I should note that when I learned about rational & irrational numbers in elementary school (I think third or fourth grade), we used a "bar" notation where we'd put a bar over the last digits in a decimal expression that repeated forever (i.e. it corresponded exactly to a geometric series with r = (1 / 10)^k where k is the number of digits under the bar, though we didn't know about that at the time). Our teachers explained that the difference between a rational and irrational number was that there would be no pattern you could ever find in an irrational number that would allow us to use the bar, which is surprisingly accurate for grade school arithmetic.
I've never considered them right at all. By saying something like
0.9... x 10 = 9.9...
and then saying that
9.9... - 0.9... = 9
you're basically just a priori defining 0.9... to be 1. In other words you're basically just defining 0.9... as a symbol to be some number x which has the property that 10x - x = 9. So you're basically just defining it to be 1.
I've never seen a proof of 0.9... = 1 using Peano arithmetic which made any sense to me. I doubt one actually exists in any true logical meaning. Unless you're making use of limits, completeness, or something equivalent I don't see how a proof could possibly make any sense.
Sure you can provide a hand-wavy argument and try to give some intuition if you'd like. That doesn't make it any sort of logical proof though. I guess it depends on what you're after.
You can redefine the "0.9..." symbol to mean something else as much as you want, you can have it meaning pi if you like, but then you are just changing the subject on the most unhelpful way.
> You can redefine the "0.9..." symbol to mean something else as much as you want, you can have it meaning pi if you like, but then you are just changing the subject on the most unhelpful way.
I _know_ that.
I think I maybe should just bow out of this conversation. I'm apparently incapable of explaining myself in a way that is understandable to people here. I'll consider this my fault.
I'll just summarize: I don't think any "proof" that 0.9... = 1 that is only expressed in terms of arithmetic operations and does not make use of limits is legitimate. In other words I claim that a proof like "0.9... = x" means "9.9... = 10x" means "9 = 9x" is illegitimate. Instead of taking "0.9... = 1" on faith it takes "10 x 0.9... = 9.9..." and "9.9... - 0.9... = 1.0... = 1" on faith. There's no proof here. It's just shifting around symbols. Of course there are logical proofs, but they make use of limits/completeness/properties of real numbers explicitly.
Feel free to disagree...
Well, I do disagree, not with the above statement, but the meaning of "0.9..." itself requires limits, so the discussion can never go anywhere if your assumptions do not include limits.
It’s worth remembering that most people who understand and agree rarely leave a reply.
I think the point is not defining 0.9... to be 1, the point is that “...” means an infinite number of 9s. If you shift the decimal point by 1, then nothing changes, there are still an infinite number of 9s. If you shift the decimal point by 5 places, there are still an infinite number of 9s to the right. And here is the logical (induction) step: if you shift the decimal point by an infinite number of places, then there are still an infinite number of 9s to the right. This works for any repeating fraction, in groups of more than 1 repeating digit.
> I’ve never considered them right at all.
Do you mean you disagree with the result, or that you agree with the result but don’t believe the proof is really a proof?
I've heard this argument many times. I understand the intuitive reasoning. I just don't find it a proof. I mean with reasoning like this why can't you have an infinite amount of 9s and then just put a 7 after that? What's keeping you from doing that? It's just a hand-wavy argument with no rules of any kind of what are allowed.
> Do you mean you disagree with the result, or that you agree with the result but don’t believe the proof is really a proof?
I've never considered that specific "proof" a proof. When 0.9... is given a proper definition of limits and considered within the real numbers, then sure of course it's true and the proof is legitimate.
You can justify the idea by defining a decimal representation of a number x as a vector x_2, x_1, x_0, x_{-1}, x_{-2} ..., with x_n ∈ {0, 1, ..., 9}. Negative indexes are digits after the comma, positive indexes before the comma. You can recover the original number simply using
x = \sum_{n=-∞}^{n=∞} 10^n x_{n} (1)
(this sum always converges as long as x_n = 0 when n > N, for some big enough N ∈ ℕ).
