“0.9999…. = 1”

This is an essential assumption of Bayes’s Calculus. If you doubt that this is common, then just take a cursory look at the mathematical community here, here, here, and here.  And here and here. Do you know what this means? It means that Calculus, like probability (see my deconstruction of probability), is false. The argument goes something like this:

0.333… is 1/3, right? Well 1/3×3=1. But surely 0.333…x3=0.999…! Therefore, by one or another form of the transitive property, 0.999…=1!

In addition to being a near-blasphemous usage of the transitive property, it is just plain false. Think about it in the following manner. 0.1 is necessarily greater than 0.0X, where ‘X’ is any countable number. 0.1 is also necessarily greater than 0.0XX. And so on. No matter how many X’s you add to the series, it will never equal or be greater to 0.1. Therefore, by mathematical induction a la carte, no amount of repetition of 0.0XXX…. could ever equal 0.1, which is what is necessary to add to 0.9 in order to equal 1. Importantly, (0.9 x / x<0.1)≠1 Λ (0.9 x / x<0.1)<1. Therefore Calculus is false. A house built on sand cannot divide itself.

Notice that all I needed to disprove this foundation of calculus was mathematical induction.

Advice to all my readers: Don’t let “math wizards” intimidate you with technobabble. And note that I am not alone.


125 thoughts on ““0.9999…. = 1”

  1. A simple proof:

    To show: 0.999… \neq 1.

    Proof: Let A = 0.999… , where A has as many 9s in it as possible. Suppose A = 1.

    As proven earlier, the greatest number of 9s possible is P, the penultimate number. So A consists of a decimal followed by P nines.

    Now let B = 0.00000…1 \neq 0 , where B has P zeros in it (including the one before the decimal). Then:

    A + B = 1
    1-(A+B) = 1-(1)
    (1-A) + B = 0
    B = 0 *contradiction*

    Therefore, A \neq 1. QED.

    Obviously 1/3+1/3+1/3 = 1; any 4th-grader knows that. But that’s irrelevant — it is inaccurate to refer to 1/3 as 0.33333… because even a penultimate number of 3s after the decimal won’t reach 1/3. Thus the .333…*3 = .999… analogy doesn’t apply.

  2. 1/3 = 0.333333 to infinity
    that mean

    0.333 to infinity + 0.333 to infinity + 0.333 to infinity = 1

    sample and stupid xD

  3. Not only has the concept of “Infinity” been dealt with in a previous post, but the equivalence – without reference to hypofinite math! – has been refuted in my argument above. In fact, your comment there is just a restatement of precisely what I refuted.

  4. A standard proof that 0.99… = 1

    Let 0.99… be replaced by a variable:
    x = 0.99…

    Multiply both sides by 100:
    100x = 99.99…

    Subtract x from both sides:
    100x – x = 99.99… – 0.99…
    100x – x = 99

    Factor and simplify:
    (100 – 1)x = 99
    99x = 99

    Divide both sides by 99:
    x = 1

    Since x = 0.99… and x = 1, then 0.99… = 1

    Mathematics is a means to discover and invent ways of finding systematic patterns. It does not have to have any correspondence to reality. The fact that it so often does was referred to by Eugene Wigner in his paper, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” link here: http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

  5. Pingback: Using Mathematical Induction to “disprove” the foundation of Calculus! « Todd and Vishal’s blog

  6. density Theorem in real analysis says

    if x and y are real number with x<y, then exits a rational number r such that x<r<y

    so, it’s clear there is no rational number between 1 and 0.99999999…, it’s mean 1 and 0.99999999… are the same number.

  7. Induction does not prove your point. Induction proves that your fact is true for any specific number of digits X that you name–that’s different from proving it true for infinitely many X’s. Furthermore, the wikipedia section you site is not a set of actual, mathematical counterarguments, it’s a set of observed counterarguments that are easily refutable, but are cited as demonstrations that students (like you, apparently) often cannot understand why .9 repeating equals 1.

