A Friendly Chat About Whether 0.999... = 1

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41

Can you explain why such a subtraction wouldn’t be valid?

Are there rules saying you can’t subtract some values?

If you are worried that some people will argue against this, well you’re right, people who don’t want to be convinced won’t be, they’ll find any way they can to wriggle out of it.

It is tricky yes, because the whole of Maths is abstract it doesn’t exist, there is no 1 that exists outside of our minds, there is nothing you can points to and say that is one, the idea of oneness is something we’ve invented to describe certain things.

So it really is down to the abstraction people carry in their head for Maths, some people may not be aware that Maths is all in their head, what they can grasp is through a natural understanding of the basic abstract concepts of Maths, but when it comes to un-natural or super natural concepts it seems wrong because for them Maths always seemed different.

Maths is nothing but a model or a system, a way of thinking about things that is useful, but it is just a model. Depending on how you use that model and what you consider to be its rules and limitations will depend on what you can and cannot believe.

Maybe the hard bit is to make people think differently about Mathematics, from the very basic concepts.

Maths models some certain real world domains really well, positive integers, basic fractions, addition. These are easy and simple to understand, it is easy to believe that Maths actually exists and is defined by these operations.

Yet negative numbers have no real world analogues. What about bank accounts? Well problem is they don’t exist, they are abstract as well, in a bank there is no pile of your money, when you go overdrawn you don’t have a negative amount of money in a pile. The amount of money you have is an abstract representation of credit.

Now many people can intuitively understand the abstract concept of negative numbers, and may never realise they are thinking, imagining and understanding such a concept. It seems natural because it is used so often by so many.

However when you get to concepts of infinity it gets tricky, why? Because many people think Maths is real, it isn’t, it is entirely abstract and whilst it has concepts with comfortably match reality, it is not real. Now if you think of Maths as some real thing that exists then you have a problem, because no one can really grasp infinity. It is, you could say, the ultimate abstract concept. However because it is not something to see to perceive to touch taste etc. it doesn’t mean it is not a useful way to think, or a useful idea. It does however exist exactly the same amount as 1 or negative numbers, as a concept in our minds. But the ultimate effects of infinity are far more profound, perhaps only because they are not as comfortable or everyday as many mathematical concepts.

This issue is one people have had a lot of problems accepting, irrational numbers, if you assume or expect that all numbers have to have exact values then you get stuck because that is not the case. Some numbers don’t even have exact values even though we call them constants, see PI or e we only have very close values to those, you could see the values we have as approaching the real value exactly as 0.999… approaches 1. Although we don’t know that PI or e are irrational, if they are then we will never have an exact value.

0.999… comes to be used because we use numbers like 0.333… a number which cannot be satisfactorily displayed in decimal. Yet the decimal system has many benefits even if it cannot display some values satisfactorily, so we work with the system, and we have to accept that in this system 1/3 is represented as 0.333… and if we accept that then we have to accept that 0.999… is equal to 1. What is important that is these numbers are representations of the true value, not the value itself, because using an infinitely recurring number in calculations is pretty difficult.

Ultimately what we write down on paper is not maths, it is a representation of Maths, the process that goes on in our heads, often it can be equivalent, and what you write are not just representing the absolute values but are the absolute values, but this is not always the case.

You cannot write an absolute value in the decimal system to represent a third. How do you deal with this? We use a representation of a third, 0.333… We don’t use that value in calculations because we can’t because its not a value that can be written down. This applied to all values that cannot be represented absolutely in decimal we use a representations. And that is fine because Maths is not defined by what is written down. What we write down is a representation of the Maths in our heads.

So in this way 0.999… isn’t 1, but it represents 1, and in that way it is 1. If you try and use Maths to prove that 0.999… equals 1 then you’ll may fail because you will be trying to use 0.999… as an absolute value when it is not.

So does 0.999… equal 1? Well no, because 1 is an absolute value, and 0.999… is a representation of a value without an absolute value.

So in maths, in the decimal system, 0.999… represents 1, 0.333… represents 1/3, we use these representations because decimal cannot give an absolute value we can use. Its not a problem because 0.333… or 1/3 is just a representation of a value which we understand in our head, we have other ways to use those values, and we often do, we can leave things in surd form, or we can accept close values, because we may not need high precision in calculations.

