More on .9999…=1

Polymathematics has posted another excellent essay on the subject of whether .9999…repeating equals one. This time he is responding, very effectively, to various counter arguments raised by commenters.

One small comment of my own, though: The name of the blog is “EvolutionBlog.” One word.

Reading the various comments left to the post reveals two kinds of skeptics. Some are people who are perfectly willing to accept that .9999…=1, but find the logic justifying this conclusion to be hard to follow. No shame in that. The idea of the limit of an infinite series is not a simple one, which is why most students find calculus to be rather difficult the first time they see it. As with most things, however, a bit of practice makes the strangeness go away.

Other skeptics, however, are outright cranks. Those are the one who say things like:

You must be a public school teacher, and I fear for your students. You don’t know enough math to teach it. Stop filling their heads with nonsense.


.999… is clearly less than 1, but mathematics isn’t advanced enough to handle infinity, so you can’t prove it. My intuition isn’t flawed, math is.

Cripes. I frequently get e-mail from creationists written at a comparable level. Do not engage. Do not engage.

Many of the commenters offered counterarguments based on misunderstandings of the role played by notation in mathematics. Polymathematics writes:

This is a common mistake among my students. You’re mistaking notation for mathematics. Notation is not mathematics. Mathematics is the study of ideas about patterns and numbers, and we have to invent notations to communicate those ideas. Just because you can write something that looks mathematical, that doesn’t imply that what you wrote has meaning. (Emphasis in Original)

Mathematicians have a little saying that captures the point being made here: Notions, not notations.

The first time I heard that .9999…=1 I didn’t believe it either. As I recall, I was a sophomore in high school at the time. I was a member of the math team, and one of the senior members of the team mentioned it during the bus ride home after one of our competitions. The following day I mentioned it to my math teacher, Mr. Manzer. I expected him to tell me that I had misunderstood what the person had said. Instead, he whipped up a quick proof to show me that .999…=1. I don’t recall which argument he chose, but I do recall that it was completely convincing. I was left shaking my head over not having realized it for myself.


  1. #1 karl
    June 22, 2006

    I had the same reaction to some of those comments:
    “I don’t believe it”
    “Math must be wrong”
    Sounds just like the ID crew.
    I wrote a LONG comment, probably just like the one I wrote here previously, about number vs numeral. But I was way down the comment list by then so I doubt that anyone saw it.
    I found that “I don’t believe it” comment unbelievable.
    I can understand it in response to Evolution. After all, we keep saying that you can’t really PROVE anything in science. But in math?! – where we PROVE everything (given a set of axioms).
    It just compounds the concern for the state of intelligence and/or knowledge in this country. No wonder people like GWB can get elected.

  2. #2 Dave S.
    June 22, 2006

    I like the guy who keeps demanding a “proof” that 1/3 = 0.333…

    And never really thought of it, but it’s interesting that say 0.265 = 0.264999…

  3. #3 Wesley C.
    June 23, 2006

    Notationally speaking, there is no value that can be greater than 0.9999… while still being less than 1. There are simply no places to increase in value such that that place will have a greater place value than the corresponding place in 0.999… Assuming that 0.99… does not equal 1, then there must be a value 0.00…001 that represents the difference between 1 and 0.999…

    As such, 0.00…001 would represent the smallest increment between any two real numbers.

    But if the set of real numbers has an increment, it can be set in one to one correspondence with the set of integers

    (Integer) < -> (Real Number)
    N < -> 0.00…001*N

    And if two infinite sets can be put in 1 to 1 correspondence, then the dimensions of the two sets are equal.

    But the dimension of the set of real numbers is larger than the dimension of the set of integers. I can’t show that here, but I think Cantor’s Slash is the generally accepted proof of it?

    So, as real numbers is not of the same dimension as integers, it follows that 0.00…001 is not a real, incrementable value?

    Which means that 0.999… = 1?

    I think so, at least… Hmm.

  4. #4 Wesley C.
    June 23, 2006

    Stupid use of non-html brackets.

