A Friendly Chat About Whether 0.999… = 1

Does .999… = 1? The question invites the curiosity of students and the ire of pedants. A famous joke illustrates my point:

A man is lost at sea in a hot air balloon. He sees a lighthouse approaching in the fog. “Where am I?” he shouts desperately through the wind. “You’re in a balloon!” he hears as he drifts off into the distance.

The response is correct but unhelpful. When people ask about 0.999… they aren’t saying “Hey, could you find the limit of a convergent series under the axioms of the real number system?” (Really? Yes, Really!)

No, there’s a broader, more interesting subtext: What happens when one number gets infinitely close to another?

It’s a rare thing when people wonder about math: let’s use the opportunity! Instead of bluntly offering technical definitions to satisfy some need for rigor, let’s allow ourselves to explore the question.

Here’s my quick summary:

  • The meaning of 0.999… depends on our assumptions about how numbers behave.
  • A common assumption is that numbers cannot be “infinitely close” together — they’re either the same, or they’re not. With these rules, 0.999… = 1 since we don’t have a way to represent the difference.
  • If we allow the idea of “infinitely close numbers”, then yes, 0.999… can be less than 1.

Math can be about questioning assumptions, pushing boundaries, and wondering “What if?”. Let’s dive in.

Do Infinitely Small Numbers Exist?

The meaning of 0.999… is a tricky concept, and depends on what we allow a number to be. Here’s an example: Does “3 – 4” mean anything to you?

Sure, it’s -1. Duh. But the question is only simple because you’ve embraced the advanced idea of negatives: you’re ok with numbers being less than nothing. In the 1700s, when negatives were brand new, the concept of “3-4” was eyed with great suspicion, if allowed at all. (Geniuses of the time thought negatives “wrapped around” after you passed infinity).

Infinitely small numbers face a similar predicament today: they’re new, challenge some long-held assumptions, and are considered “non-standard”.

So, Do Infinitesimals Exist?

Well, do negative numbers exist? Negatives exist if you allow them and have consistent rules for their use.

Our current number system assumes the long-standing Archimedean property: if a number is smaller than every other number, it must be zero. More simply, infinitely small numbers don’t exist.

The idea should make sense: numbers should be zero or not-zero, right? Well, it’s “true” in the same way numbers must be there (positive) or not there (zero) — it’s true because we’ve implicitly excluded other possibilities.

But, it’s no matter — let’s see where the Archimedean property takes us.

The Traditional Approach: 0.999… = 1

If we assume infinitely small numbers don’t exist, we can show 0.999… = 1.

First off, we need to figure out what 0.999… means. Most mathematicians see the problem like this:

  • 0.999… represents a series of numbers: 0.9, 0.99, 0.999, 0.9999, and so on
  • The question: does this series get so close (converge) to a result that we cannot tell it apart?

This is the reasoning behind limits: Does our “thing to examine” get so darn close to another number that we can’t tell them apart, no matter how hard we try?

“Well,” you say, “How do you tell numbers apart?”. Great question. The simplest way to compare is to subtract:

  • if a – b = 0, they’re the same
  • if a – b is not zero, they’re different

The idea behind limits is to find some point at which “a – b” becomes zero (less than any number); that is, we can’t tell the “number to test” and our “result” as different.

The Error Tolerance

It’s still tough to compare items when they take such different forms (like an infinite series). The next clever idea behind limits: define an error tolerance:

  • You give me your tolerance for error / accuracy level (call it “e”)
  • I’ll see whether I can get the two things to fall within that tolerance
  • If so, they’re equal! If we can’t tell them apart, no matter how hard we try, they must be the same.

Suppose I sell you a raisin granola bar, claiming it’s 100 grams. You take it home, examine the non FDA-approved wrapper, and decide to see if I’m lying. You put the snack on your scale and it shows 100 grams. The scale is accurate to 1 gram. Did I trick you?

You couldn’t know: as far as you can tell, within your accuracy, the granola bar is indeed 100 grams. Our current problem is similar: I’m selling you a “granola bar” weighing 1 gram, but sneaky me, I’m actually giving you one weighing 0.999… grams. Can you tell the difference?

Ok, let’s work this out. Suppose your error tolerance is 0.1 gram. Then if you ask for 1, and I give you 0.99, the difference is 0.01 (one hundredth) and you don’t know you’ve been tricked! 1 and .99 look the same to you.

But that’s child’s-play. Let’s say your scale is accurate to 1e-9 (.000000001, a billionth of a gram). Well then, I’ll sell you a candy bar that is .999999999999 (only one trillionth of a gram off) and you’ll be fooled again! Hah!

In fact, instead of picking a specific tolerance like 0.01, let’s use a general one (e):

  • Error tolerance: e
  • Difference: Well, suppose e has “n” digits of precision. Let 0.999… expand until we have a difference requiring n+1 digits of precision to detect.
  • Therefore, the tolerance can always be less than e! And the difference appears to be zero.

See the trick? Here’s a visual way to represent it:

visualizing 0.999...

The straight line is what you’re expecting: 1.0, that perfect granola bar. The curve is the number of digits we expand 0.999… to. The idea is to expand 0.999… until it falls within “e”, your tolerance:

beating the error margin

At some point, no matter what you pick for e, 0.999… will get close enough to satisfy us mathematically.

(As an aside, 0.999… isn’t a growing process, it’s a final result on its own. The curve represents the idea that we can approximate 0.999… with better and better accuracy — this is fodder for another post).

With limits, if the difference between two things is smaller than any margin we can dream of, they must be the same.

Assuming Infinitesimals Exist

This first conclusion may not sit well with you — you might feel tricked. And that’s ok! We seem to be ignoring something important when we say that 0.999… equals 1 because we, with our finite precision, cannot tell the difference.

Newer number systems have developed the idea that infinitesimals exist. Specifically:

  • Infinitely small numbers can exist: they aren’t zero, but look like zero to us.

This seems to be a confusing idea, but I see it like this: atoms don’t exist to cavemen. Once they’ve cut a rock into grains of sand, they can go no further: that’s the smallest unit they can imagine. Things are either grains, or not there. They can’t imagine the concept of atoms too small for the naked eye.

Compared to other number systems, we’re cavemen. What we call “tiny numbers” are actually gigantic. In fact, there can be another “dimension” of numbers too small for us to detect — numbers that differ only in this tiny dimension look identical to us, but are different under an infinitely powerful microscope.

I interpret 0.999… like this: Can we make a number a bit less than 1 in this new, infinitely small dimension?

Hyperreal Numbers

Hyperreal numbers are one system that uses this “tiny dimension” to examine infinitely small numbers. In this, infinitesimals are usually called “h”, and are considered to be 1/H (where big H is infinity).

So, the idea is this:

  • 0.999… < 1 [We’re assuming it’s allowed to be smaller, and infinitely small numbers exist]
  • 0.999… + h = 1 [h is the infinitely small number that makes up the gap]
  • 0.999… = 1 – h [Equivalently, we can subtract an infinitely small amount from 1]

So, 0.999… is just a tiny bit less than 1, and the difference is h!

Back to Our Numbers

The problem is, “h” doesn’t exist back in our macroscopic world. Or rather, h looks the same as zero to us — we can’t tell that it’s a tiny atom, not the lack of any matter altogether. Here’s one way to visualize it:

infinitesimal difference

When we switch back to our world, it’s called taking the “standard part” of a number. It essentially means we throw away all the h’s, and convert them to zeroes. So,

  • 0.999… = 1 – h [there is an infinitely small difference]
  • St(0.999…) = St(1 – h) = St(1) – St(h) = 1 – 0 = 1 [And to us, 0.999… = 1]

The happy compromise is this: in a more accurate dimension, 0.999… and 1 are different. But, when we, with our finite accuracy, try to describe the difference, we cannot: 0.999… and 1 look identical.

Lessons Learned

Let’s hop back to our world. The purpose of “Does 0.999… equal 1?” is not to spit back the answer to a limit question. That’s interpreting the query as “Hey, within our system what does 0.999… represent?”

The question is about exploration. It’s really, “Hey, I’m wondering about numbers infinitely close together (.999… and 1). How do we handle them?”

Here’s my response:

  • Our idea of a number has evolved over thousands of years to include new concepts (integers, decimals, rationals, reals, negatives, imaginary numbers…).
  • In our current system, we haven’t allowed infinitely small numbers. As a result, 0.999… = 1 because we don’t allow there to be a gap between them (so they must be the same).
  • In other number systems (like the hyperreal numbers), 0.999… is less than 1. Here, infinitely small numbers are allowed to exist, and this tiny difference (h) is what separates 0.999… from 1.

There are life lessons here: can we extend our mental model of the world? Negatives gave us the conception that every number can have an opposite. And you know what? It turns out matter can have an opposite too (matter and antimatter annihilate each other when they come in contact, just like 3 + (-3) = 0).

Let’s think about infinitesimals, a tiny dimension beyond our accuracy:

  • Some theories of physics reference tiny “curled up” dimensions which are embedded into our own. These dimensions may be infinitely small compared to our own — we never notice them. To me, “infinitely small dimensions” are a way to describe something which is there, but undetectable to us.
  • The physical sciences use “significant figures” and error margins to specify the inherent inaccuracy of our calculations. We know that reality is different from what we actually measure: infinitesimals help make this distinction explicit.
  • Making models: An infinitely small dimension can help us create simple but accurate models to solve problems in our world. The idea of “simple but accurate enough” is at the heart of calculus.

Math isn’t just about solving equations. Expanding our perspective with strange new ideas helps disparate subjects click. Don’t be afraid wonder “What if?”.

Appendix: Where’s the Rigor?

When writing, I like to envision a super-pedant, concerned more with satisfying (and demonstrating) his rigor than educating the reader. This mythical(?) nemesis inspires me to focus on intuition. I really should give Mr. Rigor a name.

But, rigor has a use: it helps ink the pencil-lines we’ve sketched out. I’m not a mathematician, but others have written about the details of interpreting 0.999… and 1 or less than 1:

“So long as the number system has not been specified, the students’ hunch that .999… can fall infinitesimally short of 1, can be justified in a mathematically rigorous fashion.”

My goal is to educate, entertain, and spread interest in math. Can you think of a more salient way to get non-math majors interested in the ideas behind analysis? Limits aren’t going to market themselves.

Other Posts In This Series

  1. A Gentle Introduction To Learning Calculus
  2. How To Understand Derivatives: The Product, Power & Chain Rules
  3. How To Understand Derivatives: The Quotient Rule, Exponents, and Logarithms
  4. An Intuitive Introduction To Limits
  5. Why Do We Need Limits and Infinitesimals?
  6. Learning Calculus: Overcoming Our Artificial Need for Precision
  7. Prehistoric Calculus: Discovering Pi
  8. A Calculus Analogy: Integrals as Multiplication
  9. Calculus: Building Intuition for the Derivative
  10. Understanding Calculus With A Bank Account Metaphor
  11. A Friendly Chat About Whether 0.999... = 1

Questions & Contributions


  1. My Calculus prof proved this for us in class.

    Let N = 0.999…

    Assume N = 1, now multiply both sides by 10
    10N = 10, now subtract 9 from both sides
    N = 1

    I *think* that’s how he did it.

  2. Why “0.999 = 1” is counter-intuitive:

    If you have “0.99” instead of “0.9”, it means that you are one step closer to 1, as close as the 10-digit-notation allows in one step, *but without reaching 1*. If you add another 9 and arrive at “0.999”, you have again stepped as close to 1 as you could in one step, but without reaching 1.

    Even if you would do this an infinite amount of times, *every step* would have the same rule: “… *without* reaching 1”.