For example, the number three is represented by x_0 = 1, x_n = 0 otherwise. 0.9... is defined as x_n = 0 for n >= 0, 9 for n < 0. Now, by using limits, we could recover the original number using the formula above. However, we can do it without them. For that, we define just enough operations for the proof.
1. If z = x - y and for all x_n, y_n we have that x_n >= y_n, then z_n = x_n - y_n for all n. 2. If z = 10x, then z_n = x_{n-1}.
For the first operation, in order to be rigorous, we need to ensure that if z_n = x_n - y_n in the same conditions, then z = x - y. The proof of this part just consists of plugging the recovery formula (1): z = \sum 10^n z_n = \sum 10^n (x_n - y_n) = (\sum 10^n x_n) - (\sum 10^n y_n) = x - y. We can perform all those operations as we are guaranteed that (1) always converges.
Now, let y = 0.9... defined as in the example above (y_n = 0 for n >= 0, 9 otherwise), and let x = 10y (therefore x_n = 0 for n > 0, 9 otherwise). Now, define z_n = x_n - y_n, so that z_n = 9 for n = 1, 0 otherwise, which yields z = 9. As we demonstrated above, this implies that z = x - y, therefore 9 = 10y - y => y = 1, so 1 = 0.9... .
PS: I don't think one can make a proof without at least using some bits of limits to be able to switch between decimal representation as a vector and the number itself. However I don't think it's a problem, because you need the same bits to be able to talk of "0.9..." as a well-defined number.
Another proof by contradiction I've heard of this is that if 0.9... != 1.0... then there exists a number in between the 2. What is it?
You can. The proof still works if you do that.
What you’re refusing to accept here is the definition of infinity.
> What you’re refusing to accept here is the definition of infinity.
I'm refusing to accept the definition of infinity? I have no idea what you mean by that. Would you make the same statement were you aware that I do in fact have a PhD in mathematics in the field of analysis? That I have in fact studied logic? Just as a hypothetical scenario.
I haven’t heard a reason yet why the logic of the proof doesn’t work, I’ve only heard that you don’t accept it. What is the reason it doesn’t work?
I wasn't appealing to authority. I never said that implied that I was correct. (Had I wanted to do that, I would have brought that up much earlier in this comment thread.) I was really just curious if you still believed that I was "failing to accept the definition of infinity" even if you knew that about me. Apparently you do.
> I haven’t heard a reason yet why the logic of the proof doesn’t work, I’ve only heard that you don’t accept it. What is the reason it doesn’t work?
Assuming that all my suspicions in that comment are correct and these proofs actually are invalid proofs (not the results which are true), then the question might become: does it matter if the proof of a fact is incorrect if the fact itself is correct? That is a philosophical question and I'm honestly not sure how I'd answer it...
> does it matter if the proof of a fact is incorrect if the fact itself is correct?
Your language isn't allowing for a notion of precision, or for multiple forms of correctness, and it's not considering audience, communication or level of expertise either. I don't think it's a question of correct vs incorrect, I think you're asking for more precision, and/or for a form that meets your own higher standard.
It does matter if a proof is wrong, if there is a step in the proof that can be shown to be false. But that's not the case here, what you want is additional definition.
BTW, reading the blog post you linked to on surreals, the "proof" looks to me to be more hand-wavy than Euler's proof that .9bar = 1. The proof begins by stating there are a finite number of 9s, in direct contradiction to the hypothesis. 10^-inf = 0, so from this blog post I don't yet see any reason why surreals clarify anything here, it feels like the opposite, it feels like obfuscation.
This could be an argument over representation and not the values of numbers. If you start by defining 0.9bar to be a different number than 1 for the specific reason that it's written down a different way, then fine. That's what the surreal "proof" tells me. Euler's proof is talking about the value of 0.9bar in the limit, not the representation. (Even if that's stated without rigorous definitions of limits.) The proof is saying the values of .9bar and 1 are the same. If the surreal number .9bar were strictly less than 1, that must mean there's another surreal number closer to 1, but there isn't, so I don't accept the surreal argument as valid logic, other than playing a semantic trick by saying 'look I defined they way we write a number to be meaningful, therefore 0.999... is by definition different than 1.'