    You are under no obligation to believe the centuries of mathematical theory that claims that .9 repeating equals 1. However, you are under a moral obligation to do so with a little humility, which ought to make you think twice about saying that those centuries of mathematical theory by some of the smartest minds in history completely missed a simple induction argument. It’s not that they missed it; it’s that it’s incorrect.

  8. Chebotarev’s density theorem is a deep theorem from number theory and has nothing to do with Aria Turns’s ‘density’ theorem. It appears that you typed ‘density theorem’ into Google or Wikipedia and name-dropped the first item that you saw without inquiry into its relevance to this farcical debate. This action should speak for itself.

    Ms Turns was referring to the fact that the rationals are topologically dense in the reals – that is, they ‘get everywhere’ and if you paint all of them black it will look as if you have painted the whole number line – and I don’t know a construction of the reals that doesn’t yield this as an immediate property.

    I ask you this question: Suppose 0.9999… is different from 1. What, then, is half of 1.99999… ?

  9. Molly Toff,

    Nice try. Instead of proving that 0.999…=1, you have instead proven that it cannot. Do you think I am so foolish as to allow in a question-begging assumption such as “99x = 99”? I am not.

    Furthermore, you state that “[Math] does not have to have any correspondence to reality.” No further comment should be necessary.

    And Eugene Wigner is not a credible source since he has a vested financial interest in (contemporary) mathematics being a legitimate field. On the economics of writing, please see the famous tome by intellectual Charles Beard, “An Economic Interpretation.”

  10. Aria Turns,

    Your post is very challenging. My first instinct is to say, “So much the worse for Chebotarev and his co-conspirators.” After all, the following is not a valid argument form:

    1) Noted Scholar’s thesis says X
    2) X is inconvenient for math
    3) X is false

    However, because of the intuitive appeal of the Density Theorem, which you so parsimoniously explicate (see Molly’s egregious subterfuge above), I am tempted to try and salvage it. Unfortunately I am in no position to do so now, as I have piles of research yet to do. But I will point out this: The traditional approach of calculus, that 0.999…=1, fairs no better under Density Theorem. After all, there is some number (so the story goes) before 0.999…., but I ask you: What comes between *that* number and 1? Quod errata, my dear, quod errata.

  11. How come you didn’t give your comments on Todd and Vishal’s blog about your conclusion being misleading since Inf is not an element within the set of all Natural numbers?

  12. oh by the way, in terms of the words you used in this entry, let me rephrase it: Infinity is NOT a countable number. X can only be a countable number as you said.

  13. Dear hi second-rate dumbass,

    Insofar as I can even discern what you are trying to say through the drivel, it is still not substantial. Please take your negative attitude elsewhere.

  14. I wandered here from another blog and saw your math, and I had to comment…

    Your argument is this, in your own words:

    (Premise) 0.1 is necessarily greater than 0.0X. 0.1 is also necessarily greater than 0.0XX. And so on. No matter how many X’s you add to the series, it will never equal or be greater to 0.1.

    (Conclusion) Therefore, no amount of repetition of 0.0XXX…. could ever equal 0.1.

    Note that your conclusion is identical to your premise.

    And your premise is false, because 0.0999… is equal to 0.1, by the same mathematics used by Molly Toff above.

    If you disagree, then please show exactly where Molly Toff has erred in her math.

  15. Hazel,

    Just because you don’t agree with something, doesn’t mean it’s not even serious. That’s pretty arrogant if you ask me.

    And anyway, I’m not the only one who has doubts about this particular part of calculus. Some of my other posts are way more controversial!!

  16. That’s my point. I don’t see anything on your site that is reasonable. For instance your post on the fact that the speed of light is constant shows virtually no understanding of the experimental work that has verified Einstein. Your arguments are simple and uneducated. Furthermore, all the proofs people have shown here that 0.99.. = 1 are solid. The fact that you think you are bringing down such basic math and science seems so sophomoric that my first thought was that your site is a spoof. If it’s not it seems to me you should concentrate on your education more.