  3. i really like your explanations of tricky math concepts. do keep posting more good stuff like this. looking forward to your post on approximating functions :)

    also from what i understand, dark matter is different from anti-matter. When matter meets anti-matter they annihilate and release energy. Anti-matter is well understood while dark matter is not. It is simply conjectured to exist to explain the speeding up of the expansion of the universe when it should really be slowing down. So you can make that (infinitesimal) correction into your excellent post :)

  4. I think you meant anti-matter where you said “Dark matter destroys regular mass when they come in contact […])

  5. The hyperreal case is a little bit more subtle than that.

    See, the object 0.999…, as you understand it, doesn’t quite exist there. The hyperreals add infinitesimals to the real line by also adding infinitely large numbers, including infinitely large integers. And since {0, 0.9, 0.99, 0.999, …} is a sequence on the positive integers, it gets a lot more terms when it gets embedded into the hyperreal system; it becomes a hyper-sequence, for lack of a better term. Canonically it corresponds to a hyper-sequence whose length is unbounded even in the hyperreals, and still has 1 as a limit.

    Now, we could also look at the hyper-sequence which started the same way, but stopped getting bigger at some hyper-integer w. The difference between 1 and that limit would be 10-w, which is positive in our system. (This is what the arXiv paper you cited is talking about.) There are many sequences which are increasing like 0.999… on the standard integers, then take a constant value x on most of the nonstandard integers, but x could be anything.

    So the real problem is that 0.999… isn’t well-defined in the hyperreals—it doesn’t really equal anything. I know of no context where 0.999… has a clear resolution and it isn’t 1.

    (And for the record, we aren’t rigorous because we like to be. We’re rigorous because the subject demands rigor. Intuition fails a lot.)

  6. some bla guy — I think you’re right about why 0.999… = 1 is a counter-intuitive fact, but there’s an easy counter. Each finite term 0.99…9 is larger than the previous term; so the “infinite term” 0.999… is larger than all the finite terms. So the fact that 1 is larger than 0.99…9 isn’t an obstacle; in fact, it’s a requirement.

  7. Oh, since I’m here anyway:

    Aaron — Any proof that assumes N = 1 to prove N = 1 is dead in the water. The usual “proof” goes like this: If (1) N = 0.999…, then (2) 10N = 9.999…; subtract (1) from (2) to get (3) 9N = 9, from which N = 1. I say “proof” because first you have to establish that arithmetic with infinite expansions makes sense, and it’s usually easier to do some other proof instead.

    ram — Your broad point is correct, but what you’ve described is “dark energy”. Dark matter is mass that we know must exist, because of its gravitational effect on visible objects near it, but can’t see, because it doesn’t interact with the electromagnetic field. I’m pretty sure it would actually slow the expansion of the universe, but don’t quote me on that.

  8. I always use this quick explanation to the layman:
    1/3 (one third) can be represented by 0.333…
    If you take each thirds and add them up (0.333.. + 0.333… + 0.333…) they add up to 1.0, not 0.999…

  9. A very simple way to solve this:

    Take two numbers 3 and 5; now, to see if they are equal we can try to find a number between them. Well, a number like 4 or 3.75 is between.

    So, now let’s take 0.999… and 1.0. Can you find a number that is between an infinitely repeating set of 9’s before it goes to 1? No, there is no number between 0.999… and 1. If you truly figure 0.999… as an infinite string of 9’s then there is nothing before you would have to round up to 1.

    However, for practical purposes we have to round to a finite number. A finite string of 0.9999’s is only equal to one because humans can’t work/comprehend an infinite string of 9’s.

  10. You object that, when asked whether .99999…=1, we view .99999… within “our system”, whatever that means. Well, of course we do! How else could we possibly interpret the question or try to answer it?

    When someone asks whether .999… equals 1, they are most certainly asking within the context of the real numbers. Switching gears and trying to interpret the question within the hyperreals is as arbitrary and evasive as choosing to instead view it in the p-adics, where another different (but equally valid) answer could be given.

    This is silly. This “argument” really needs to be put to rest. Mathematical rigor exists precisely for this reason.

  11. Ah, I knew this would this would invite the curiosity of students and the ire of pedants! :)

    @Aaron: Great question. These types of proofs make assumptions about how addition and subtraction would work with these infinite decimals (does 9 * 0.999 = 8.999…?), but they do work for the regular number system (see http://math.fau.edu/Richman/HTML/999.htm).

    @some bla guy: I’d love to have a word with Cauchy — I bet he’d be interested in learning about new number systems that can rigorously approach the same problems differently!

    @mau: Neat — I’ll have to see how well Google translate does at math :).

    @ram: Whoops, thanks for the correction! Yes, I meant anti-matter :).

    @Chad: Great points, thanks for the discussion! I guess it depends on the meaning of 0.999…, which is indeed ambiguous. I think the better phrasing may be “The hyperreal number uH=0.999…;…999000… with H-infinitely many 9s, for some infinite hyperinteger H, satisfies a strict inequality uH < 1" (from Wikipedia). I think the higher meta-point is figuring out the question behind the question -- the layman isn't asking about 0.999... as constructed in the real number system. They want to know what happens when one number gets "infinitely close" to another -- can this be represented? 0.999... is the most convenient form of this question (also see 1/infinity -- does this equal 0? Yes, if you take the limit approach, no if you take the hyperreal). @haileris: Does 1/3 = .333.. exactly, or is 0.333... different at the infinitely small level? :) @Adam: Great point -- because our current number system cannot represent the difference between 0.999... and 1 (there's no number in-between), in our current system they are equal. However, other systems allow it, so I take the approach of "it depends". @Jeff: Thanks for dropping in, but I disagree that it needs to be put to rest. Transform the problem: if it's 1600 and someone asks 1600s Jeff what does sqrt(-1) mean, what do you say? That they are asking this question in the context of the real numbers, and the answer is undefined? How else do you answer it? The alternative is to explore a new number system (complex numbers, hyperreal numbers) and see if it has interesting properties. You can't take the question at face value, it's really about exploring the nature of infinitely small numbers.

  12. Hailis has the answer and yes, Kalid, 1/3 _does_ equal 0.333… The numerator is really 1.000… and the division continues ad infinitum. To say that 0.999… does not equal 1 is to say that neither 3/3 nor 9/9 equal 1. I would love to hear your explanation as to why the rules for reducing 9/9 are different than those for reducing 8/8 or, for that matter, 1/1.

    Additionally, I don’t think there is much assumption involved when considering how addition might work with these particular infinitely repeating decimals. Try adding 1/3 and 1/7. When expressed as a decimal, each has an infinitely repeating sequence; yet we can identify a very specific and uncontroversial answer: 10/21.

    If we were discussing 0.12341234… it might be a different story; that number is not rational. 0.989898… comes close to 1 but never touches, which makes it an interesting candidate for the tolerance and accuracy portions of this discussion. But nobody is proposing that 0.989898… equals 1.

    0.9… is indeed a special case, but it is not the number-line equivalent to infinitely-close-but-not-touching (like the way my fingers don’t actually touch my keyboard as I type – there is a tiny gap between the atoms). 0.9… is 9 * 1/9. It’s a concept we can imagine and denote, but it doesn’t really exist as a unique number. It is, in truth, 1.

  13. @Ogre_Kev: I agree that in the current real number system, .333… = 1/3. But what this means is this:

    “The infinite sequence (.3, .33, .333, .3333…) converges to the limit 1/3”, which is another way of saying “We can make an element of (.3, .33, .333…) as close to 1/3 as we wish”.

    You might want to check out http://math.fau.edu/Richman/HTML/999.htm:

    “Perhaps the situation is that some real numbers can only be approximated, like the square root of 2, whereas others, like 1, can be written exactly, but can also be approximated. So 0.999… is a series that approximates the exact number 1. Of course this dichotomy depends on what we allow for approximations. For some purposes we might allow any rational number, but for our present discussion the terminating decimals—the decimal fractions—are the natural candidates. These can only approximate 1/3, for example, so we don’t have an exact expression for 1/3”.

    So, as long as we stay in the real number system, 1/3 is the limit of .333… [which is fine, but we don’t have to stay in the real number system; others can capture the idea of what we mean when we say infinitely close].

    As a side comment: if 3/10 is not 1/3, and 33/100 is not 1/3, at what point does another digit make it exactly 1/3? This is a bit like Zeno’s paradoxes, which have not been fully resolved :). The meta-point is that we can make that sequence as close to 1/3 as we need, which in the real number system means they are equal in the limit.

    @Igor: I think it’s possible to sketch out ideas intuitively and return with rigor to cement the foundation — Calculus developed this way, did it not?

    @Michael: Great question — I think all numbers may be equal abstractions of the mind. The real number system may be “less real” because it’s more limited than others.

  14. “The infinite sequence (.3, .33, .333, .3333…) converges to the limit 1/3″, which is another way of saying “We can make an element of (.3, .33, .333…) as close to 1/3 as we wish”.

    Not exactly. It means that most of the elements of {.3, .33, .333, …} are close to 1/3. Here “most” means “all but finitely many”, and “close” means “within any predetermined positive distance”. That’s how you get that limits are uniquely determined. The terms of a sequence can’t all be clustered around a and all be clustered around b.

    “As a side comment: if 3/10 is not 1/3, and 33/100 is not 1/3, at what point does another digit make it exactly 1/3?”

    There isn’t one. The limit of the sequence is not (generally) a term of the sequence. See comment #9 (my response to “some bla guy”).

    “This is a bit like Zeno’s paradoxes, which have not been fully resolved.”

    Sure they have, in large part by the limit concept. (If they hadn’t been resolved, even Newtonian physics wouldn’t be possible.)

    “The meta-point is that we can make that sequence as close to 1/3 as we need, which in the real number system means they are equal in the limit.”

    Limits are unique in the hyperreals as well. You might have infinitesimal separations, but you also have infinitesimal resolving power (if that makes sense). Be careful here: a lot of “standard” sequences with limits, such as {.3, .33, .333, …}, don’t have limits in the hyperreals unless you make some canonical extension to a hypersequence; if you do that, the extension will still have the original limit (in this case 1/3).

    “The real number system may be “less real” because it’s more limited than others.”

    Not so much. See, the construction that Robinson applied to the reals to get the hyperreals can also be applied to the hyperreals. If we call the result the hyperhyperreals, well, we can apply the construction again, to get the (hyper)^3-reals, and so forth. Each is “less limited” than the previous, but none of these can be the “real” system, because each is “more limited” than the next. But really, none of these is more limited than the others, because the same expressible facts are true in all of them.

    To me, the “real” system is the simplest system which easily models the phenomena we’re interested in—which in this case is the ordinary real line.

  15. By the way, you might read Fred Richman’s article a bit more carefully. The system he creates is one in which 0.99… and 1 resolve differently, but it’s also one in which negation and multiplication don’t make sense. So, fair enough, such systems exist, but I wouldn’t want to work in any of them.

  16. @Chad: Thanks for the info & discussion! I’m not rigorously versed in the details, so am learning as I go along :). As far as how “Does 0.999… = 1?” is interpreted by most mathematicians, here’s my guess:

    * 0.999… means “continue .999 in the obvious way”
    * It is not common to define real numbers as a sequence of decimal digits (though not impossible). We prefer to construct a real number as a Cauchy Sequence of rationals (for example).
    * 0.9, 0.99, 0.999, … is the obvious Cauchy Sequence representing that infinite decimal expansion
    * Now that we have a sequence, I see you are using the equals operator. Like a compiler doing integer to floating point conversion, I’m going to “cast” the sequence into a real number (if possible) by taking the limit of the sequence, and compare that to 1.

    So, as long as we’re staying within the real number system, 0.999… interpreted this way means 1 (and .333… = 1/3). But is that the only interpretation? If we interpret 0.999… as possibly referring to a hyperreal number (1 – h) then what conclusions can we draw?

    I think there’s a notion of an “infinitely small gap” that’s we cannot describe with real numbers that leads to interesting approaches.

    It’s interesting to me that early physics was developed with the use of “non-rigorous” infinitesimals; clearly there is a concept there (being able to manipulate dy and dx independently, not taking them as an operator) that was not captured in the current real number system. If there’s a number system (the hyperreals) which can explicitly capture that idea (vs. breaking the rules in the current one) I think it’s more useful for that purpose.

    So, by “limited” I don’t mean less capable, but not as innately useful/expressive (you probably know, but most programming languages are equally powerful (Turing Complete) but differ vastly in how useful/usable they are). I agree about the hyper^N reals, I had suspected that too :). But I don’t know of situations where we’re trying to solve problems by relying on 2nd-order infinitesimals and having to work around it in the current one — if we were, I’d suggest that system as the most expressive.