By the way, that blog post claims "The set of real numbers contains no infinitesimals." Wikipedia claims: "the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers..."
> I don't necessarily understand your use of the word "invalid", when what it seems you mean is incomplete and/or too informal for your taste.
> [...]
> It does matter if a proof is wrong, if there is a step in the proof that can be shown to be false. But that's not the case here, what you want is additional definition.
I'm going to try to be more formal, but not entirely formal since (1) the details are almost never-ending and require a lot of formal logic and (2) I'm not 100% sure about the reasoning myself.
What I mean by a "proof" is a sequence of logical steps that start with axioms of your logical system. We are obviously not looking at things that formally, but I actually think it is still an important point. The rational numbers can be thought of as being defined by certain axioms of arithmetic. E.g. you have the natural numbers as well as the minimal extra values so that you can add, subtract, divide, multiply, etc.--in other words you have a field. Let's call these the arithmetic axioms. Then when you go to the real numbers you basically extend the rational numbers in such a way so that you have completeness and retain all the previous properties. So basically for real numbers you have the prior arithmetic axioms and you have the completeness axiom.
Next comes the question of what is actually to be proved. For that we _must_ make some sort of definition of what we mean by "0.9...". Let us define it as the limit of the sequence of partial sums (all of which are rational numbers) _if_ it exists. So to prove "0.9... = 1" means to prove that the limit exists and equals 1.
So lets say we believe we have a proof that the limit equals 1 using an argument like this:
0.9... = x => 9.9... = 10x => 9 = 9x => 1 = x => 0.9... = 1
This is not a formal symbolic proof, but there is one thing we can see immediately: this proof does not make use of the completeness axiom. Therefore it should logically be the result of a sequence of logical steps starting from what I earlier referred to as the arithmetic axioms. Now here is the point where the surreals come in. The point with the surreals is that they contain rational numbers and the arithmetic axioms still apply. (To be clear, I haven't thought this through 100%, but I am almost certain this is true modulo my hand-wavy reference to "arithmetic axioms".) That means that that same proof should work inside the surreal numbers to also prove that the limit of the partial sums is 1. But here is the key important point. Within the surreal numbers, the limit of the partial sums is _not_ 1. So what does this tell us? The original supposition that there exists a proof only making use of the arithmetic axioms cannot be true. So the proof must make use of the completeness axiom. Well the surreal numbers is _not_ complete so that axiom doesn't exist there and therefore the fact that the proof exists and works within the real numbers and not the surreal numbers is not a contradiction.
Okay this is a bunch of logic mumbo jumbo and no grade student should be expected to worry about things at this level, but we can still give a more simplified version of the proof that actually is in essence correct. Some people in this thread use the argument: "Well since we know the number must be less than or equal to 1 and we also know it must be bigger than any number smaller than 1, then it must be 1." That argument (while a priori assuming convergence) is at least implicitly using an intuitive idea of completeness. Is it logically formal and 100% rigorous? Of course not. But it is explicitly making use of a property of real numbers while the first proof is not. I think if students are to be taught anything about the real numbers, then they should get some general intuition for this property. The algebraic version of the proof is invalid (my claim, which hopefully is at least a little more well-supported given this comment here) and it doesn't give the intuition they should (hopefully) get anyway.
In any case, hopefully this does some to help clear up what I've meant throughout these posts.
P.S. Finally, in response to this:
> By the way, that blog post claims "The set of real numbers contains no infinitesimals." Wikipedia claims: "the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers..."
So what? Are you saying those statements are contradictory? If so, how? And if not, what are you saying?
How did Euler actually write his proof, do you know, or have a link? I’ve poked around online but can’t find it.