    (Integers) to (Reals)
    N to 0.00…001 * N

  5. #5 confluence
    June 23, 2006

    It got posted on Digg. Last time I checked, there were about 800 comments. About half of them are the kind that would make a mathematician cry – written by people who are convinced that their special personal theories about the way numbers work are flawless, while those craaaazy mathematicians of “the establishment” are conspiring to oppress right-thinking people and sell them obvious rubbish. Because these are clearly two different numbers, duh! *I’ve* figured that out in two minutes using common sense; how can these mathematicians not have noticed it for hundreds of years? Ha ha, silly mathematicians.


  6. #6 JF
    June 23, 2006

    Does it make much difference if we change from decimal to a different, higher-than-10, base? We’re only writing all those nines because “9” is the highest meaningful symbol in that number system, you simply can’t go any higher in any position than 9 so that’s what we’re stuck with. But if we can use a symbol that stands for a higher number, will that affect anything?
    What I’m thinking of is:
    In decimal, 0.9999…9 = 9/10 + 9/100 + 9/1000 + …
    But in, say, hexadecimal then the ‘granularity’ (for want of the correct term; I’m no trained mathematician) gives us:
    0.FFFFFFF = 15/16 + 15/256 + 15/4096 + …
    Does that get you closer (“closer” because of the finer “granularity”) than the decimal version of “0.999…9 = 1” or does it all not matter once you bring infinity into it?

  7. #7 Bob
    June 23, 2006

    I’m just a lay person, only a semester of college calculus, and I had never heard this before.

    The “proof” that helped me understand it was the 1/3 + 1/3 + 1/3 example.

    Secondary was the “an infinite number of zeros and then a 1 doesn’t work” argument to show that there’s nothing you can add to .999… to make it one.

  8. #8 Dave S.
    June 23, 2006

    I don’t think it matters JF. I mean, 4/6 is not closer to 1 than 2/3 is, even though the steps of the former (1/6 + 1/6 + 1/6 + 1/6) are ‘finer’ than the latter (1/3 + 1/3). 0.999… is not a dynamic number that’s approaching 1, it’s just a number that is 1. There is no hole between 0.999… and 1.

  9. #9 JF
    June 23, 2006

    Thanks, Dave S. I’m not disagreeing with the idea of “0,999…9 = 1” but I am curious about what makes it so.

    About your 2/3 and 4/6 analogy, that’s not quite what I was thinking about. In that case, the numbers are equivalent, but I was trying to show the difference between 9/10 and 15/16. Consider:
    0.5 = 1/2
    That’s as near as you can get to 1 if you use “2” as the denominator. However, if you can use finer divisions — say 3 — then you can get a little near:
    0.666…6 = 2/3
    With each increase in the fineness of the “granularity” you get nearer by virtue of having finer ‘chunks’ to play with.
    So, with the infinite series 0.999…9 the fineness of the series is limited by the highest number of ‘chunks’ available to you, i.e., 9. It’s like, imagine you’re trying to get to 1 in base 10. You can get 9/10 but that’s as high as you can get. But you still have 1/10 left to make up so you go to the next position and here you plop down a 9 yet still you’re not there; you’ve made up 90% of the last position’s deficit but you’ve still 10% to find. This carries on forever, obviously, and you’ve always got a 10% deficit to make up.

    Now, with a fixed number of positions, you will eventually get to the point where you know the value of your missing 10%. But with the infinite series of positions, that leftover 10% from your last position will always be eluding you.

    But if you use, say, base 20, then you’re only 5% from your target at every position along the series because the ‘chunks’ available to you are finer.

    So, in summary, what I’m trying to say is I wonder what makes one series equivalent to 1 and the other always short of that figure, yet the only difference (that I can see; maybe I’m overlooking something really obvious!) between the two series is that one is finite and the other infinite.

  10. #10 Dave S.
    June 23, 2006

    JF –

    I’m not a mathematician by any stretch, and don’t want to make any authoritative sounding but silly replies, so take whatever I say in this field with a grain of salt! 🙂

    But it sounds to me like you’re describing a version of Zeno’s Paradox. Imagine two giants in a race. One covers 90% of the distance of the course remaining in a single stride, while the other covers 95%. Say the course is 10m long. Now after the first step, Giant 1 is 900cm along, while Giant 2 is 950cm. Giant 2 in the next step again covers 95% of the remaining distance (he’s now at 997.5cm), while Giant 1 only 90% (he’s at 990cm). It’s true Giant 2 remains in the lead after each step for any finite number of steps and so is always closer to the goal. But Giant 1 get’s closer and closer to Giant 2 with each step. At an infinite number of steps the two converge on the same distance covered, and both hit the finish line at the same time.