    I appreciate the clarification on the Richman piece — he does say it’s an open problem. I’m interested in going through http://www.jstor.org/pss/2316619 which expounds on infinitesimals and their representations further (http://en.wikipedia.org/wiki/0.999…#Infinitesimals).

  17. No problem—this whole discussion is helping me clarify a lot of these ideas as well.

    Here’s the thing: if you want to do calculus with infinitesimals, first you have to do arithmetic with them. And that leads to problems, if you also try to do arithmetic with infinite decimal expansions. If you want .99… to resolve to 1 – h, with h infinitesimal but nonzero, then does 1.99… resolve to 2 – h or 2 – 2h? Both make sense. (And don’t say that 2 x .99… is 1.99…8. That has an 8 in the “last place”, and there is no last place.)

    Interestingly, though, if we let go of decimal expansions and consider arbitrary sequences of numbers, we get awfully close to hyperreals. In one “hyper” construction, the hyperreals are precisely the sequences of reals modulo a certain equivalence relation. Sequences which have equal terms at most indices are considered equal, and statements which are true at most indices are considered true.

    For example, {1,2,3,…} represents an infinitely large number (on account of most natural numbers are larger than x for any fixed real number x); call this number w. Its reciprocal sequence {1,1/2,1/3,…} represents the infinitesimal 1/w, and {.1, .01, .001, …} represents the infinitesimal 10^(-w) (which we’ll call h), and {.9, .99, …} represents 1 – h; but {0, .9, .99, …} represents 1 – 10h, so there’s some nasty ambiguity in (.99…). Also {1.9, 1.99, 1.999, …} represents 2 – h, and {1.8, 1.98, 1.998, …} represents 2 – 2h. So we can get enough infinitesimals to do calculus, but we have to go beyond decimal expansions to do it.

    “We prefer to construct a real number as a Cauchy Sequence of rationals (for example).”

    Since we’re being technical here anyway: Cauchy sequences of rationals only represent real numbers. We still have to specify when two Cauchy sequences represent the same real number; and {a_n} and {b_n} do this precisely when {a_n – b_n} converges to 0 in the rationals. In particular, {1, 1, 1, …} and {.9, .99, .999, …} represent the same real number (which is 1), because {.1, .01, .001, …} converges to 0.

    I’ll stand by my original position, more or less: there’s no unambiguous way to interpret the infinite decimal expansion of a fraction x as “infinitesimally less” than x, and still be able to do arithmetic with those decimal expansions. The Wikipedia article you cite backs me up on this, at the end of its introduction: “[S]ome settings contain numbers that are ‘just shy’ of 1[, but] these are generally unrelated to 0.999…”.

    “So, by “limited” I don’t mean less capable, but not as innately useful/expressive (you probably know, but most programming languages are equally powerful (Turing Complete) but differ vastly in how useful/usable they are)”.

    That’s exactly what I’m talking about. The statements which are true on the real numbers are exactly those which are true on the hyperreals — if you’re careful about how you interpret those statements. Nonstandard analysis—that is, analysis with the hyperreals—hasn’t really caught on, and I suspect it’s because proper interpretation is just as difficult to deal with as epsilon-delta argument, with no real gain.

    It might be useful to write a nonrigorous calculus textbook based on nonstandard analysis; in fact, I think it’s been done.

  18. (Geniuses of the time thought negatives “wrapped around” after you passed infinity)

    I am not sure if this is a statement of derision or fact. The idea that the negatives “wrap around” is not that far fetched. The 1 point compactification of the reals is homeomorphic with the circle. In that context it makes perfect sense to think of the negatives as wrapping around at infinity. I allow my students to use the analogy frequently with asymptotes.

  19. I have a few questions. They may sound like objections, but they’re definitely in question form because I am admittedly slightly out of my depth here. These are the main trip-ups that are keeping me from wrapping my mind around what you’re saying:

    How does the fact that we’re working all this stuff about 0.999… in base 10 effect the issue? If it’s not equal to 1, then it’s obviously not a rational number, but can it be expressed, or even approximated, in other bases? In this alternate number system you propose, is 0.777… in base 8 equal to 0.999… in base 10?

    And I know this has been touched on in the comments already, but I’m still wondering about the (1/3)*3=1 angle. I take it that in this new system, (1/3) wouldn’t be equal to 0.333…, for the same reasons as with 0.999… and 1. So does this mean that (1/3) cannot be calculated?

  20. Stephen: To an extent, base doesn’t matter. If we take 0.9… in base 10 as infinitesimally less than 1, then the same is true of 0.1… in base 2, and 0.7… in base 8, and so forth. And you’re right about 0.3…; under this scheme, it would be infinitesimally less than 1/3. The problems show up when you try to nail down that infinitesimal and do arithmetic. Consider the argument I make in #22: if 0.9… = 1 – h, do we have 1.9… = 1 + 0.9… = 2 – h, or do we have 1.9… = 2*(0.9…) = 2 – 2h?

    Come to think of it, this is a better argument for 0.9… = 1 than the 3*(1/3 = 0.3…) idea, because it eliminates the possibility of “infinitesimal separation” entirely. These last two equations are consistent precisely when h = 0.

    I’m not quite sure what you mean by “1/3 cannot be calculated”. It would mean than no decimal expansion evaluates to precisely 1/3. However, since “infinitesimal separation” falls apart when we try to be precise and do arithmetic with it, this is an academic concern.

  21. @Chad: Thanks for the discussion and for helping out with the questions! Yes, Keisler’s book seems to be an excellent resource, and I’m going through it to really understand calculus at a deeper level than when I studied it originally (many proofs seem to fall into place using “algebra”, like the proof of the product rule).

    All of this has got me thinking about analysis — I’m sure I’ve made some technical errors in the post that I need to correct. The goal is to start the discussion and embrace the idea of infinitesimals :).

    @Jason: Not meant to be a statement of derision, but more “The geniuses can have trouble thinking about new when they are first introduced, too”. That’s really interesting that the negatives can take that interpretation — though I’d be very surprised and impressed if that’s what Euler and others had in mind. The meta point being that we present math all neatly packaged up, even though it took centuries of debate and revision to get there [like pretending Shakespeare wrote his plays in a single draft, a single sitting, and implying to students “that’s how poetry is done”].

    Actually, I wonder if this has come full circle: negatives were thought to wrap around, this interpretation was ignored/found not useful by many (the majority of people do not know this interpretation, I posit), but later found useful. Infinitesimals were first thought to be useful (Leibniz), later thought nonrigorous by the majority, and then later found to be useful and rigorous.

  22. It’s simply hard to envision hyperreal numbers doing to mathematics what complex numbers did. Chad’s examination solidifies my view.

    On the other hand, I have read much of the Jerome Keisler book. I think that someone who truly understands Calculus through the conventional approach is on equal footing with one who understands it through the infinitesimal approach. However, understanding comes more easily, at least for me, from the latter. The major benefit is that “infinitesimal calculus” comes with the visual interpretation.

  23. I love that the article is titled “a _friendly_ chat…” and starts with lambasting the supposed ire of pedants.

    Who finds infinitesimals useful now – how were they used recently?

  24. I was going to post that same proof!

    I don’t get the “if you use a new system, 0.999… does not equal 1” bit. If they weren’t equal then there would be a fault with that simple proof.

  25. #31: There’s no “fault” with the proof, but consider all the hidden premises. In order for #30 to work, we must have a unique interpretation of integer numerals and infinite decimals as numbers, and notions of +, -, *, / that do what we expect. If anything there fails in a given number system, the proof tells us nothing about that system.

    Give you an example: Let Q be the set of rational numbers, and let Q* be the set of all downward closed subsets of Q. That is, if X is an element of Q*, then it is a subset of Q, and if p ∈ X and q < p is rational, then q ∈ X. Q* then is ordered by the subset relation ⊂.

    What sets exactly does Q* contain? Well, the empty set qualifies: everything it contains is a rational number (because it doesn't contain anything), and it's downward closed for similarly silly reasons. Also the entire set Q qualifies: it's a subset of itself, and downward closed (because it contains all rationals, less than p or not). Beyond that, take any real number x; then the sets xlow = { p ∈ Q : p < x } and xhigh = { p ∈ Q : p ≤ x } are in Q*. xlow and xhigh are distinct precisely when x is rational. It can be shown that every element of Q* takes this form, and that they’re all distinct.

    Now: let’s say we interpret every integer or rational numeral n as the set nhigh. And let’s say we interpret the infinite decimal 0.abcdefg… as the set of all rationals p where p ≤ 0.a or p ≤ 0.ab or p ≤ 0.abc or … (which would be well-defined). Then the interpretation of 0.999… would be precisely 1low, which is distinct from 1high, which is our interpretation for 1. Similarly 0.333… would evaluate to (1/3)low.

    This whole interpretation is at least superficially reasonable. And #30 doesn’t apply, because we haven’t even defined +, -, *, /, let alone verified that they behave sanely.

    If we tried to do that, we’d quickly have to eliminate ∅ and Q as valid numbers, and identify plow with phigh, at which point we’d essentially have Dedekind’s construction of the real line. But if we don’t bother with arithmetic, we can interpret rationals and infinite decimals reasonably and still not have 0.9… = 1.

  26. Apparently this blog doesn’t recognize the <sub> tag. I hope what I write is still comprehensible.

  27. @Chad: Thanks for the details! I totally agree about the hidden premises.

    Intuitively, I also see the argument like this:
    “Can x^2 = -1? Well, if x is positive, x * x = positive. But if x is negative, x * x = positive. And if x = 0, x * x = 0. Therefore, sqrt(-1) cannot exist”.

    There’s a hidden premise about what x is allowed to be.

    So, looking at the argument

    x = 0.999…
    10x = 9.999…
    10x – x = 9.999… – 0.999…
    9x = 9.0

    We need to take a break and see what’s happening. Does 9x = 9.0? Hrm. Let’s multiply it out
    9(.9) = 8.1
    9(.99) = 8.91
    9(.999) = 8.991
    9(.9999) = 8.9991

    And so on. So is 9 really the same as 8.999…1? :).

    In fact, if we look at the limits involved, it’s a restatement of the first equality. Let’s assume each limit has an implicit n->inf.

    x = lim[ 1 – 1/10^n]
    10x = 10 lim[1 – 1/10^n]
    10x – x = 10 lim[1 – 1/10^n] – lim[1 – 1/10^n]
    9x = lim[(10 – 1) – 10/10^n + 1/10^n]
    9x = lim[9 – 9/10^n] => this is where we get 8.1, 8.91, 8.991…
    x = lim[1 – 1/10^n]

    It seems the argument is a bit of a tautology, and reduces again to x = lim[1 – 1/10^n]. The question is then whether this is exactly 1. It is, if we disallow the idea of a number too small to detect with the reals (like disallowing an imaginary number). But if we allow the possibility of a difference in our premises, then we can state that x = 1 – h [where h is that tiny infinitesimal difference we couldn’t notice with the reals].

    I’m not sure how rigorous this is (it probably isn’t) but it’s how I’m starting to see the implicit assumptions in the 10x – x argument.

  28. I usually would explain this by saying, lets find out the difference between 1 and 0.999…


    1.000… –

    If you follow the subtraction to infinity and beyond your answer of the difference is 0.000…

    So if the difference between 0.999… and 1 is 0.000… that means there is no difference between the two!

    0.999… is an artefact of the decimal system, because some values cannot be represented by terminated decimals. e.g. 1/3 or 0.333…

    If you argue about whether 0.999… is or isn’t equal to 1, then you need to argue if 0.333… is or isn’t equal to 1/3. Along with a whole bunch of other numbers which don’t have terminating representations in decimal.

    The problem comes from infinity being involved. Damn you infinity! And people’s ideas that a number is an exact thing, how can there be more than one way of representing 1?

    But we have n^0, n/n, cos(0).

  29. Dark matter vs Anti-matter

    You say that (Dark matter destroys regular mass when they come in contact, just like 3 + (-3) = 0).