  17. Hazel,

    Are you running for President of Patronizing?

    These are blog posts. You can’t expect long tracts like Principia Mathematica. These arguments will develop over time. See: Kuhn’s Structure of Scientific Revolutions. Furthermore, the “proofs” you refer to are frequently disputed on this very thread.

  18. There is no meaningful difference between 0.99999 with infinite repeating 9s and 1. What don’t you get about that? I think you use your calculator too much and your brain too little.

  19. Is this blog a parody, or is its author seriously espousing crank mathematics?

    For those of you convinced that .999… is not the same as 1, in spite of the numerous mathematical proofs of said fact: if that’s the case, then what is the number halfway between the two?

  20. Dear Davis:

    Is this blog a parody, or is its author seriously espousing crank mathematics?

    I call false dichotomy. It’s only “crank mathematics” if you presuppose against it. As for “parody,” a few people have responded to me with words like that. As I understand parody, maybe I sometimes do it (like having my name be notedscholar – a parody of academia). Similarly my blog’s subtitle is humorous. BUT if you mean my fundamental opinions, then it’s not parody. It’s polemics, and it’s also minority opinion (which leads some people to say “troll”, and idiotic term), but not parody.

    For those of you convinced that .999… is not the same as 1, in spite of the numerous mathematical proofs of said fact: if that’s the case, then what is the number halfway between the two?

    Okay. First of all it’s a nonsense question. That’s like asking “where is the fifth corner of a square?” But I’ll humor you. Your very question can be turned around on traditional mathematics. Observe:

    What is the difference between (0.999…-0.000…1) and 1? The same supposed “paradox” applies. Does it bother anyone? No, it doesn’t bother anyone.

    But in any case – and I’ve already answered this over at Topological Musings (http://topologicalmusings.wordpress.com/2008/11/12/using-mathematical-induction-to-disprove-the-foundation-of-calculus/#comments) – the answer to your question can be given symbolically. The difference between the two is precisely 0.000…1, where the notation “…” indicates Penultimate (P) decimal places. If you need background on P, see my much earlier post on Infinity.

    Hope this clears things up.


  21. Dear notedscholar:

    Thank you for not addressing my response at all.

    I ask again: Where, exactly, has Molly Toff erred in her math?

  22. Dear Digdug:

    Molly Toff erred on three logical accounts, one of which was mathematical in nature. You can feel free to reread my response to her. Until then, it’s been fun.


  23. I’m 99.999999999999999% sure this guy is a fake. On the off chance that I’m wrong, he’s the dumbest mf’er that ever lived.

  24. @ Notedscholar

    It might not make much sense to you that there should be a number bewteen every two distinct real numbers, but is is actually one of the axioms for the real numbers (in most axiomatisations). This is one of the deeper properties of real numbers. It makes my head spin just thinking about how many real numbers there are. I just can’t wait until next semester when I will take a nice real analysis course and get to understand how they really are constructed.

    See axiom 2 of: http://en.wikipedia.org/wiki/Tarski%27s_axiomatization_of_the_reals
    or http://en.wikipedia.org/wiki/Dedekind-complete for more information.

    As for your counter argument, 0.000…1 is not a real number. It is one of the characteristic things about infinite sequences, that they actually are infinite, and has no end. Thus there cannot be a 1 “after” all the others.

    To be a bit more precise there is exactly one decimal in every real number, for every natural number (ofcourse infinitly many might be 0). For instance the 4th decimal of pi is 1 as (pi = 3.1415…). So in 0.00…1 what natural number gives you 1? The answer is that none of them do. So 1 could not be a part of the decimal expansion. So your counter argument falls short on the fact that is is not talking about real numbers. Perhaps you could fit it into the theory of hyperreals, but I doubt it.


  25. Dear Epsilon,

    I try not to delete other people’s comments, because I support discussion.