  11. #11 JF
    June 23, 2006

    Yes, that’s what I had in mind when I started thinking about this. But there’s a difference between Zeno’s Paradox and the infinite series of recurring digits.

    In the Paradox, it’s a rather arbitrary set of figures that you can work with. As long as Giant 1 is slower than Giant 2, the Paradox works as described. The actual figures aren’t constrained by anything else.

    But with the infinite series of digits you’re limited by the maximum value symbol able to be used in your number system. The aim is to reach 1. In an ideal world, that would be 1/1 or 12/12 or 7645/7645 or n/n. But that’s not how number systems work. You don’t have a symbol in a single digit-position to represent n, where n = the base of the system. So, the next best thing in achieving n/n is to go for (n-1)/n. That’s valid and it’s how we arrive at 0.9999999999… and my example above of 0.FFFFFFF…

    So if we replace your giants with number systems, at every point along the way, the number system that has the greatest value of n will be ahead (i.e., nearer to 1). But then, it seems, that all counts for nothing as soon as we announce that the series will be infinite. As if by magic, they both become equivalent! That’s the bit that fascinates me. It’s almost like as soon as we introduce infinity then the rules change.

  12. #12 Karl
    June 23, 2006

    There is no “closer”. .999… is not “getting” closer to one. It is not a train, it is not a giant taking steps. It is a numeral that represents one particular number.

    The history of mathematics is replete with invention of new numbers and new numerals (symbols for numbers). At some point someone realized that the decimal representation of certain numbers (fractions whose denominator had a factor of something other than 2 or 5) would be infinite with a repeating block. They invented the symbol “…” to represent that fact. Then they noticed that 1/3 as a decimal is .333…, and 2/3 is .666… So to be consistent, for the symbol to work with all the other properties of mathematics, 3/3 must be .999… But, of course, 3/3 = 1. SO, .999… is DEFINED as ONE. It is a definition. There is no arguing with it.
    What you are having a problem with is infinite sequences and limits. You need to take a calculus course.

  13. #13 Todd
    June 24, 2006

    Wow. That’s all I can say. Wow. Poststructualism lives and breaths. When people cannot even agree something something as rigorous as a mathematical proof, especially one as simple as 1/3 * 3 = 1, 1/3 = 0.33…, therefore 0.33… * 3 = 0.99… = 1, it’s time to seriously consider scrapping the entire educational system and starting ever.

  14. #14 Dave S.
    June 26, 2006

    Everyone knows that 0.999… = 1.

    Just like 0/0 = 1.

    *Dave S. ducks further flying debris*

  15. #15 Dale D.
    June 27, 2006

    Isn’t it obvious by inspection that the limit of 0.999…=1? Am I missing something here?

  16. #16 Ken Winniford
    July 6, 2006

    The number “1” is a rational number. That is there is at least one equation that is equal to one and can be expressed in the form of p/q where both p and q are nonzero intigers. If p=6 and q=6 then 6/6=1 or p/q=1.

    the number 0.999… is not rational because it cannot be stated as p/q = (some integer). You cannot divide some integer by some other integer and get a quotient of 0.999.

    The fact than one can affix “…” to a number does not change the status of the original number.

    “1” is rational, “0.999…” is not. A number cannot be both irrational and irrational.

  17. #17 Dave S.
    July 6, 2006

    Ken –

    A number cannot be both irrational and irrational.

    One of those “irrational”s is a “rational” I presume.

    5/5 = 1 = 0.999…


    Were not talking about 0.999, we’re talking about 0.999… . The “affixing” of “…” makes a big difference. 0.999 does not equal 0.999…. Furthermore, irrational numbers have decimal expansions that do not terminate or become periodic. Since 0.999… is periodic, it doesn’t qualify.


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