, change them and then see what the implications are. But even if those new axioms lead to a concept of 0.9r and a concept of 1 and you find those concepts do not satisfy 0.9r=1 that does nothing to stop the 0.9r=1 statement from the other axiomatic system being valid.

For example, in arithmetic we all know that a*b = b*a, ie 3*5 = 15 = 5*3. Fine. But suppose someone said "I don't want a*b = b*a, I want them to be different!". This is exactly what is done in quantum mechanics, operators do not satisfy a*b = b*a. However that doesn't change the fact that for usual numbers a*b = b*a.

No one with any grasp of logic and the principles of mathematics and physics would ever claim mathematics leads to physically valid conclusions as a matter of course. Rather they would say that if we state something true about a physical system, associate that to a set of appropriate mathematical axioms, deduce the implications of those axioms (as much as we can, there's infinitely many things which follow from axioms of sufficient complexity) and then convert those deductions back into a statement about physical reality then we should obtain a statement about reality which is physically viable. This is an extremely convoluted way of saying "If Nature behaves logically and we can formalise that logical behaviour then we can predict what Nature will do".

The mathematics isn't doing the thinking for a physicist. A physicist should be thinking about how to formalise the structures seen in Nature. Mathematics is then a way of converting those statements into new statements. That conversion is valid regardless of whether or not the statements themselves reflect reality. Mathematicians construct many many more mathematical concepts and formalisms than will be seen in Nature. It is the job of a physicist to link these two kinds of structure, mathematically conceptual and physically manifested, in as precise a manner as possible. When someone says "You're being driven by the maths!" what they really mean is "You have assumed that the mathematical axioms you began with are the correct formalisation of the structures seen in Nature.". A mathematician will say "I don't care, the maths is valid regardless of its physical manifestation or not, I can carry on" while a physicist will say "Okay, so doing more maths on this system will not get me to a physically sound conclusion. Time to find a new mathematical structure to associate to this physical system".

0.9r=1 is purely mathematical in nature (nature, not Nature), you cannot be lead by the mathematics to a false conclusion, only one which is not physically valid. After all, what does 'false conclusion' actually mean? Mathematically it means a conclusion which doesn't follow from the axioms, which is too often conflated with the layperson meaning, which is "physically valid". Hence why your comment might be applicable in some places within physics within the bounds of mathematical conceptual formalism the 0.9r=1 statement constructed from standard arithmetic is true. No ifs, no buts, no maybes, nothing, it is true.

I wouldn't agree with that. It is widely used and standard approaches for understanding chaotic or just complex systems are insufficient but statistics leading our understanding of them? Not something any self respecting scientist should be saying.

What on Earth are you talking about?

Seriously? 10/3 is not 3.33333, that is an approximation to the actual value of 10/3. You posted only 5 decimal places, ignoring the rest of them. This is a mistake children make all too often when they think their calculators are somehow perfect. That is having the maths lead you without understanding as you've (to use your example) used a hammer on a screw. Calculators and any physical process in general are not going to be able to properly manifest many mathematical structures, as the concepts are idealised, beyond the ability for reality to manifest physically. A computer cannot do infinite precision calculations, hence the whole double precision thing in programming languages and hence why 3*(10/3) may not be equal to 10 on a computer, the computer has finite precision error. Dealing with such imperfections and the resultant implications for computational models is a serious area of research known as numerical analysis.

Someone one another forum, during a discussion about 0.9r=1, said "But you'll run out of 9s! What happens when you've used up all the 9s in the world!". This was a point of hilarity for myself and a few friends. Do you see kids putting their hands up in the middle of a maths test and say "I've run out of 7s, can I have some more?"? The person saying that failed to realise the conceptual nature of the topic at hand, tying mathematics too closely to physics. Mathematics is a tool for physics but it is so much more as well, it is a study of logical structures. Infinitely many viable but pairwise contradictory mathematical formalisms and structures can be made. Somewhere in all of that are structures which mirror structures seen in Nature. Mathematicians explore this mathematical universe while physicists make the conceptual link from it to the structures seen in Nature. When someone 'circles the wagons' about mathematical physics it is not to defend the mathematical structures but the links made between maths and Nature, which we call Physics. Severing a link doesn't negate the mathematical structure.

Where does energy and mass come into 10/3 = 3.33333? Energy and mass are labels we give to particular physical structures. What is the energy of 5? Does -3.54 have mass? Of course not, they are abstract concepts. By adding in more abstract concepts we can start combining them and building up more structures ie there is a difference between the set of all numbers and the field of all numbers, as the latter includes rules for combining and working on the former.

In physics mass and energy come up in regards to the Hamiltonian, the H thing mentioned earlier. If H(t) = H(t+T) then we say H could represent a physical system whose energy is conserved. If we one day find out energy isn't always conserved then H(t) = H(t+T) remains mathematically true but we cannot use the structures within it to describe the relevant physical phenomena. Instead we'd find some new mathematical concept, K, whose properties are such that the K(t) - K(t+T) mathematical structure better models the physical structure in Nature observed in regards to energy not being conserved.

What do skyscrapers have to do with 0.9r=1 and mathematical physics?