    But try a little less of the sarcatic ad hominem and a little more of addressing actual arguments!


  26. Dear “Logician”,

    I doubt you are really a logician. I’ve known several logicians!

    Simply appealing to “axioms” doesn’t help. We can make just anything an axiom. And citing old dead people for the already subjective judgment doesn’t help your case.

    You say,

    There must be a number between every number. This is axiomatic.

    But, this results in paradoxes about infinity itself. What is between the Penultimate and Infinity? Even worse, what is between 0.999… and 1? What is between the last “infinite ordinal number” and one? At some point, real numbers stop (I have called these “countable numbers”), and fake numbers begin. You know, there’s a reason we have the term “irrational numbers.” But the sense of these terms has obviously been lost on some people.


  27. In your response to Davis (on November 26), you say that asking what is halfway between .999… and 1 is a nonsense question.

    So, let’s put it differently: what then is (.999… + 1)/2? If you still consider the question nonsense, then either you believe that some pairs of numbers in your system cannot be added, or that some numbers cannot be divided by 2.

  28. Dear Todd Trimble,

    First, I would like to paraphrase David Lewis: Nonsense is nonsense even when you talk it about mathematics.

    To address your point, it is clear that 0.999…+1 = 1.999… where “…” means “penultimate decimal places.” Now keep in mind that the “Penultimate” is axiomatically the last number in a series. This axiom does not preclude there being – for the purposes of representation only – “more” decimal places in order to produce half of 0.999… than there are places in 0.999… itself. However, this difference is clearly illusory, since “more decimal places” very clearly does not necessarily yield larger numbers. But what is of mathematical interest in my theory is that 0.999… could not have decimal places added to it. So the formulation 0.999… + 0.000…1, for example, is nonsensical, because there is no decimal place after P decimal places. Of course to represent the division you ask for we may “pack in” more decimal places before we hit P decimal places, but this is surely a limitation on mathematical notation and not my theory.


  29. (1) In a comment on my blog, I asked you what is the difference between 1 and .999… and you responded, “.000…1”. I took that as your asserting that

    1 – .999… = .000…1

    (whatever that might mean exactly). Now you tell me that the “formulation”

    .999… + .000…1

    is nonsensical. Why would the first equation be sensible but this formulation not? What prevents you from saying it equals 1, according to the first equation?

    (2) “Of course to represent the division you ask for we may “pack in” more decimal places before we hit P decimal places, but this is surely a limitation on mathematical notation and not my theory.”

    I asked you a simple question, which you avoided answering directly. What in “your theory” is 1.999…/2? Vague references to “packing in” more decimal places doesn’t cut it.

    In order to have a mathematical theory, you need to have a precise language in which to express it. So “limitation on mathematical notation” necessarily means a limitation of “your theory” — how are we to make sense of your theory if you can’t or won’t supply adequate notation in which to express it?

    “First, I would like to paraphrase David Lewis: Nonsense is nonsense even when you talk it about mathematics.”

    You said it!

  30. By your logic, a person could never cross a room, because he would first have to traverse nine-tenths of the distance across the room (0.9), then nine-tenths of the distance that remains (0.09), and so on.

  31. Dear Todd,

    I’m afraid you’re losing me – and I can’t help suspect that you’re trying to.

    I’m also afraid that you and I, in our cross-discussions across blog, have been equivocating on the decimal notation. This should clear up some of your troubles. I may have assumed traditional notation once or twice on your blog. Some signals got crossed.

    “Packing in” is not a vague reference. What I mean is that the decimal places are more or less illusory since they do not add to the size of the sum, and still “fit” into the larger number. Hope this is clear.

    You must be able to express yourself perfectly before you can have a theory

    This is just snobbish. Are you telling me that Newton had to have already formed his theory before forming his theory? Ludicrous. In any case, Wittgenstein showed that there can be such a thing as “wordless” knowledge and experience.


  32. No, not trying to lose you; just trying to get straight answers to the straightforward questions I asked above in (1) and (2). I’m still waiting.