The argument about 0.999… equalling 1 isn’t a mathematical argument it is a semantic argument about the way we represent Maths in a written form, as long as you understand the benefits and draw backs of using this it is not a problem.

So back to the original point, is my subtraction valid? Not really because we aren’t using absolute values but representations, the calculation can never be finished because it goes off into infinity.

However how much understanding does the person you are telling this to need to know, want to know? And does it matter anyway? Is the argument about 0.999… equal to 1 that significant in maths?

42

One simple comment. If we define a number system that allows 0.9999999… < 1 then that also implies that in the same number system the normal decimal expansion is not necessarily an equality. Or more precisly

0.99999… 0.3333333 < 1/3. I'm not going to prove it, but the line of reasoning follows from the equality proof provided in a previous comment.

I personally find this fascinating, because I'm currently learning about number theory and divisibility. And once you accept the entire set of real numbers most of those theorems go out the window. But if you accept infinitesimals, then you get em all back again.

43

Could the experts comment on how this argument fits into this discussion:

The formula for summing a geometric series is:

1 + x + x^2 + x^3 + … = 1/(1-x)

So

.99999… = .9(.1 + .01 + .001 + … )
.99999… = .9(1/(1-.1))
.99999… = .9(1.111…)
.99999… = 1

Does this add anything, or does this just suppose the premise in question? And if so, what implications would this have for the formula of a geometric series?

44

@ Chad Groft… you said in #32, “since “infinitesimal separation” falls apart when we try to be precise and do arithmetic with it, this is an academic concern.”

According to wikipedia, “The hyperreal numbers satisfy the transfer principle, which states that true first order statements about R are also valid in *R. For example, the commutative law of addition, x + y = y + x, holds for the hyperreals just as it does for the reals.”

If the commutative laws (and others) truly hold, then what’s the problem? If we can add them and multiply them and do “arithmetic” with them, then they qualify as Number, correct?

What am I missing? Have the hyperreals been given a rigorous treatment or not?

45

@41: You’re supposing the premise in question, namely that a limit exists. Try doing it with x = 2 and see what happens.

@42: I wasn’t talking about the hyperreals. Those are well-defined and contain infinitesimals. It’s just not clear which infinitesimal 1 - 0.999… should be.

46

If 0.999… = 1, then 1 is the sum of an infinite series. Are all (real?, rational?, integer?) numbers therefore likewise?

47

@43 Thanks. I really appreciate all you explanations…

so am I to understand that we can do arithmetic with the hyperreals but not with the infinitesimals?

I was under the impression that (one of) the point(s) you were making was that we could not apply the rules of arithmetic to the infinitesimals, or that they did not behave in a way that was consistent or something along those lines. Perhaps I was mistaken to think that if rigor had been applied to hypperreals then rigor had been applied to infintesimals?

For me, the question of “do infinitesimals exist” in a manner that qualify them as “number” as defined by the commutative and other arithmeitc laws should settle the question of does 0.999… = 1.

So, do infinitesimals rigorously qualify as numbers or not?

48
If 0.999… = 1, then 1 is the sum of an infinite series. Are all (real?, rational?, integer?) numbers therefore likewise?

Well, yes, since every real number C can be represented as C/2 + C/4 + C/8 + …

One thing you have to understand is, there’s no such thing as the infinitesimals. There are many contexts—for example, the rational numbers, the real numbers, the hyperreal numbers, the rational functions on one variable, the complex numbers—and infinitesimals exist (or don’t) within each context.

Every real number (hence every rational number) is less than some natural number n (dependent on the real); so every positive real, no matter how “small”, is greater than 1/n for some natural number n. Thus infinitesimals don’t exist in Q or R; thus also the set {.9, .99, .999, …} has a least upper bound in Q and R, and it’s 1.

On the other hand, consider the rational functions on one variable: all the fractions f(x)/g(x), where f and g are real polynomials (and g ≠ 0). These can be added, subtracted, multiplied, and divided (except by 0). We can also put an ordering < on them, for example by considering their behavior at +&infty;. Then the rational function 1/x is positive, but less than r for every positive real number r (because 1/x < r once x gets bigger than 1/r). We say that 1/x is an infinitesimal. Similarly the function x is greater than any real number, because once x gets large enough (i.e., greater than r), x > r. (Tautological, but still true.)