By the way I wouldn't hold it against him if he did write the proof that way. I'm pretty certain all the logic/model theory that comes into play came long after his death. The surreal numbers certainly did.
>0.9bar7 is a completely nonsensical number
For the same reason that 0.9...7 isn't a meaningful number, you cannot move the decimal 'an infinite number of times' and then after this, look at what number you have left and see it still has infinite 9s left. It's like you're trying to perform transfinite induction on the set of numbers generated by moving the decimal point. You can only move the point a countable number of times, so there is no sense in which the property can still be true after infinity many times.
Yes you absolutely can, for exactly the same reason. 0.9bar7 is nonsensical precisely because you can move the decimal to the right an infinite number of times, and still have an infinite number of 9s before the 7.
> You can only move the point a countable number of times
Not true, the implicit definition of “...”, the very statement that there are an “infinite” number of 9s, means exactly the opposite of what you claim, it means you can move the decimal an infinite number of times.
>And here is the logical (induction) step: if you shift the decimal point by an infinite number of places, then there are still an infinite number of 9s to the right
This is not how induction works. The induction shows that you can shift the decimal point any finite number of steps to the right and there will still be infinite 9's after it. If you want to show something is still true after infinity steps, you require transfinite induction, but this doesn't make sense because the '...' decimal representation only represents a countably infinte number of 9's.
This is the same reason 0.9bar7 doesn't make sense - because decimal representations only have countably many digits.
This should rather say that it's because decimal representations have digits indexed by the natural numbers I guess, rather than by any larger countable ordinal
You say
> you're basically just defining 0.9... as a symbol to be some number x which has the property that 10x - x = 0.
Okay, well what is an alternate definition that makes more sense intuitively?
0.333...., for example, is one that seems pretty intuitive. We can get to .333... by iterated long division of 1 by 3.
3 | 1
0 (3 * 0 = 0) => 0.
10 (add zero)
9 (3 * 3 = 9) => 0.3
10 (add zero)
9 (3 * 3 = 9) => 0.33
...
So does this trick always work? If we have a repeated decimal, can we always multiply by 10 ^ (length of repeated sequence), subtract off, and get the value of that repeated decimal? If so, then it is reasonable to say that 0.999.... is equal to 1.
We can't really go in the forward direction without cheating (that is, going from 1 -> .999...); the best we can do is to modify long division to allow us to do it:
3 | 3
0 (3 * 0 = 0) => 0.
30 (add zero)
27 (3 * 9 = 27) => 0.9
30 (add zero)
27 (3 * 9 = 27) => 0.99
...
Obviously this isn't in Peano arithmetic exactly, but I think it holds together. If we allow repeated decimals in general to be valid representations of rational numbers, then we have to accept 0.999.... is equal to 1.
Peano arithmetic only covers nonnegative whole numbers, so one will never exist.
Thank you for the pedantism. How about I replace "Peano arithmetic" with the "operations of multiplication/addition/division/etc. expressible upon the rational numbers"?
(Disclosure: I have no idea what Peano arithmetic is.)
Regardless, the point has been clarified, but discussing it further is kind of pointless.
edit: It just occurred to me that the original pedantic comment essentially breaks the following Hacker News guideline:
Please respond to the strongest plausible interpretation of what someone says, not a weaker one that's easier to criticize. Assume good faith.
I just found it a bit interesting considering it is my response complaining about the pedantry and not the pedantry that is receiving downvotes.
Real numbers are defined as an equivalence class such that if the differences of two infinite sequences of rationals tend toward zero, then they are equal. The difference between 0.999... and 1.000... clearly tends towards zero as it heads of to infinity, and so they are equal.
If you want to argue that it doesn't then you have to come up with some other definition for numbers which have an infinite decimal expansion.
(Technically, of course, 1 is a rational number, but if you're using 0.9999... to represent it, you're using a real number representation, so you're bound by the definition)
> Real numbers are defined as an equivalence class such that if the differences of two infinite sequences of rationals tend toward zero, then they are equal. The difference between 0.999... and 1.000... clearly tends towards zero as it heads of to infinity, and so they are equal.