    What I mean is that the decimal places are more or less illusory since they do not add to the size of the sum, and still “fit” into the larger number. Hope this is clear.

    No, it’s not. But it might help if you would just answer the concrete questions posed in (1) and (2).

    And better still if you would not use insert made-up stuff between the blockquote tags: I never said

    You must be able to express yourself perfectly before you can have a theory

    and you know it. To use that as the basis of an ad hominem attack against me (calling me “snobbish”) is dishonest, to say the least.

    As you know, NS, Newton was not out to defeat math and science. The burden of proof that science and math are defeated rests on you, and I’d advise you to get used to the fact that any such “proofs” had better be precise. Otherwise they convince no one.

  33. You idiots. Think about it for one second. 1-0.1=.9, yes? extend the number of decimal places to infinite, and the difference between two becomes zero. Therefore they are equal. That is the short greatly abridged answer, you are concerned to greatly with the decimal representation of a rational number, which is actually an approximation of 0.999…
    The person who stated that the example of 3*1/3=1 does not prove this has made this mistake. 1/3=/= 0.333 it equals 0.333… if one can appreciate the difference. An irrational number cannot be completely described in a decimal and hence trying to approach this problem from that angle is doomed to a conclusion as misguided as the approach that formed it.

  34. Dear Noam,

    Thank you for PROVING THE POINT.

    This isn’t, by any chance, Noam Chomsky is it? Because he wouldn’t be so careless with his citations.


  35. I stumbled upon this today and I just want to say something:

    If we assume that 1 is an absolute value that is nothing else but 1, then nothing else can be 1, than what already is.

    You see two apples. They are identical. They = 1

    You bring in a third apple that is as close to identical to these two as possible, with the most meaningless smallest possible thing left out. You could say it was identical and none would really care since it made no difference in the world and it could be used as it was…

    Except the knowledge that there is a flaw there, makes it not the same.

    It is totally irrelevant to anyone in any possible situation in real life, whether it was exactly the same, or the closest to the same as anything can possibly be. But the truth is that if it’s not the same, it’s not the same. And why even bother trying to argue that something what is not something, is something, when it’s clearly obvious that it cannot be.

    You don’t need fancy equations to understand this. You just need to admit to yourself, that if two things are different, they are not the same. Even if it made no difference in the world, it’s obsolete to claim so.

    I know someone will pop up and say: “Well if it makes no difference in the world, then it is the same, as there is nothing that the other apple can that the other cannot”

    The only thing they cannot do, is be the same.

    The problem with this lies in semantics. And if you claim that two different things are the exact same, then you could just say that everything is the same at the same time. Because it just takes the ground from under the possibility of actually handling matters as individuals with differences. And it also makes it obsolete, since if everything is the same, there is no point in trying to differentiate things from one another.

    I took this too far and almost confused myself to insanity, but prove me wrong and I’ll give you a cookie!

  36. Consider this:

    In regards to .999… (point nine recurring), I hope we can say that .999… > 1 is a false statement without much argument. Therefore, the converse must be true: .999… 1. No matter how many zeros we put in between the decimal and our nonzero digit, .999… will always have more nines beyond it. If A plus any positive P is greater than B, we must conclude that A cannot be less than B. Thus, .999… cannot be less than one. .999… >= 1.

    We’ve already established that .999… cannot be greater than 1, and since we now know that it cannot be less than 1, that leaves only one option:

    .999… = 1

  37. Consider this:

    In regards to .999… (point nine recurring), I hope we can say that .999… > 1 is a false statement without much argument. Therefore, the converse must be true: .999… <= 1.

    For any positive P, .999… + P > 1. No matter how many zeros we put in between the decimal and our nonzero digit, .999… will always have more nines beyond it. If A plus any positive P is greater than B, we must conclude that A cannot be less than B. Thus, .999… cannot be less than one. .999… >= 1.