    Assuming you mean Anti-Matter this also isn’t correct.

    Matter is not destroyed, it is converted to energy.

    The equation is more like

    3 matter + 3 anti-matter = 6 energy.

    I don’t think anyone has managed to find a case for when the law of conservation of energy is not true.

  30. Andrew: That’s what I was arguing at first. Here’s the problem: how do you know that subtraction from the left, as you’re doing, is valid? It’s one thing to explain an idea, and quite another to defend the same idea against a skeptic. You have to go back to common ideas, possibly to first principles, and then you have to defend those principles intuitively.

    Kalid: No, 9 ≠ 8.99…91, because there’s no such number as 8.99…91. Not with infinitely many 9’s hidden in the “…”, anyhow. However, we do have 8 < 8.1 < 8.9 < 8.91 < 8.99 < … so the two “intertwined” sequences should have a common limit (if they have one at all), and your argument can be taken to show that 9*(0.9…) = 8.9….

    For the real numbers, the intuition is that of “continuous quantity”, or “length”. We can add and subtract lengths; we can agree on a unit length and use it to multiply and divide (otherwise length*length = are and length/length = ?); and we can compare lengths. Moreover all these operations are compatible in ways that are familiar to anyone who made it through high school math (commutative, associative, distributive, etc.).

    But that’s not quite enough to get all the possible lengths. There should be a quantity x where x^2 = 2; we can even construct it with compass and straightedge. But there is no rational number which satisfies the equation, even though the rationals have +, -, *, /, and <. For that matter, there are lots of numbers—e.g., 2^(1/3), π, e—that qualify as lengths but can’t be constructed by compass and straightedge. In general, if we have a normal, continuous function on an interval, and it’s negative at one end and positive at the other, then it should be zero somewhere in between—no line-jumping.

    The way we get that—pretty sure the only way we can get that—is as follows. Say we break our quantities into two sets P and Q. Every quantity is in exactly one of P or Q. Moreover P is downward closed: if x ∈ P and y < x, then y ∈ P also. This makes Q upward closed by default. Essentially we’ve split our line into two coherent halves.

    The intuition we appeal to—and by “we” I mean “originally Dedekind”—is that any such split should correspond to an actual quantity. That is, there should be some quantity x which is either the greatest element of P (and less than everything in Q), or the least element of Q (and greater than everything in P); and the split is taken to happen at x.

    So, for example, take an increasing but bounded sequence like x_n = 1 – 10^(-n), and take P = {x : x < x_n for some n} and Q = {x : x > x_n for all n}, and find the splitting point y. y is in Q, because if in P it would have to be the greatest element of P but still less than x_n for some n; but it is the *least* element of Q (that is, the least upper bound for {x_n}). We take y to be the limit of x_n. If there is no least upper bound, then there can be no limit. (There’s a similar idea for decreasing, bounded-below sequences; general sequences are trickier.)

    Let’s look at the sequence 1 – 10^(-n). Certainly 1 is an upper bound for this sequence. Is there a smaller one, say 1 – h for h positive? Well, if 1 – h is an upper bound, then so is 1 – 10h, since 1 – h ≥ 1 – 10^(-(n+1)) implies h ≤ 10^(-(n+1)) implies 10h ≤ 10^(-n) implies 1 – 10h ≥ 1 – 10^(-n); and this goes through for all n. And 1 – 10h < 1 – h. Thus (rewinding a bit), if 1 is not the least upper bound for x_n, then there is no least upper bound (every upper bound less than 1 can be shown not to be least, and upper bounds greater than 1 obviously don’t work). So either .9… = 1, or .9… doesn’t equal anything (or arithmetic is broken).

    So I guess I’ve come full circle on this issue. It’s true that any argument in math rests on certain assumptions about the context, but if we use infinite decimal expansions, we’ve internalized all those assumptions. We must either reject arithmetic, reject the .9… notation, or accept that .9… = 1.

  31. 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?

  32. 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.

  33. 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)


    .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?

  34. @ 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?

  35. @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.

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

  37. @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?

  38. 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.

  39. @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! :)

  40. @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.

  41. @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.

  42. 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

  43. @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.

  44. @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.

  45. 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.

    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…

  46. @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? 😀 ) 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.

  47. 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.

  48. @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.

  49. 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?

  50. I like this view:
    let X=0.999…
    then 1+X=1.999…
    also X+X=1.999…
    so 1+X=X+X and thus 1+X UNLESS we want to junk the rules of arithmetic that we like to use.

  51. Okay, let’s say the error margin concept is true.
    Then I might as well say that a particle has no mass, since we can’t possibly measure it – nor has it any volume since we can’t measure that either.
    I don’t think that makes it matematically correct to assume things, based on the fact that we can’t measure it.
    Hence, 0.999′ will never equal 1 either. It’s convenient for mathematicians, no doubt – but that’s about it! c”,)

  52. Oh, and by the way…haileris wrote in #11:

    1/3 (one third) can be represented by 0.333…
    If you take each thirds and add them up (0.333.. + 0.333… + 0.333…) they add up to 1.0, not 0.999…

    This argument assumes falsely that 1/3 can be represented by 0.333′.
    The flaw in this one is, that 0.333′ is NOT 1/3, only an approximation(!)

  53. I don’t think it should be a question of wether 0.999… = 1, but a question of wether 0.333… = 1/3.

  54. Yes, 0.999… = 1. OK, fine, but will someone please tell me what the next real number is???

    A Student

  55. I found this website after the uncomfortable explanation on what the e (constant) truly meant. And the moved to some other posts, as I loved the intuitive grasp of e that now makes perfect sense. I think the central issue is communication, where I define it as understanding the true meaning of each message exchange. I see a lot of people struggling with understanding other people all the time, UNKNOWINGLY. When you don’t know the underling (or hidden) premises, both can be talking about a different topic, and thinking they are talking about the same thing.

    For an amusing (at least to me) example, see the airplane in a conveyor belt myth…will it take off? See the comments to the myth buster video here:


    If you read the more than 80 pages of comments, you’ll realize that those that where “not right” are those that understood the problem different. And that’s probably several thousands of them. But until they saw the video, they couldn’t stop arguing that the plane cannot take off. And rigthly so: the instructions somehow implied to most that the plane remained stationary, probably using the wrong analogy with cars. They though (myself included) that the plane couldn’t advance, as part of the “assumptions”. There are several webpages with hundreds of posts and interactions. Then there where those that had it right for the wrong reasons (they started agreeing that a stationary plane couldn’t take of…there’d be no lift), yet they developed all kind of magic because they believed it could take of anyhow. Then some others interpreted all kind of weird ideas about the setup. For a sample discussion at ~280 comments (closed to new posts for sanity), you can go to:


    This is the problem description:

    “A plane is standing on a runway that can move (some sort of band conveyor). The plane moves in one direction, while the conveyor moves in the opposite direction. This conveyor has a control system that tracks the plane speed and tunes the speed of the conveyor to be exactly the same (but in the opposite direction). Can the plane take off?”

    What’s amazing is that when Mythbuster did the test, many people argued (as I already said) that the setup is not what they expected: that it OBVIOUSLY takes of, as it can advance in the belt.

    WHat I mean with all this is that there’s one thing I like about better explained: the analogies, the relationships approach, the examples and the illustrations are really helping a lot of people get on the same page, and communicate on the same level. And also (example: this thread) to challenge our axioms as something natural, that might even lead somewhere. I like it a lot. Thanks Kalid.

  56. About 4 years ago, I was a bit stressed and took a month off. I am a bit stressed as well, so I tried to understand the meaning of e, pi and i, in a way that would make me understand them (use them intuitively in new forms, as if those were tangible objects).

    I loved the explanations and then, by curiosity, I stumbled on this post, which reminded me of a week I spend looking at prime numbers just for fun 4 years ago.

    I was looking at the meaning of prime numbers beyond the obvious “divisible by 1 and itself only”, and felt that it was very much like asking if what Color is 42 (in truth, 42 is not a call….it’s actually THE answer to the meaning of life and everything….eh).

    In my quest to try to make meaning, I started to think about concepts like notation limitations (is the notation lacking to get at where I want to get to). About what is the relationship of infinity to unity, how does one traverse from systems that “add up” (1,2,3,…) to systems that make groups (1/2, 1/3, 1/4, …). Is it possible? (I liked the concept of hyperreals which I wasn’t aware off). I also though about periodicals. For example, if I can say 0,999… where there are infinite digits that can take me to 1…why can’t I say 1…… (infinite string of 111…, no decimals). If .9999 is close to 1, is 999… equal to infinity? And is that infinity some kind of “1” in a higher level view of things? What is that “1” plus 45? At some point I didn’t know what I was getting at, but I assumed that if there’d be a way to bridge systems that go like (1,2,3,…) and systems that did (infinite/2, inf/3, inf/4) where infinite you can see as a higher order “1” (hyperreals looked a bit like this and catched my attention), then I would have answers to some questions.

    My conclusion at reading 0.9999… is that it is the same as asking if 9999… exists. And if it exists, then “1” is of some kind of higher order…and what we could think as the “universe of ways to break apart that higher order number). 0.9999… is the way to express infinity in the lower order system (in terms of units starting with 1,2,3…).

  57. @Federico: Thanks for the comment. You’re completely right — communication is key. All those unstated assumptions and premises can mean we completely miss the point of what each other is thinking. In my head, the process goes “I have an idea. I write it down. You read the words. You recreate the idea.” If my description was fuzzy or left out key information, the idea you recreate will not be the same one I had, leading to endless confusion (great example with the conveyer belt).

    I find it really helpful to talk about the intuition of things (i.e., describe the idea in your head) vs. focusing on facts/details, which are the *results* of that idea, which you hope the person reading will re-create properly. Thanks again for the thought provoking comment.

  58. if would like to answer this question through simple divisibility.
    consider 1/9
    as we can not directly divide 1 by 9 as it is greater than 1.so we adjust a decimal point.at first division we get numerator=0.1 and remainder 1with decimal value of 1/10 which equals it to 0.1
    we know that dividend=divisor*quotient+remainder
    applying this we get 1=9*0.1+0.1 equation is satisfied

    now similarly at the next iteration we get quotient 0.11 and remainder 1.this time as we should divide this remainder with 100(10^2 as it is 2nd iteration) again applying same rule ie dividend=divisor*quotient+remainder we get
    1=0.11*9+0.01 equation is satisfied again.

    now if we generalize this equation then what we get is
    1=(0.11111…..n)*9+10^(-n) that is why we can state than 1/9 is not exactly 0.1111…. but it is 0.1111….n+((10^-n)/9)to be accurate
    now you wont find any difficulty.

  59. I would like to answer this question through simple division.
    consider 1/9
    as we can not directly divide 1 by 9 as it is greater than 1.so we adjust a decimal point.at first division we get numerator=0.1 and remainder 1with decimal value of 1/10 which equals it to 0.1
    we know that dividend=divisor*quotient+remainder
    applying this we get 1=9*0.1+0.1 equation is satisfied

    now similarly at the next iteration we get quotient 0.11 and remainder 1.this time as we should divide this remainder with 100(10^2 as it is 2nd iteration) again applying same rule ie dividend=divisor*quotient+remainder we get
    1=0.11*9+0.01 equation is satisfied again.

    now if we generalize this equation then what we get is
    1=(0.11111…..n)*9+10^(-n) that is why we can state than 1/9 is not exactly 0.1111…. but it is 0.1111….n+((10^-n)/9)to be accurate
    now you wont find any difficulty.

  60. From the Internet Encyclopedia of Philosophy article on Bernard Lonergan…

    “Each mode of knowing has its proper criteria, although not everyone reputed to have either common sense or theoretical acumen can say what these criteria are. A major impediment in theoretical pursuits is the assumption that understanding must be something like picturing. For example, mathematicians who blur understanding with picturing will find it difficult to picture how 0.999… can be exactly 1.000…. Now most adults understand that 1/3 = 0.333…, and that when you triple both sides of this equation, you get exactly 1.000… and 0.999…. But only those who understand that an insight is not an act of picturing but rather an act of understanding will be comfortable with this explanation. Among them are the physicists who understand what Einstein and Heisenberg discovered about subatomic particles and macroastronomical events – it is not by picturing that we know how they function but rather by understanding the data.”