> If you want to argue that it doesn't then you have to come up with some other definition for numbers which have an infinite decimal expansion.
> (Technically, of course, 1 is a rational number, but if you're using 0.9999... to represent it, you're using a real number representation, so you're bound by the definition)
I'm not sure why you think I don't know the difference between rational and real numbers, but I assure you I do. What I said was I don't see how a proof involving the standard arithmetic operations found within the rational numbers, but not including any concepts of limits, completeness, etc. is invalid. Let me know if you still don't understand my point.
Trying to rearrange it and remove as many negatives as possible, I started with your statement:
> I don't see how a proof involving the standard arithmetic operations found within the rational numbers, but not including any concepts of limits, completeness, etc. is invalid.
I think what you mean is that any proof that does not use the concepts of limits and completeness is going to be invalid.
That seems clear to me, the reason being that one needs to define what one means by the sequence of symbols "0.9999...".
You can say "It's infinitely many 9s stretching off to the right", but that doesn't tell me what it means.
People seem to think it does, but when I dig deeper, they usually don't have any sense of what it means. And therein lies the problem (as I see it). People blithely write the glyphs, but don't have a concrete interpretation.
If someone really wants to understand it then I'll explain current mathematical thinking, including non-standard analysis and the surreals. But most people don't want to put in the work to understand how these issues have been resolved, and just want to argue from their intuition.
https://thatsmaths.com/2019/01/10/really-0-999999-is-equal-t...
I don't quite know how to formalize it, but I'm pretty certain that if these proofs logically worked (in the "theory of proofs sense"), then they should work in the surreals as well.
Anyway it's just intuition. My main point in this thread is that I don't really accept the proofs of this that don't use completeness as a step. Though I do suspect that proofs not making use of it are actually incorrect proofs in their own right. If I were curious enough I'd think back about formal proofs and models and all that jazz, but I probably already have spent more time in this thread than I should. :)
edit: The more I think about it I feel like someone actually explained to me this (i.e. why this proof is wrong using surreals as reasoning) a long time ago and I'm just remembering echos of it in my mind. Wish I could remember something more useful...or that I were a logician...
The rules are essentially the same.
Anyway it's still an artifact of representation. If rationals were represented as fractions, then it is unrepresentable. 1/1 X 10 == 10/1?
We do longhand division of 9 by 9, but start by putting a 0 in the first place.
0.99
9 | 9.0000
0
9 0
8 1
90
x = 0.8...
10x = 8.8... = 8 + x
9x = 8
8/9 = 0.8...
Technically it’s an infinite series (sum of an infinite sequence), which is a finite number in certain cases like this one.
Sure, but the point of "elementary school" arithmetic is not "elementary" arithmetic, as a mathematician would define it :-)
The goal is to teach people to reason by matching patterns. Deductive/Inductive reasoning can slowly proceed from that, as they try to frame their intuition for patterns into increasingly more general abstractions.
what are the properties that we would lose?
0.999… = 1 is a property of the way we write some rational numbers, not of the number system itself.
0.999... with infinite nines is equal to 1.
That is, for any number system I've seen, 1 = 1 + dx, and infinity = infinity + 100.
The point is that in elementary school arithmetic, you define addition, multiplication, subtraction, division, decimals, and equality, but you never define "...". Until you've defined "...", it's just a meaningless sequence of marks on paper. You can't prove anything about it using arithmetic, or otherwise.
What the "arithmetic proofs" are really showing that if we want "..." to have certain extremely reasonable properties, then we must choose to define it in such a way that 0.999... = 1. Other definitions would be possible (for example, a stupid definition would be 0.999... = 42), just not useful.
What probably causes the flame wars over "..." is that most people never see how "..." is defined (which properly would require constructing the reals). They only see these indirect arguments about how "..." should be defined, which look unsatisfying. Or they grow so accustomed to writing down "..." in school that they think they already know how it's defined, when it never has been!