    We’ve already established that .999… cannot be greater than 1, and since we now know that it cannot be less than 1, that leaves only one option:

    .999… = 1

  38. So , if 0.(9)=1 can you tell me why…
    1.0000..-0.7777…=0.33333…,but 0.9999…-0.7777…=0.22222..

  39. .777…=7/9

    1-(7/9)=2/9=.222…, NOT .222….222…..222…..2223!!!

    Which is an example of why people shouldn’t try to subtract INFINTELY REPEATING decimals without converting them to their fractional form first. Also, calculators round and should not be taken literally.

  40. Funny, as one of the top contributors on the early inquiry site WHquestion, I held the same position—thas ·’9′ ≠ 1|1·’0′. Thus their differend is ·’0’1, a quasicomplex—true, it’s not mootly “real”—infinitesimal or differential, the iota in surreal algebra. Its half is either indefinite, or ·’0’05—it’s all arbitrary.

  41. My dear Math Chick,

    You make an interesting point, that calculators should not be taken literally. I wonder if electrical engineers would agree with you? In what way do calculators produce metaphors? Compare the sentence “.777…=7/9” with the sentence “Life is a house.” What can we learn from this? Do you really wish to insult the achievements of engineering in this fashion?


  42. notedscholar:

    Do you know what Cauchy sequences are? Defining real numbers in terms of equivalence classes of Cauchy sequences of rational numbers makes this equality really, really simple to understand. Read a book on analysis to learn more. Good luck!

  43. Str,

    I don’t know what str stands for, or what you are trying to say in your comment. If Professor Cauchy has made “this” easy to understand, great! But ease of understanding does not yield truth.


  44. El Suscriptor Justiciero,

    First of all, Noam Chomsky is not dead. Are you confusing him with his wife? She recently passed away, which is very sad.

    Second, I agree that the link proved the point, so with whom are you arguing?


  45. 😦 Disappointed again.

    1/3 is not equal to 0.3333…. in your world. And I think you know why. You stipulate that you do not believe infinity to be an actual concept. You are satisfied that you have disproved the concept of infinity. Unfortunately to say that 1/3 = 0.3333…. you have to be willing to say that 1/3 = 0.3333(followed by an INFINITE number of 3s). As a scholar you should know, that you can’t have it both ways, either infinity exists or it doesn’t. And as long as it doesn’t exist you simply need to refer to 1/3 as one third. And I don’t think you’ll disagree that 1/3 + 1/3 + 1/3 equals 1.

    It is completely incorrect to say that 1/3 = 0.33333… with a FINITE number of 3’s following. If you cannot see that I strongly encourage you to sit down and do this problem, long division style. 1 divided by 3. Now when you’re done come back and give us your answer. Ohhhhhh wait, you’d die before you finish, but yeah, that’s totally not infinite. 🙂 Good day sir.

  46. Next to Last:

    As a scholar you should know, that you can’t have it both ways, either infinity exists or it doesn’t.

    Noted Sholar is like those modern geocentrists who say “Relativity means that any given object may be considered the center of the universe. Therefore geocentrism isn’t wrong. Therefore the Earth is the center of the universe, everything orbits around it, and Galileo and relativity are wrong.”

  47. Noted Scholar (I misspelled last time) said:

    Do you think I am so foolish as to allow in a question-begging assumption such as “99x = 99″? I am not.

    That’s not a “question-begging assumption”, it follows from the premises. “X” doesn’t even have to come into it, if you don’t like it. It is obviously true that if

    (100 * .999…) = 99.999…


    (100 * .999…) – (1 * .999…) = 99.99… – (1 * .999…)

    and therefore (here comes the factoring on the left side, is this the part that’s actually disputed?):

    (100 – 1) * .999… = 99.99… – (1 * .999…)

    or 99 * .999… = 99


  48. Another error I feel the need to correct. plamen said:

    but 0.9999…-0.7777…=0.22222..
    0.2222..2223 is not 0.22222…

    The result “1.00…-0.7777..=0.2222..223” is wrong. I’m guessing it was mistakenly derived from induction, probably with a calculator.