  61. The first thing to realize is that we are not talking about a number, but a process, and asking whether this process is equivalent to a certain number, which it is. That this is hard to accept is merely a reflection of the fact that we abstract our abstractions into concreteness (forgetting their abstractness) and then complain that our abstractions do not act concretely. 0.999… is not a number. It is a symbol that represents a process that never ends. This is something that cannot be imagined. But it can be understood. Like all paradoxes, however, we have to be very careful (rigorous) with how we describe what we are trying to understand. That is why the first thing we have to realize is that 0.999… is not a number, but a representation of a number. An abstraction of an abstraction. It is the symbol itself which causes the confusion. 0.999… is just a shorthand way of writing “9(1/10) + 9(1/100) + 9(1/1000) …” The conjecture that follows is, all (whole? rational? real?) numbers are the result of an infinite process. The interesting thing is this relationship and whether it is general. ?

  62. I only got to scroll down 1/3 of the bar (no just kidding, 2/5ths of the bar), I’ve got school tomorrow.

    I quote only myself and noone else, and will post no sources. BUT.

    1/3 exists only as a fraction. In decimals it immediately becomes an imaginary number. No 3 equal parts can be added to make one whole.

    3333+3333+3334=10000 etc.

    So in practice, 1/3= 3…+3…+3…+n where n is 1-3…+3…+3…

    Having a loads of decimals doesn’t make it any more difficult, it just becomes a lot more to write down. Heep hooray for n,x,y,a,b,v,t etc:)

    The interesting part of this is the conception of infinity. Can you comprehend?
    Hypothesis: The universe never ends.
    Can you imagine? Do you understand the concept?

    Some people ask “when you reach infinity, what’s beyond that?”
    If there’s something beyond that, it’s not infinity.

    Try to imagine infinity. If you can partially understand the idea and get butterflies or similar in your stomach, you are one step closer to understanding near-infinite numbers 😀

    They do exist, they only near-infinitely doesn’t.

    Hope you liked my 1-(3…)^3 cents 😀

  63. “So in practice, 1/3= 3…+3…+3…+n where n is 1-3…+3…+3…”

    n is 1-3…-3…-3…


  64. “No 3 equal parts can be added to make one whole.”

    (…) to make 1.


    Well you probably understand.. sorry I am tired, long day today, long day tomorrow :) Math can be fun 😀 (considering near-infinite possibilities, noone can prove me wrong ^^ Or.. can they?? But then we’d have to include hypocrisy into the algorhythm! Or… would we??) Ok I really have to go to bed now

  65. Being a freshman in high school, I’ve never heard the word “Rigor” applied to math before. (Not at all a favorite subject of mine, despite it forming the basis of engineering/most other scientific disciplines.) Though it would certainly explain the common question of “Why can’t I just use a method that gets me the ANSWER?” *Ah, the silliness of youth who expect it to be easy.*

    So, if for the sake of consistency we have an abstraction of an abstraction (Of an abstraction, of an abstraction……) at what point do we find our discussion to become founded in the roots of philosophy rather then math? (A very abstract field.) As I think this one has crossed that line. Assuming this is true, from a purely philosophical standpoint, an infinite never ends. (By it’s very definition, and prefix.) So if we are to say that 0.999 always fills another place with nine, then it cannot become a *WHOLE* one no matter how close it becomes to the number in question.

    Just because one cannot reasonably observe a difference between infinite .999 and one does not mean that no difference exists between the two. Once again, from a purely philosophical point of view I see three obvious answers.

    One, 0.999 does not equal one because an infinite 0.9999 will never reach a conclusion that would create a whole, complete, one.
    Two, 0.999 equals one because from the human perspective no device could be invented from our current collective knowledge that would calculate to infinity, thus making the observable answer one.
    Three, The problem is unsolvable because a device cannot be created that can calculate to infinity and as such it is a fools errand to make assumptions about the inherently unknown.

    All three of these answers run on various assumptions about the state of our universe and knowing the conclusion to most other things I’ve written on the net, all three of those answers are almost guaranteed wrong.

    Because this is math, not philosophy.

  66. @irrelevant: Interesting comment — I’ll take a crack.

    Whether 0.999… = 1 depends on what types of numbers we allow to exist.

    The common case is that infinitely small numbers do not exist — i.e., a number is either measurable and comparable to other numbers, or zero. In this case, 0.999… = 1 is more of a symbolic equivalence, similar to 4/2 = 2.

    0.999… is saying “What number is implied by the sequence 0.999…” and in more formal terms “What number is the limit of this sequence?”. Because we’re assuming we exist in a number system where infinitely small numbers do not exist (or rather, are 0), then 0.999… = 1 because there cannot be a difference between them.

    If we do allow infinitely small numbers to exist (and hey — we allow negatives, imaginaries, irrationals, and other “strange” numbers to exist) then yes, we could represent the difference between 0.999… and 1 (as the infinitesimal “h”, say) and now we have an answer: 1 – 0.999… = h, a number which is smaller than our real numbers but still not zero. As others have mentioned, this introduces other issues, but the general insight is that the answer depends on whether you allow infinitely small numbers.

    On philosophy: Math makes assumptions/axioms and explores their consequences. Certain mathematical models map better to our present understanding of the universe than others. A thousand years ago, negative numbers would have been baffling (they were only invented in the 1700s, by the way). Today, I don’t think the world would work without them. They started off as mathematical curiosities and we found ways to incorporate them into the problems/situations we face.

  67. IT makes me sad to see all of this useless talk
    1 does not equal finite number 0.999, there is no tolerance in finite numbere.
    if you make the following statement that 1/9=0.111 is again wrong! 1/9=0.(1)
    it is a never ending calculation of 1/9.. Ex:{10/9=1 r 1}
    witch ca be expressed as 1/9. So 0.(1) cannot be multiplied with any number unless expressed as a fraction! The same works for any finite periodical non-complex number. Artifices like error tolerance are needed for complex numbers that cannot be expressed in simpler terms. In other words: Do not try to complicate things. You become smarter by simplifying complicated things

  68. I am putting marks on the number line like this
    (d is the distance between 1 and the position of the mark):

    time 0: I am putting a mark at position 0 (d = 1)
    time 0,9: I am putting a mark at position 0,9 (d = 0,1)
    time 0,99: I am putting a mark at position 0,99 (d = 0,01)
    time 0,999: I am putting a mark at position 0,999 (d = 0,001)

    Where am I putting a mark at time 1?

  69. @Andrei: I’m not sure I understand what you mean by “finite number 0.9999” — the idea is to figure out what the closest number to “0.9999….” (i.e., infinite nines) would be. If we assume infinitely small numbers cannot exist, then the number that statement refers to is 1 since there cannot exist a number between 0.999… and 1. If we allow infinitesimals, then we can represent that “gap”.

  70. We have 1/3 = 0.33333 recurring;
    0.333(recurring) * 3 =>1 (exactly)

    So what stops the division of 3/3 => 0.99999…

    We should be able to get either of these two proper answers from the same division, so what do we have to unlearn about long division to get the answers.

    We need that, in the long division, three into three (without any remaining digits to pull down) is not one, but zero with three left over.

    We drop down the unlisted trailing zero to get three into thirty, which then gets an answer of nine remainder three, and repeat.

    Now these are correct, but unconventional, ways of having the remainders happen, and it is by this method that we can get both
    3/3 => 1 (exactly), and
    3/3 => 0.99999(recurring)

    Without a simple long division example that ‘normal’ folks can do, they will continue to have difficulty with all the logic reasoning and theorem proofs.

    Philip Oakley

  71. I missed a step. It (#80) should have read:

    We have 1/3 = 0.33333 recurring;
    0.333(recurring) * 3 => 0.99999 (recurring);
    while 1/3 * 3 => 3/3 =>1 (exactly).

    And this leads onto defining a suitable long division for each case.


  72. Heh, from everything I read I concluded that definition 0.99999…=1 itself is absurd because it all boils down to quantisation of principal for counting for any integer radix. In essence there can only exist integer numbers because dividing 1 allready means that that so called selected 1 is only group of smaller individual even integer units that represent quanta of one unit that holds meaning for practial use.
    I may offend every mathematition in the world, but whole math always operates (sum,sub,multi,div)only with integer numbers, its only result that may seem noninteger, all that left is to find smallest (whith what we can practialy operate to put the result for use) radix, where radix can be n->infinity. Maybe someone can prove otherwise.

  73. the answer for 0.9… = 1 stands in ONE question:

    now if you prefer 0.9… questions, here we go:
    what is water, cat, tree, air, oil, words, egg, chicken, man, what, where, when, how, who, W aka double you,

    one there’s just one, but dead can’ t see.

  74. between any two real numbers we can find infinitely many real numbers. But here between .999999999…. and 1 we cannot find a number . therefore 0.999999….=1

  75. I’m not a mathematician so excuse me if this comment appears naive or irrelevant.
    How can any infinite number (eg. 0.3recurring) have any mathematical process applied to it when the concept of infinity is beyond human comprehension? Surely a person can only apply a mathematical process to a number of which it knows its totality.

  76. @Joseph: No worries, all questions are welcome! That’s a great question — personally, I’m not sure it’s easy to put things like infinity into a category of things we cannot understand. Concepts like zero, negatives, imaginary numbers, etc. befuddled us for a while (are are extremely unnatural), but we managed to make sense of them. Similarly, there are ways to think about infinity but we may not have found the simplest explanations for them yet :).

    Numbers like .333… are only infinite because we count in terms of tens (there is no simple way to represent 1/3 when your counting system is based on 10). But, the babylonians counted in terms of 60 (how we have 60 seconds in a minute, and 60 minutes in an hour!) and when counting with “60”s you can get 1/3 without an issue (and no repeating decimals).

    So, I think sometimes our confusion is the result of using an awkward system vs. a limit to our understanding.

  77. Thanks Kalid. That’s a perspective I hadn’t considered, and one to which I’m going to give some thought.

  78. It can possible to find the answer if we device a method to
    square,cube,etc. 0.9999… .

    If it tends to Zero it should be less than One. If it does not it is equal to One.

    Also,we can only logically understand a number if we know its neighbors(Yes even Numbers succumb to relativity!).

    Lets call 0.99999… “p” [ “p” for promisingly progressive]

    And lets call 0.000……0001 “i” [ “i” for infinitesimally interesting] :p

    Now trying to find the “left side neighbor” of p.

    lets call it “n”.

    p – (1-p) = n

    2p – 1 = n

    1 – p = i

    [Let me be frank over here.. i has to be non zero for n to exist AND n has to exist if we want p to make sense in our realm of rationalization]

    (1-p)^2 = 1-2p+p^2 ( If we assume 1=p)

    (1-p)^2= 0..that would help if we want to allow to make i^2=0 too. Which will not help as mentioned above.

    Thus more helpful(logically) than not is that:

    | 0.99999…. < 1|

  79. the multiply/substract with infinite numbers is too difficult for my brain

    let’s try it with the first finite decimal (where those operations are defined):
    n = 1 (. 10) n = 10 (- 9) n = 1
    n = 0,9 (. 10) n = 9 (- 9) n = 0
    therefore 1 = 0

    multiplying an infinite number is not defined, you either fix it and get a quantity 10 times larger then it’s all finite numbers otherwise the ‘multiplied’ result is another infinite number that you can’t compare and meaning is lost.

    if it’s about limits and approximations then it should be written explicitly, not using an equal sign, right ?

  80. The number of replies here, without resolution of determination.. imply “absolute truth” is a consequence.

    The consequence is not absolute.

    If it were.. there would be determination of absolute truth prior to absoloute truth being determined..

    I post because somewhere else i asked what progress actually meant. and so far as close as i can get.. it includes point of reference.

    If the points of reference are different in consequence, then progress can be anything.. it depends on the points of reference and the consequence.