The hyperreals are basically the above example on steroids. In addition to adding infinitely large (and infinitely small) numbers, we also import all sorts of higher-order structures from standard real-number theory. That lets us do analysis with infinitesimals, then translate the results back to real numbers. (Except nobody really bothers to anymore.)

So yes, it’s possible to work with infinitesimals rigorously, if we’re in a context where they make sense. If you’re going to ask “[D]o infintesimals … qualify as numbers?”, I’ll have to ask “What kind of numbers?”

What we’ve hashed out in the previous comments is, IF we are in an ordered field (that is, a context where +, -, x, ÷, and < are defined and behave as we expect), and IF the set {.9, .99, .999, …} has a least upper bound (which is the only sensible interpretation for .999…), THEN that least upper bound must be 1.

We can have no infinitesimals, and say .99… = 1; we can have infinitesimals, and say .99… doesn’t resolve to anything; or we can drop the field property, and do some weird Dedekind-cut construction to have .99… resolve as just slightly less than 1. But we can’t have full arithmetic and .99… defined as something less than 1.

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[…] Everything written here was inspired and taken from here and here. Check out Kalid’s site. It is […]

50

@46 Thanks again. I remember having a sense from “studying” such things years ago that to “qualify as number” meant that the basic laws of arithmetic applied. Infinitesimals sound too messy to me.

The problem with the Dedekind cut is that you have to let the process of adding the series finish before you cut. I would think Dedekind would simply point out that 0.999… is infinitely close to 1, which is another way of saying the “gap” is infinitely small, which is a pretty good definition of zero, is it not? If the gap is zero… no cut. The numbers are one and the same.

To Kalid, I would say that this point kills all the intuitive attempts to inject a number in the gap. There can only be a gap at some discreet (location? moment? point?) in the series. And despite the fact that we can measure a gap at all such DISCREET points, this seems akin to the act of collapsing the wave function - by taking a measurement we have to stop the process!

The most essential fact about an infinite series, however, is that the process NEVER stops. Any time we imagine a gap, we are imagining something discreet, which is to say we are imagining some OTHER number.

It is as if we are saying, “add this to this to this to this and never stop, oh and when you are done tell me what the sum is so I can subtract from 1 and ‘measure the gap’”

OK, tell you when I get there. Stand by! :slight_smile:

51

@Student, @Chad: Awesome insights! There’s so much I need to learn about the details of analysis. I think one of the biggest questions is “What does ‘0.999…’ really mean when someone asks about it?”

Technically, we can interpret it as a limit, but it may not be what the student had in mind. They may be asking “What is the number closest to 1 but not one?” The traditional answer is “none – there is no such number, you are either 1 or not 1” or alternatively “there is an entire class of small numbers infinitely close to 1 (appear like 1 to us, but are not the same at a different level of detail)” and then you explore the implications if that were possible (akin to exploring the idea that a square root of a negative number is possible).

Our analysis of what 0.999… means may be like a linguist analyzing the sentence “I ain’t never gonna do that again”. Technically there’s a double-negative which means they do want it again, but a step back realizes the intent of the statement. So there can be an impedance mismatch between what the asker is asking for and how the more knowledgeable answerer interprets it.

That said, there’s 2 really interesting avenues: the analysis of the intuitive notion of infinitesimals (what is the person asking about?) and also the construction of other types of numbers (0.999… which may not be a “valid number”, etc.), which Chad summarized [i.e., your statement can be interpreted in the following ways…]. But I love the discussion! It’s a great chance for me to learn more about this topic.

52

@48 Kalid: Thanks for commenting on my blog! Your site inspired me to start my own. I have the mathematicians lament framed above my desk! You should also read David Foster Wallace’s book, “Everything and More.” It is all about this very topic. He is a talented author and communicator. His book is essential to this discussion for all the lay people out there who seek an intuitive sweep of the subject.

And thank you Kalid for talking about this on your site, for this problem is the very core of which all of mathematics - and philosophy - revolves.

I am by study a philosopher, not a mathematician, but if you boil it down, what is the difference? Just ask Bertrand Russell!