  81. My idea is the closest positive number to 0 is: 0.0000…01
    It would have to include a 1 since our decimal number system we start counting with a 1 and not some other number. This seems to be the only way to represent the infinetly small. I don’t think
    0.0000…1 is the same as 0, since is 0 the same as 0.0000…1? It is if to say since integer 1 is so close to zero in our counting system, that it is practically 0 when you consider the infinity of integers.

    The two closest numbers to 1 would be:

    higher = 1.0000…1
    lower = 0.9999…9

    If you take the average of the two values above, you get 1.0

    Please leave some comments.

  82. (continued)

    To say 0.9999…9 = 1 is like saying 1.0000…1 = 1
    and therefore you are saying 0.9999…9 = 1.0000…1
    which are the closest numbers to 1.0, but their difference
    is 0.0000…2 which is greater than the smallest number or
    infinitely smallest number of of 0.0000…1

  83. Those doubting Thomases or less gifted ignoramuses or devil-may-care Johns who question the veracity of the elegant equation .99999999…=1 would not have the derring do’s to write 1/3 = .33333…
    whereupon their wisdom of writing such equation gone?
    Accept it or deny it but you can not ignore the charm of it.


  84. This is confusing for me and my type of people. The comments that Chad Groft wrote just might be true you never know whose right and whose wrong. IF ANYONE CAN TECH ME HOW THIS PROBLEM WORKS pleas let me know bye=)?

  85. Hey Kalid I read this somewhere
    how do you explain this

  86. Why is it that we were raised to just simply round the number?
    When doing a mathmatical euqation that .00001 that you take out can make all the difference. If decimals and negative numbers are really not numbers at all, why must we deal with them?

  87. @Madison: Great question. I think this gets to the heart about what a number is. Traditionally, numbers were a count of something (fingers, rocks, sheep, etc.) so everything was straightforward, or in chunks. Over time, numbers evolved to be fractions, decimals, and real numbers — becoming “continuous” in a way. The problem with continuity is that you can keep subdividing it (seemingly forever), and things like 1/3 = .33333… means that “well, we can keep subdividing our smallest counting unit (tenths, hundredths, thousandths) into 3”. Rounding is an easy way to “deal” with this (“just call it .33 and ignore the rest”) but it brings up issues with what we mean by precision, etc. It’s a really interesting issue and something I want to explore more! Appreciate the comment.

  88. Take two real numbers, for example 1 and 2. There are an infinite number of real numbers between 1 and 2. So, you can always increment 1.5 to get a new number that is also between 1 and 2. e.g 1.5 + 0.01 = 1.51. Take (1.999 …) what real number can you add to (1.9999 …) to get a number that is also between 1 and 2? There is no number. Therefore (1.9999 …) is either less than 1 or greater than 2. We know that both of those cases are absurd, so it must equal 2. Right?

  89. I am not a particularly math-centered person and I thoroughly enjoyed this article. It is a great leg up over that stumbling block that lays before Calculus, where people must come to terms with math not being an infinitely precise, binary thing.

  90. @15 0.12341234… is not irrational. It is equal to 1234/9999, and .989898… is 98/99, or 9898/9999, etc.
    Likewise, any repeating decimal is a rational number which can be expressed as a fraction, with the numerator being the repeating digits, and the denominator being the same number of nines, or to put it a bit more mathematically:
    0.abc…nabc…n… = abc…n/10ⁿ-1
    where n is the number of recurring digits, or the length of the numerator in the fraction.
    Similarly, 0.999… is equal to 9/9 or 99/99 or 999/999, etc., all of which = 1

  91. My biggest issue with saying 0.9… doesn’t equal one is the base thing (how would this apply to binary 0.1… and hexadecimal 0.F…). My second-biggest issue is that any “new” arithmetic dealing with 0.9… would have to allow for numbers of the form 0.5…2, etc. (That is, five-tenths plus five-hundredths plus five-thousandths, going on forever, somehow followed by a two. And it’s somehow a “bigger” number than the same thing followed by a one instead of a two.)

    If we allow for this, then all sorts of patterns in decimal arithmetic break. For example, one can normally derive one-twelfth by multiplying the decimal forms of one-third and one-fourth, and this in turn may be done through extrapolation. 0.3*.25=0.075, 0.33*.25=0.0825, 0.333*.25=0.08325, 0.3333*0.5=.083325, and so on, adding a “3” to the middle of the answer. By extrapolation, we find that one-twelfth should equal 0.83333…25. Under normal arithmetic we quite correctly ignore the “25” part (it could be replaced with any other finite sequence and remain equally meaningless), so one-twelfth simply equals 0.83333…

    But in a hypothetical new arithmetic, we would be wrong in ignoring that “suffix”! Which would in turn mean that there is an actual difference in the “value” of one-twelfth in decimal, depending on how we calculate it! For example, ordinary long division applied to 1/12 never involves any sort of “25 at the end”. Yet if 0.999… is less than, not equal to, 1, than 0.83333… would be less than, not equal to, 0.83333…25, right? Madness! :)

  92. Lenoxus, you’re partially right.
    The “alternative system” talked about here is called ‘trans-finite arithmetic’. It’s about as crazy as Cantor’s cardinal calculus. Anything relating to infinitesimals is rather crazy. But it’s terribly useful. The normal rules of arithmetic aren’t completely broken in this language, but they take more care.
    And it is very useful when doing calculus and functional analysis. Suppose you have a function y(x). If you think of dy and dx as an ‘infinitesimal’, rather than as a limiting process, you suddenly allow yourself to do thing like 1/(dy/dx) = dx/dy and other tricks which physicists have been doing for more than a century, with great ease. True, it needs care. But it can be made rigorous, and it’s as useful as it is entertaining.

  93. @Anonymous: Sort of. Do we have infinite precision in anything, and if you do, do you have infinite computer storage space to store that infinitely-precise number, like 1.20000000… (exactly)? If we can get 2 numbers beyond our knowingly-finite error tolerance, very, very close (too close for us to measure) = there.

  94. A simple proof to disprove that 1 is NOT equal to .99999…

    If you assume that .9999… = 9/10, and 1 = .9999..
    It must also follow that 9/10 = 1

    However, that is not true because claiming that 9/10 = 1 would lead us to believe that 9 =10 . That is of course FALSE and mathematically unsound, therefore 1 is NOT equal to 0.999…

    The weakness in all the proofs presented so far to “prove” that 1 =.999.. lie in the FALSE and erroneous assumption that fractions can always be exactly and perfectly represented in decimal form without degrading their true values.

  95. Somehow this discussion has some similarity or relation to the equation:
    1/0 = infinity.

    Of course it has been settled to be UNDEFINED because it is absurd to come up with the result that 0 x infinity =1 . How is it possible to have so many nothings(0) and come up with one? That is only true in the world of fantasy and magic!

    Same thing with infinitely repeating decimals when converting fractions. No matter how many times you repeat it you would never arrive at the true exact value of the original fraction the decimal was derived from. Infinity by definition cannot be arrived at and it cannot reach its end.
    0.9999… therefore can never reach the value of 1.

  96. I’m sorry, I just realized that through my haste I made a fatal false assumption in my previous “simple proof”.

    However, I’m still not buying the claim that 1=.999…
    Infinitesimally close maybe, but never equal to 1.

    I need to make sure next time that I don’t post after I drink my beer. :-)


  97. Earlier I argued why repeating decimals cannot “end” with a “different number”. For example, there can’t be such a number as .0…1, the presumed “nonzero difference” between .9… and 1.

    I feel like my specific argument wasn’t adequately countered. I’d be very suprised if there were a number system in which one-twelfth is not equal to one-third of one-quarter, and that the difference between the two amounted to “0.0…25” or some such.

    I’d like to re-iterate the point in a new form: Are certain rational numbers only expressible in certain bases? Mathematicians say no: If it can be expressed in base 10, it can be expressed in any other base, from 2 upwards – so long as repeating decimals are permitted.

    However, if .9… does not equal 1, then this becomes false, at which point major issues arise. Ordinarily, we might say that .9… is equal to “one minus the smallest rational number”, which in turn means “one minus zero”, or simply “one”. But if we accept the “unequal” argument, then instead we are saying something like “one minus the smallest rational number greater than zero and expressible in base ten“. Which leads to more madness.

    In binary, the number 0.1… is ordinarily equal to 1. If it’s not, then it can’t be equal to 0.9… either; in fact, no binary number would be equal to .9…. This is especially strange because ordinarily a “binary number” isn’t actually a kind of number but a way of describing a number, rather like a language.

    Ultimately, if you reject the equality, you reject repeating decimals in general as meaningful or useful. That’s something you have every right to do (they’re merely a tool for which fractions can usually substitute). What you can’t do is say that 0.9… is a coherent, actual number and that it’s not equal to one. It just doesn’t work.

  98. Also, the problem with the graph with an infinitesimal difference between the .999… graph and the 1 graph, is that you didn’t go on to infinity. You stopped at a finite value. This is analogous to saying 1/2+1/4+1/8… ≠ 1 because you’ll never reach one. Clearly this is false.

  99. This is analogous to saying 1/2+1/4+1/8… ≠ 1 because you’ll never reach one.

    Well, most 0.9… “deniers” would consistently deny that equality as well, for exactly that reason. Above, I posited the question (in different terms) of whether these two sums are unequal to one another, in addition to being unequal to one? I think a consistent denial would involve those being three separate numbers, instead of the same number expressed three ways.

    Another problem is saying you’ll “never” reach one implies a process linked to time, or to some other variable. If 0.9… will “never” reach one, can we say say far it is as of noon today? Has it reached ten digits, or 10^10^10 digits, or more than that even? Obviously, that’s silly. 0.9… is “all the way there”, just like 7/14 is “all the way” to 1/2.

  100. The fact that we say it’s “infinitely” repeating implies that it would never end and it would never reach finality. Coming up with a final result is just an imaginary concept of infinity.
    0.9999… is just an approximation of 1 in pure terms and it is no different from saying 1.000…1 is an approximation of 1.
    This problem arise because in every number system not all fractional forms can be exactly represented in decimal form just as not all decimal forms can be exactly represented in fractional form.
    Just because we can round off a number to its closest value when we calculate real world problems does not mean the rounded number is exactly equivalent to its closest value form.
    This discussion is really more of a discussion on how much tolerance in number
    value is good enough for a calculation process.
    Lastly, is it even possible to come up with an exact value when a number that is dependent on the concept of infinity is multiplied with by number?
    After all, any nonzero number multiplied by infinity would still be infinity.

  101. Infinity is undefined and any process of calculation involving infinite concept is likely undefined.
    Infinite floating numbers can only be approximated with the smallest tolerance possible but it cannot be exactly represented as a whole number or as a fractional form of number. Even when dealing with numbers in different bases some numbers that are exactly represented in one base system would turn out to be not exact when converted in another base system representation. That problem is also true when dealing with different number forms. The fraction 1/3 (base 10) is exactly 1/3 in fractional form but it is only approximated as .3333… in decimal form. Same problems occur with other fractions that result in repeating decimals when we attempt to convert them.
    2^1/2 is exactly 2^1/2 but when converted to decimal form it can only be approximated to be 1.41421…
    It is therefore ridiculous to say that .999… is exactly equal to 1. The correct statement should be .999… is APPROXIMATELY equal to one and we should just be fine with that in our real world calculations where numbers are applied.

  102. 1.00…1 doesn’t mean anything. Increase it by 0.00…9, and it’s 1.00…1 again. So
    1.00…1 + 0.00…9 = 1.00…1? That makes no sense.

  103. And your statement, “not all decimal forms can be exactly represented in fractional form” is absolutely ludicrous.

    “2^1/2 is exactly 2^1/2 but when converted to decimal form it can only be approximated to be 1.41421…
    It is therefore ridiculous to say that .999… is exactly equal to 1.”

    2^1/2 is not rational and cannot be represented by a fraction. .999… is a repeating decimal and therefore IS rational.

  104. Rational or irrational, INFINITELY repeating numbers are just approximations of the fractional number form it’s trying to represent.
    .999… has no exact fractional representation in our base ten numbering system and it is NOt equal to 1

    Repeating numbers should only be treated as the final approximated answer value in any calculation. Using it to process an algebraic equation and assuming( falsely) that it could be an equivalent representation to some other number in order to simplify or cancel out terms would result in an incorrect and imprecise proof.