Here is a thought for you that you can take to Pythagoras, Plato and Aristotle, who were the first giants to address these issues before we separated math and philosophy…

I posit that all paradox comes from one source: the category mistake. We ask, “what color is 42?” and scramble for an answer, thinking we have disturbed the beast at the center of the universe when we have but made fools of ourselves.

It was from Pythagoras that Plato got the idea of the Forms. But skip that. You can look it up. The point I am coming to is what Aristotle said in resolving Zeno’s paradoxes: there is the actual, physical world of 5 apples and then there is the abstract, potential world of the integer 5. Number. Form. Infinity. These are not concrete things. They are manifestations of the human mind. To ask. “are they real” is to make the category mistake, for in this question we transfer them from the category of potential abstract (integer) to the category of actual physical (apples).

Similarly we can say things like:

1: curiosity killed the cat.
2: the world’s largest ball of twine is a curiosity
3: the world’s largest ball of twine killed the cat

This is akin to the category mistake. In philosophy we wrestle with the meaning of “to be,” “existence” and what it means to mean something. But we do not escape these problems in mathematics. We create the rigor as if we had changed professions and became lawyers, laying out the laws that prevent us from falling into these philosophical holes. But this is like saying that killing is wrong because it is illegal. In so doing, we simply avoid the real problem.

Thus we make the category mistake. We treat the infinite like the discreet. We treat number like apple. We talk of points on line that take up no extension and then define a line as an extension, dense with points.

Draw a circle, then draw another with twice the radius through the same center; smaller circle inscribed by larger. Now extend a line from this shared center so that the line extends through both circles, intersecting each at a point. If you were to rotate this line so that you cross each point of the inner circle you would also cross each point of the outer circle. Which is to say that there is a 1-to-1 correspondence between the “number” of “points” in each circle, despite the fact that the circumference, or “length” of one is greater than the other.

In other words, given a line of any length, there are exactly the same count of “points” or “numbers” on this line as any other line, no matter what the “length.”

This is set theory. Cantor please save us!

Cantor says this infinity is larger than that infinity? This is how we are saved!?

Abstraction of an abstraction of an abstraction… until we are integrating integrals and developing topological tautologies, will it ever end?

We forget during all of this that we left the concrete world behind long ago. We have built a house of cards.

But it works so well, you say! We have electricity! But what do we mean, “it works?” We mean the relationships work, which makes sense only insofar as we can apply these relationships to something concrete. The rules work, the equations work, but that does not mean that the abstract things we use as placeholders of the concrete are therefore real!

Just because 5 of anything added to 5 of anything equals 10 of something does not mean that the integer 5 exists!

And this is the crux of our conceptul problem with 0.999…

It is not anything concrete. It represents a hypothetical idea. It is not real. What is real are concrete things and how they relate. Ideas merely help us do the relating, they are not that which we are to relate to!

This is the category mistake. We ask, what color is 42?

This is the source of all philosophical paradox and this case of the repeating decimal is one of the oldest, thorniest of them all.

I find when I reach these impassable points that it helps to rename all my cliches. Instead of “number,” say “abstraction” for example. As in, “the abstraction, ‘0.999…’” or “the abstracton ‘1’”

This helps remind me what number is. It is not 1 apple we are talking about. It is not something concrete we are discussing. It is not something actual that is causing us confusion. It is an idea (that we are trying to treat as if it were a concrete thing like an apple) that is causing us problems.

When we talk of the “integers” we speak as if these abstractions are concrete, and thus we make the fundamental philosophical misstep, the category mistake.

53

Yes, it is a truly mesmerizing question:

“What is the number closest to 1 but not one?”

Instead of trying to answer it, perhaps we should try to explain why it makes as much sense as asking:

“What color is 42?”

And what I mean, is does this question really make sense?

By “number” do we mean “idea?”

By “closest” do we posit some “unlimited divisibility of space”

These questions are puzzling because the idea of a point and the idea of a line is puzzling, but no more so.

When we imagine the continuum, we dream of things we cannot touch

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@Student: I can only encourage you to keep writing – you write very well and it’s a pleasure to read! I’m adding that book to my toread list :).