  105. This IS really interesting, let’s see if I can add something even more mind breaking:
    so 1/3= 0.3…+1/3h using an arbitrary close precision, then
    also 1.9…=1+0.9…+h and is different from
    2*0.9…=1.9…8 where the difference is exactly h to get 1.9… and 2h to exact 2.
    Seems to work:)
    Even when using infinite numbers, look:
    h is an infinitesimal to 0.9… and multiplying them by infinite gives an infinite number and h(infinitesimal)*infinite=a number, the practical difference is still infinite among them.
    There’s now the problem that 1/3h or any other of is fractions is smaller than h, a supposedly infinitely small distance…

  106. Mike on March 2, 2013 at 3:14 pm said:
    “Also, the problem with the graph with an infinitesimal difference between the .999… graph and the 1 graph, is that you didn’t go on to infinity. You stopped at a finite value. This is analogous to saying 1/2+1/4+1/8… ≠ 1 because you’ll never reach one. Clearly this is false.”

    t = 0: I am shifting the line [0, 1] to position [-0.5, 0.5]
    t = 0.5: I am shifting the line from position [-0.5, 0.5] to position [-0.75, 0.25]
    t = 0.75: I am shifting the line from position [-0.75, 0.25] to position [-0.875, 0.125]

    This is a list of infinitely many steps (I am not stopping at a finite value). What is the position of the line after execution of all of the steps on this list, at t = 1? Is there a step on this list shifting the line to position [-1, 0]?

  107. I’m replying to a comment that appears in the email alert but has yet to show up here. netzwelter asked about the position of xir line after executing “all” of an infinite number of steps, “not stopping at a finite value”. The position is indeed -1, 0], not just “close to” -1. Xe also asked “is there a step on this list shifting the line to position [-1,0]?” The answer is that it depends on what is meant by “a step”. A finitely-numbered step, no. All the infinitely-numbered steps, yes.

    This gets at the core strangeness of infinity. There is no finite number of steps for which these principles hold, but they do hold for an infinite number.

    And nearly all objections to 0.999… miss this. They say, in effect, “But we’re not there yet!” when this whole concept of “yet” hides an assumption of finiteness. In short, the sum of a truly infinite set of numbers of the form 0.9, 0.09, 0.009, etc is exactly 1.

  108. @Lenoxus on June 20, 2013 at 5:16 pm:
    What do you mean by “finitely-numbered step” and “infinitely-numbered step”? As far as I can tell, there are only finitely-numbered steps on this list, and none of these infinitely many finitely-numbered steps is a shift to position [-1, 0].

  109. Hi Khalid. I very much like your site. It’s great to see somebody explaining maths ideas simply.

    When it comes to this question, I agree with you that we usually give the wrong answer. It’s much better to discuss how we deal with infinities than to assert that 0.999… = 1 and then blind the other party in the conversation with a barrage of analysis.
    I think it would have been nice if you could have concluded that 0.999… is 1 in the standard decimal system that people learn in school at the end, only because it’s a little annoying when people come up to you and try to tell you that 0.999… isn’t 1!

  110. Hi Conor, thanks for the note! Good point, I’ll see if I can clarify the article. Correct, in our current decimal system 0.999… = 1 (because we cannot represent infinitesimal differences), but the deeper response is that we *could* represent it in another system. A little like how the Romans didn’t have a symbol for zero, but it doesn’t mean the concept is inherently impossible :). I’ll need to see if I can make this subtlety more clear though!

  111. But it’s not just a matter of notation.

    0.999… necessarily means “Nine-tenths plus nine-hundredths plus nine-thousands, etc, forever.” There’s no disagreement with this, right?

    Well, mathematicians find that the sum of these fractions is exactly equal to one. There’s no point in the process where our means of representation come into play. We don’t say “Well, there’s an infinitesimal part that we lack the means to express, so we’ll round it away.” We simply find that the number “one” is equal to that infinite sum.

  112. If I am mixing 9 liters of red paint and 1 liter of green paint, the mixture ratio is 9:1. If I am mixing 99 liters of red paint and 1 liter of green paint, the percentage of red paint is 99%. If I am not stopping increasing the percentage of red paint, the percentage of red paint is 99.999…% (even if this means that the whole infinite universe is filled with red paint). But a universe filled with red paint and 1 liter of green paint is different to a universe solely filled with red paint, and no green paint in it. How do we point out this difference using standard decimal notation?

  113. An excellent question, netzwelter. My understanding is this: It depends on what you want to express. If you’re asking for the actual amount of each type of paint, we would say there are “aleph naught” (or “aleph null”) liters of red paint and one liter of green (or no liters of green). There isn’t exactly a decimal representation of aleph naught, it’s just the word for the number of integers there are (and also the number of rational numbers).

    If you’re asking about the ratio of red paint to all paint, that would be 1:1, despite the fact that not all the paint is red paint. I know it makes no sense, but I believe that’s the mathematical answer. Likewise, the ratio of green paint to red paint (or of green paint to all paint) is 0:1, even though the actual amount of green paint is not zero. This also seems strange, but it’s the way it has to work. In fact, the ratio of any finite amount of green paint to an infinite amount of red will be 0:1.

    All that said, I think there are mathematics in which you could make that distinction better, by dealing with infinitesimals and similar concepts. The math of infinitesimals doesn’t actually change the equality of .9… and 1, but it does deal with other distinctions that standard arithmetic does not.

  114. Hi, thanks a lot for this post. My mathematical philosophy course was talking about the 0.999=1 idea and I couldn’t believe how just because something was infinitesimally small meant that all the sudden it jumped to becoming another number. But you showed me the idea of hyperreals. Also, depending on which base is chosen, then the number that goes to its equivalent “1” is different. And since this number is different for each base, then it just seemed arbitrary to me… And that didn’t seem right. Seems like math still has a long way to grow.

  115. @Netzweltler, If I understand what you are asking correctly, then this is my answer :)

    If the amount of green paint amongst the red paint is measurable, then you can measure it and write down the value. 0.000000000001 perecent is fine to write down if you can measure it as such.

    If though you are the point where the amount of green paint is no longer measurable, you can not assign a number to it, it is equivelent to zero. In other words, if the amount of green paint is so small you can not measure it, how do you know it is not infact orange paint? Or unicorns? If you can’t measure it, it can be anything, inlcuding nothing. So we repreent that as zero: 0. If you CAN measure it, then you represent it withthe measure number: 0.00000001 percent. Hope that helps :)

  116. @D-Physicist on September 13, 2013 at 12:27 am:
    My question on June 20, 2013 at 2:46 pm (#119) might be a better fit to your answer than the red paint to green paint question (since we are dealing with one liter of green paint – which isn’t immeasurably small and which is always one liter of green paint).

    This list does not contain a step shifting the line to position [-1, 0]:

    t = 0: I am shifting the line [0, 1] to position [-0.5, 0.5]
    t = 0.5: I am shifting the line from position [-0.5, 0.5] to position [-0.75, 0.25]
    t = 0.75: I am shifting the line from position [-0.75, 0.25] to position [-0.875, 0.125]

    So, we are missing the target by an immeasurably small segment, which is 0, if we are trying to use standard decimal notation to describe the size of the segment. Nevertheless, these infinitely many steps don’t have the same effect as moving the line from [0, 1] to [-1, 0] in one step.

  117. The question of whether .999… =1 can be easily addressed without recourse to philosophical nit-picking.

    .999… is short hand for the series 9/10 +9/100+9/1000…, which everyone agrees has the limit 1. The question people are asking is whether the series really adds up to 1 or whether or not it just barely misses it by some infinitesimal result.

    The problem is that addition is defined for finite inputs. The axioms behind addition don’t allow you to calculate an infinite number of additions. In order to make a sense of an infinite series, you have to define what the sum of an infinite series is, and that definition cannot be derived from the base principles of addition alone.

    For convergent series, we define the sum to be the limit of the partial sums. Doing so has useful properties, it preserves linearity and other properties that we associate with addition. But, it is merely a definition. You can’t prove that the infinite series actually does , when carried out completely, add up to the limit, because the sum of an infinite series has no meaning unless you give it a definition.

    You are free to not to define the limit of the partial sums of a convergent series as its sum. If you can find an interesting alternate definition, feel free to explore the mathematical consequences. But, if you reject that .999… = 1, then you must reject the use of the limit as the sum of any convergent series, which wipes out a large portion of very useful and interesting math.

    I wonder how many people who doubt whether .999… = 1 also doubt whether 1+1/2+1/4+1/8… = 2, or whether e = 1 + 1/2! +1/3!…

  118. With regards to Peter post #70

    I’ve long regarded (and still do) ‘numbers’ like 1/3 to be a process and not a ‘number’. Even in denoting 1/3 we have not denoted a number, we have denoted 2 integers and a process (division). Still, we like to conceptualize a platonic concept of a number, n and set it equal to 1/3 then ask ourselves “What is n? Is it a process, or the result at the end of a process?”

    For me it is similar to ‘looking at’ a quantum particle and asking what is it? Is the particle a quantum wave function (a process)? Or is the particle the result at the end of a process? E.g. does this marble on the desk have a position, or can I simply apply the position operator to the wave function describing the marble? Gary Zhukav wrote a fascinating book in 1979 entitled ‘The Dancing Wu Li Masters’ addressing this issue of perception quite eloquently. He often says ‘the dance is the dancer, the particle is the wave function’. It may sound odd at first, but consider the two following recursions:
    – does the dance exist if there is no dancer to perform it?
    -can the dancer be so called if there is no dance to perform?

    -Is 1/3 a number without a process to use as an approach vector?
    -Is 1/3 a process if there is not a target to point to?

  119. The ole Euler proof for infinite series is based on mathematical induction for finite partial sums, then , with a grand leap of faith, POOF! We can conclude something for infinity!

    Sorry to burst this bubble, but Euler incorrectly subtracted the 2 “numbers” in order to attain his desired result, fooling many of the mathematical community.

    The Limit concept prevents this equality in the first place (as a sum). That is besides the point for now. Let’s show how the proof is flawed, shall we?

    First, let’s try and apply the “proof” using infinity:

    \displaystyle{S =   a + ar + ar² + ... + arâ¿ + ... ar^â}
    \displaystyle{rS = ar + ar² + ... +  arâ¿ + ... ar^(â+1)}

    Now what usually happens here is the magical INCORRECT subtracting of infinite series terms. You see, here is the correct way of subtracting infinite converging series: \displaystyle{ Σ { a_i - b_i } = Σ a_i - Σ b_i }
    But you see in the proof, they subtract like this: \displaystyle{a_i+1  -  b_i:}
    \displaystyle{S - rS = (a - 0) + (ar - ar) + (ar² - ar²) + ... }
    This forces the results to be what is desired along with asinine things such as 2 different decimal numbers being the same decimal number, ie, 0.(9) = 1
    The particular detail here is that rS has one more term in the sequence at all times since you are subtracting the “next term” in S from the “current term” in rS.
    However, when done correctly, you get consistent results:
    \displaystyle{S = Σ a_i}
    \displaystyle{rS = Σ r x a_i}
    \displaystyle{S - rS = Σ {ra_i - a_i}}

    Apply to 9.(9) where a = 9, r = 1/10:
    \displaystyle{S = 9 + 9/10 + 9/100 + ...}
    \displaystyle{rS = (1/10) x S = 9/10 + 9/100 + ...}
    \displaystyle{S - rS = (9 - 9/10) + (9/10 - 9/100) + ... = 8.1 + 0.81 + 0.081 + ... = 8.999...}

    You are led to believe that S – rS = a, but clearly above:
    \displaystyle{S - rS = (a - ar) + (ar - ar²) + ...}

    If a = 9, and r = 1/10, then S – rS = 9
    But if we do it correctly, S – rS = 8.999…

    As we can see, the proof is flawed using incorrect infinite series subtractions to achieve the results that were desired. The proof was not a proof at all since it is invalid… unless you assume 9 = 8.999… which is completely circular, that is, assuming \displaystyle{S - rS = a} which is the basis of the entire “proof” of 0.(9) = 1.