I like that distinction between what’s happening in the real world vs. what’s happening with our abstractions – are the questions even sensible? I love stepping back like this and questioning our basic assumptions about what an idea means. Often times we’re working in a framework which just makes it impossible to understand (my favorite example is assuming that a number (one abstraction) must be one-dimensional (another abstraction), while imaginary numbers throw those assumptions for a loop).

I think the crux of the 0.999… issue may be if we allow the existence of another abstraction, the idea of infinitesimal numbers relative to our own.

55

@51 Thank you very much. That is quite a compliment coming from you! I remember when I first realized that imaginary numbers were of “higher dimension.” That may have been the very thing that cemented my love of mathematics.

Another interesting thought on your point of “the closest number” to any other number is that this question underscores the difficulty with an infinitely dense set. Because there are an infinite “number” of “numbers” between any 2 (real) numbers, can we ever even understand what it means to ask of one of these numbers, what the “next one” is? Or do we have to accept that the concrete concept of “next” has to be left down here on the ground while we ponder these ideas of the heavens?

Lastly, in researching the nature of a repeating decimal, I have been reminded of the fact that a decimal is not a “number” but a special notation, or representation of a number! An abstraction on top of an abstraction. We started with those 5 apples and moved up to the abstract number 5, brought it back down to the world and pretended it was as concrete as those apples. Then we took a geometric series, went another level of abstraction up and invented decimal notation to make our abstraction less messy to write. We made a map of our map.

We just keep forgetting that our maps are in fact maps! And in the case of the repeating decimal, a map of a map of a map.

Somewhere along the way, I think we lose the ability to make certain concrete denotations of all these maps of maps.

We are simply reminded of this when we ask a meaningless question like what color is 42, or what is the next real number.

It is merely our own semantics that are failing us.

At least that is my intuitive conjecture. Perhaps I should write all of this as a question. I would love to be shown right or wrong.

56

wow a very lively discussion on this one. Haha looks like it’s really the simple yet profound problems that can get people from all backgrounds involved. Just the perfect combination of math, philosophy, reasoning, etc.

@Kalid
Yes please look into analysis and post some articles on it. The article on the imaginary unit as a rotation was great and now I’m not so scared of complex analysis. But analysis in general scares us students…

57

@52

I think you have the point exactly.

Maths is very interesting, as it is something we thought up inside our heads, I was thinking. Are we the only animals to have done this? I think not, consider the evolutionary benefits for an animal that can do this, if it can count, if it can estimate volume, or distances, by knowing this information it can make decisions. If its outnumbered it can run away, if its opponent is bigger it can run away, or smaller it can fight, it can tell if it can reach something or not.

In this way we can see that an understanding of maths of some kind is present in many animals, but the understanding is natural, rather it is part of their way of experiencing and understanding the world.

The difference with humans is that we have externalised this internal thought to a degree, rather than just have it as a tool that works to help us think and react in a subconscious way we have found a way to translate those ideas into a structure we can apply to a vast number of other situations. And a tool that is standardised between people.

When you think about that is pretty impressive, whilst we all may think differently, and talk different languages mathematically we are all the same. (or similar, or are we? :smiley: ) We have taken a basic fundament of consciousness and mapped it out.
However the externalisation of this way of thinking is not the thing itself, it is a representation of our thought or mode of thought. And I guess here is where we get into philosophy. Simulation and Simulcra by Jean Baudrillard comes to mind, when you have a representation, an icon for something people eventually begin to focus on that as being not a representation but the actual thing. A representation is easier to deal with in the mind and so it comes to be a thing, the thing people think of. What is the meaning in the iconography of maths of 0.999… if you try to consider the meaning of a pure abstract idea as a real solid thing, considering a infinitely recurring number as an actual value such as an integer, how do you reconcile that with the rest of the iconography of maths which is more solid.

The issue is one that affects more than this however as 0.999… is the tip of the iceberg as you move deeper into mathematics things become more abstract and as things start to become difficult to reconcile with the solid iconography of absolute numbers, you have to start accepting that the iconography of maths is not as solid as you thought, it is not absolute, which brings to mind Gödel’s work, essentially it was all about the validity of the iconography of the mathematic system, questioning its absoluteness. And the uproar it caused is basically a similar issue, the maths world had to reconsider the basic assumptions that maths was based on, instead of it being a solid system it is not, it can’t be proven absolutely. There are special cases and abstractions, abstractions based on abstractions. Trying to reconcile what you once thought as something solid and absolute with something which is not.

Maybe if you want a nice safe place, thinking maths is absolute is great, you have a solid system to fall back on. You can ignore the fact it isn’t, to a degree, but in reality it’s not safe at all, it’s all about risks and reward, maths is dangerous, what assumptions should you make, should you accept, do they invalidate the system. Is the benefit of this way of thinking more useful than the problems the drawbacks create. Without a rigorous understanding of the limitations, assumptions, and fundamentals, you could fall into infinity, a circular argument, a paradox and never get out. Eventually you have to accept that Maths is a model, a system that we can use to model existence, but there is no such thing as a perfect circle, a perfect integer, or infinitely recurring or irrational numbers, at least not a meaningful representation in reality. The fuzzy world starts to encroach on the logical precise world.

One thing I thought about when I was thinking about 0.999… how would such a result occur in maths, how could we get a results which gives that value? What possible inputs to lead to such an irrational output. Only irrational inputs. our mathematics is limited by precision as such the only time we can get a number like 0.999… is when we use other numbers like 0.333… and so on. The artefact of 0.999… is produced by the use of other equally irrational and non-absolute values, it isn’t a result of real world calculations, so to understand the answer we have to understand the inputs. So the question isn’t whether 0.999… is 1 but the question is, Is this the true question? In what cases is that answer received and why, it is not a real world answer, or maybe it is, can you add two rational numbers to get an irrational number? What two rational numbers do you add to make 0.999… What real world situations do we experience this as an answer? Or is it something which only occurs in abstract problems?

If I get a result of 0.9999999999, I’ll call it 1 anyway because it is unlikely I need such precision. And ultimately 0.999… is even closer, or infinitely close to that value, not calling it 1 is just ridiculous, it is easy to get caught up in the belief of an absolute iconography of maths, but pull back, think, is 0.999… such a big deal, I don’t think it is, it is a storm in a teacup caused by people who are too busy trying to prove the iconography of absolute maths. Rather than examining maths itself where it exists and where it came from.

58

It’s funny listening to people talk at cross-purposes. Some of you sound like Nigel in Spinal Tap: “But these go to 11.” It’s what attracted me to contract law, legislation and eventually judicial review.

And the art of communication, Chad, is to address your audience in a way that makes sense to that particular audience. Increasingly lengthy and arcane responses aren’t usually effective. See #31. It’s a good guess poor Nicole and Rick had no idea what you were saying. You can’t communicate with people by going over their heads.

This is where Kalid shines. As he nicely makes clear, and as we eventually learn about most things in life, the answer depends. It’s why I become hyper-vigilant when an attorney asks the witness to answer “yes” or “no.” As though there were no shades of gray.

At the risk of revealing myself to be the caveman that I am (aren’t we all looking for moments of clarity, a large black monolith that can help us advance?), what sits well with me is the idea that 0.999… and 1 are separate numbers. But they are so close to each other that there is no number between them. You can’t subtract 0.999… from 1 because the difference is, in my cave, undefined. So my number line isn’t perfectly continuous, and sometimes operations such as subtraction break down. But enough about me.

59

@Wheatgerm: Well, it would be nice if there was a shallow end to this pool; but there just isn’t one. When you’re talking about what an argument is and what makes it valid, you have to think about all the explicit and implicit premises. (As someone who’s studied law, you probably understand that.)

The rest of the response is to show that all those premises are necessary, by showing that the conclusion fails if any of the premises fail. The only way to do that is to construct counterexamples, and that takes a while. Perhaps my intent could have been made clearer.

Kalid’s writing is in fact very clear, and I read this blog to improve my own exposition; but I’d rather express a true idea poorly than a false idea well.

As for your (probably very common) idea that .999… and 1 are distinct numbers but have no numbers between them — where do you think (1 + .999…)/2 lies? If two real numbers are distinct, then their average must lie strictly between them.

60

I should moderate that last paragraph.

Your “number line” sounds a lot like the construction I outlined in comment 32, where one gets the sort of resolution you talk about at the expense of being able to do arithmetic. If we’re talking about real, physical magnitudes, though, we want the “common” real number line.

Did comment 46 make any sense?