  120. Lets try that without the latex:

    Let’s try and apply the “proof” using infinity:

    S = a + ar + ar² + … + arⁿ + … ar^∞
    rS = ar + ar² + … + arⁿ + … ar^(∞+1)

    Now what usually happens here is the magical INCORRECT subtracting of infinite series terms. You see, here is the correct way of subtracting infinite converging series: Σ { a_i – b_i } = Σ a_i – Σ b_i
    But you see in the proof, they subtract like this: a_i+1 – b_i:
    S – rS = (a – 0) + (ar – ar) + (ar² – ar²) + …
    This forces the results to be what is desired along with asinine things such as 2 different decimal numbers being the same decimal number, ie, 0.(9) = 1
    The particular detail here is that rS has _one more_ term in the sequence at all times since you are subtracting the “next term” in S from the “current term” in rS.
    However, when done correctly, you get consistent results:
    S = Σ a_i
    rS = Σ r x a_i
    S – rS = Σ {ra_i – a_i}

    Apply to 9.(9) where a = 9, r = 1/10:
    S = 9 + 9/10 + 9/100 + …
    rS = (1/10) x S = 9/10 + 9/100 + …
    S – rS = (9 – 9/10) + (9/10 – 9/100) + … = 8.1 + 0.81 + 0.081 + … = 8.999…

    You are led to believe that S – rS = a, but clearly above:
    S – rS = (a – ar) + (ar – ar²) + …

    If a = 9, and r = 1/10, then S – rS = 9
    But if we do it correctly, S – rS = 8.999…

    As we can see, the proof is flawed using incorrect infinite series subtractions to achieve the results that were desired. The proof was not a proof at all since it is invalid… unless you assume 9 = 8.999… which is completely circular, that is, assuming S – rS = a which is the basis of the entire “proof” of 0.(9) = 1.

  121. Let x= 0.99999……
    Hence 10x = 9.999999……
    10x – x = 9.9999……. – 0.9999999
    Therefore 9x = 9
    so, x=1,
    But x=0.9999……..
    Therefore, 1 = 0.99999999….
    this ways 10 = 9.999999……
    but 9.9999….. is completely divisible by three ( the quotient is 3.33333…… )
    Therefore, 10 is completely divisible by 3 and it leaves the remainder 0 when divided by 3

  122. Eric V: If 1/3 is really a “process” more than a number, then are all fractions processes? For example, what about 1/5, expressible in decimal as 0.2, but in binary as 0.00110011…?


    The proof was not a proof at all since it is invalid… unless you assume 9 = 8.999… which is completely circular, that is, assuming \displaystyle{S – rS = a} which is the basis of the entire “proof” of 0.(9) = 1.

    There’s no reason to label that a circular assumption. For one thing, there’s no reason to approach it with the assumption that they’re different numbers either. For another, the equality of 1 and .9… doesn’t rely on 8.999… = 9 as a part of the proof. What you’re saying is somewhat like to “if 3/6 = 1/2, then 10/5 = 2, and to say the 10/5 = 2 is an assumption based on the principles that are assumed true so that 3/6 can equal 1/2”.

    Tanmay Bhore: You can’t say 9.999… is “completely divisible by three” in any meaningful way, because there’s a remainder of .333… (which you called the “quotient”). So so conclude that 10 is divisible by 3 with a remainder 0 is obviously incorrect.

  123. The whole article is founded on a flawed principle. If someone asks whether 0.999… = 1 [b]without any qualifiers[/b], then I wouldn’t jump to the conclusion they were asking about the equation in a hyperreal number system any more than I would jump to the conclusion they were asking it in a hexadecimal number system. Clearly the equation is false if we are using hexadecimal numbers. But to argue the equation is false on those grounds is ludicrous. The real, decimal number system is the OBVIOUS default for any general discussion.

    The author would have been better served to make it clear that THEY are the ones deviating from the norm and into alternate number systems instead of trying to claim it was the layman asking the question that must have been asking about alternate number systems.

  124. This stuff can easily be proven. it stems from why rational numbers can be expressed as fractions (and potentially integers).

    0.3333333333333333333… equals 1/3 as we already know
    multiply by three
    0.99999999999999999999… equals 3/3 or 1

    i.e… if you want an algebraic proof
    x = 0.5555555555

    Step 5:

    Your two equations are:

    10x = 5.555555555

    x = 0.5555555555

    10x – x = 5.555555555 − 0.555555555555

    9x = 5

    Divide both sides by 9

    x = 5/9

  125. (Edited)
    i.e… if you want an algebraic proof
    x = 0.5555555555…

    Step 5:

    Your two equations are:

    10x = 5.555555555…

    x = 0.5555555555…

    10x – x = 5.555555555 …− 0.555555555555…

    9x = 5

    Divide both sides by 9

    x = 5/9

  126. I already disproved the algebraic “proof”.
    It is based on incorrect mathematics of infinite series. (Comment 133).
    if “x” = 0.555…
    then 10x – x is
    10 Σ 5/10ⁿ – Σ 5/10ⁿ = Σ 50/10ⁿ – Σ 5/10ⁿ = Σ 45/10ⁿ

    Next is the question: “How can we ACCURATELY describe Σ 45/10ⁿ as a “number” ?
    4.999… ?
    But what about the ‘5’ that is generated at every iteration?
    Do we take a leap of faith and say it doesn’t exist?
    s1 = 4.5
    s2 = 4.95
    s3 = 4.995
    For any ‘n’, there exists a 5, and the 9’s are inserted between the 4 and the 5.
    But “looking” at the 4, there are only 9’s after and we “never reach the 5” ?
    What if we looked at the 5? Would there be only 9’s in FRONT of it with no 4 at the beginning ?
    An endless decimal is not something that we can fathom. It has no “value” since the finite value that we think we can assign to it, is not that value since there is always another digit after that.

  127. “When writing, I like to envision a super-pedant, concerned more with satisfying (and demonstrating) his rigor than educating the reader. This mythical(?) nemesis inspires me to focus on intuition. I really should give Mr. Rigor a name.”

    Call him Mortimus Rigor. Or Morty.

  128. Why do we need to “represent” the real number \displaystyle{one} with \displaystyle{0.999...} ??
    \displaystyle{0.999...} means “The limit of the sequence of partial sums derived from the series of (infinite) 9’s”
    If we understand Cauchy sequences and definition of real numbers, why do we choose to represent “the real number” alternatively with a shortened version of one particular Cauchy sequence (of infinte sequences possible) but any other abbreviated Cauchy sequence is disallowed ??

  129. Hey Big K–

    The insistence by many mathematicians that 1=0.999… has always made uncomfortable, not only because I find it philosophically indefensible but because we doubters are usually dismissed as dabblers and dilettantes. When I first read this article, I admired the courage of your position, although at the time I didn’t fully grasp the nature of the hyperreals.

    However, as @49 Student observes, David Foster Wallace’s book “Everything and More” addresses this very problem. (You, by the way, don’t need to read it, since you already know the field.) Wallace doesn’t commit himself to one side or the other, but he nicely summarizes the concept.

    In Wallace’s formulation, which is just another way of saying what you said, 1>1-1/H>0.999…, where H is equal to infinity and 1-1/H expresses the eternal difference, the infinitely small gap between 1 and 0.999…

    For emphasis, let me say it again: 1>1-1/H>0.999…

    I don’t know what the pedants would say, but, philosophically, it makes me very happy.

  130. Hey Tim, thanks for the pointer! I didn’t know David Foster Wallace wrote about this issue — and there’s always more to learn. I like that approach and want to check out the book.

  131. I enjoy the book immensely, and if you had the time to read it, I think you would as well. Like the material on this site–still the most lucid site around–it’s written in a conversational style, always highly readable and accessible to the layman.

  132. The reason “0.999…” = 1 is because we say it is. Nothing more. It is an invention of the human mind or if you prefer, like a religion.
    We already have addition of the reals.
    Then we try to add infinite objects, which turns out to be impossible.
    Then we define an “infinite addition process” as the limiting value for which none of partial sums can ever attain.
    Limits are not addition and addition is not a limit. Basically it comes down to “ah screw it, the only value this series can be that makes any sense is the limiting value so we’ll just call it that”.
    In reality, the series doesn’t “add” up to anything since it never ends.

  133. And just to be clear, I understand that the notation “0.999…” is short for “The limit of the sequence {9/10, 99/100, 999/1000, …} which is what the 9’s represent” which is obviously the number 1.
    What I disagree with is:
    \displaystyle{\frac{9}{10} + \frac{99}{100} + \ldots = \lim_{N \to \infty} \sum_{n = 1}^N \left[\frac{9}{10} + \frac{99}{100} + \ldots \right]}

    In that equality, we should be able to evaluate BOTH sides of that equation and obtain the same value.
    We cannot. That is my issue.

  134. let s beleve it is true but it means line is 2-d.
    in dx d can’t be define.
    the number line is actully a point.
    and so on

  135. This should be the amended version:

    I agree with those who say we’re talking about abstraction (or processes) as they were concrete objects we could apply our naive intuition to. The way I see it \displaystyle{0.999\ellipsis{}} is NOT a number per se (\displaystyle{0.9} is, or \displaystyle{0.99}, or \displaystyle{0.999} and so on and so forth), it’s simply something we defined to deal with the idea of an infinite process. So when we’re saying “the number \displaystyle{0.999...}” what we really mean is “the series of sums of the form \displaystyle{9/10^n}, for \displaystyle{n \rarr \infinity} converging (another term we defined) to one. In other terms, 0.999… is \displaystyle{\emph{enumerable}} (i.e., in this context, you can approximate it as precisely as you’d like), but it’s not \displaystyle{\emph{computable}}. The same holds for \displaystyle{\frac{1}{3} = 0.333...} we’re just defining one of the terms, e.g. 1 or \displaystyle{\frac{1}{3}}, and expressing it in a form which trick you in believing you’re dealing with an actual number.

    To sum it all up, we read the (dis)equation \displaystyle{0.999...&lt;1} as "there's some number, made up of a 0 integer part and an infinite decimal part of all 9s, which is strictly less than 1", i.e. there's a point there on the real line, just as close to 1 as we can think of, just an infinitesimal "gap" from 1. But from the very definition of reals, think of it, you can always add one more 9! That's to say, there's no possible "gap" between \displaystyle{0.999.\ellipsis} and, from which follows their equality. Yet again the point I'm trying to make is that what we should really read in our mind is this: "in the reals, the number 1 can be defined as an infinite sum of fractions ever-closer approaching 1". We're only definind another mean to compute 1, not another number per se.

  136. Hi Daniele, thanks for the note! I think the confusion is that when a mathematician sees “0.999…” they think of the limit of an infinite series, but a layman thinks “The closest number to 1.0 (that isn’t 1.0)”.

    Now, this might be a meaningless or imprecise statement (similar to “the largest number over 1.0”), but saying “0.999… = 1, by taking the limit” misses the underlying (unarticulated) question about how close we can get (which depends on our number system, an interesting question about how we can evolve our point of view).

  137. “when a mathematician sees “0.999…” they think of the limit of an infinite series,”
    That is the issue. They think of the limit of the series AND the series as being equal, that is:
    9/10 + 9/100 + … + 9/10ⁿ + … = 1
    That stuff on the left will never equal 1, as proven with the definition of limits theory. There is always an epsilon value for every step along the way. Thus we can induce that there is an epsilon for every index, and all indices.
    For the same “logic” they use to say “for each partial sum you are correct, but 0.999… is not any of those” and “all the 9’s are there” yadda yadda, I can use the same logic, that “I am imagining the epsilon value for every single one of those 9’s AT THE SAME TIME, and not one-by-one”

    An endless string of 9’s is just that. It is not equal to 1. We can analyse that string and use it to DEFINE the number 1, or construct it, but that doesn’t mean we also have to say it is equal. There is no gain by doing so.

  138. 0.33333…+0.77777…= 1 and 0.66666666… Doesn’t equal 0.777777…so 0.999999…does not equal 1

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