    It’s true that 1-.777 = .223, and that however many (finite) 7s you put in there, you will get a bunch of 2s followed by a 3. But it’s still not the case that 1-.(7) contains an infinite number of 2s “followed by a 3”; it’s simply an infinite number of 2s, or, more simply, two-ninths.

  49. Dear Notedscholar,
    In what field are you noted, and whom has noted you, precisely?

    If you are so noted, why are you not using your real name and degree and in which field is this degree?

    As an aside note, degrees in theological studies are not relevant to making one a scholar in mathematics, and so make you nothing more than an ignorant fool in what you try to disprove.

  50. Pingback: Homework for 7/16 « Dr. Beavers' Blog

  51. Pingback: Assignment 6: July 18 « codymitchell

  52. Your proof attempts to use the inductive property, but that can only apply to finite numbers. An arbitrarily large finite number of 9s following 0.0 will always be less than 0.1, but that isn’t the same as an infinite number of 9s. Technically, an infinite number of 0s is a theoretical construct which would be analogous to the limit of the sum of X*10^-n from 1 to n as n approaches infinity. Your inductive argument shows that any finite sum will be less than 1, but that says nothing about the limit.

    Consider 1/x where x>0. By the archimedian property, there is always a number between 1/x and 0 (for instance, 1/2x). What you’re essentially arguing is that the limit of 1/x as x approaches infinity isn’t 0 because any finite value of x will yield a 1/x greater than 0. It’s a stupid claim, and betrays a lack of understanding of what a limit is, and of calculus in general.

  53. “At some point, real numbers stop (I have called these “countable numbers”), and fake numbers begin. You know, there’s a reason we have the term “irrational numbers.” ”

    Do you have a similarly fun description of, say, imaginary numbers? Or hyperreals? I’d love to see those!

    One of the best jokes I’ve heard in a loooong time. Thank you!f

  54. Next to Last,

    I am sorry that you are disappointed! Hopefully you can be more optimistic and satisfied now.

    Okay, I see what you are trying to do. You are trying to link my critique of calculus to my denial of infinity. Nice try. But you know, even though my refutation of infinity corroborates my critique of calculus, it is independent. But I guess you are right, that the convergence of evidence supports my position!

    Also, from “you will die before you finish” it does not follow that “the task is infinite.” This inference is completely squirrilous.


  55. Lenoxus,

    You write, “(100 * .999…) – (1 * .999…)”. But one of the paradoxes of infinity is that you cannot subtract one from the other. For example, Hilbert’s Hotel.

    Best wishes,

  56. Dear Sane Individual,

    I am noted in the sciences and maths [sick].

    I do use my real name in social networks, but I don’t want to associate with my home institution… after all it has been bad for Behe who has done so.

    Thank you,

  57. Dear Greg,

    First, despite your claim that I have stupidity, thank you for at least admitting that my argument proves something. Now, you make the same mistake as Lenoxus above, by assuming that infinity does not involve paradoxes. Yet you also admit that your view is nothing more than a “construct,” like gender or race. How can you hope to refute my argument based on this?

    As for calculus, I am criticizing calculus! As for limits, I of course am pulling the limit back to the penultimate in general, as you can see in my famous post on negative numbers.

    Next time,

  58. Persilja,

    You are correct that imaginary numbers and irrational numbers are a joke! In fact, that is why they were invented! I think I have a post on imaginary numbers that you can find. Try Google!


  59. Wow… just wow.
    I refuse to believe this is a real argument you’re making. This, and every other post you’ve made, is riddled with mistakes and mathematical face-palms.

    But congrats, you just got a bunch of pageviews out of me

  60. Hi Brett. Welcome to the blog. I’m confused about which argument you have in mind. Like many of my posts, this one has generated a long stream of serious discussion. Also, I do not know what a “face-palm” is (at least not in this context).

    For what it’s worth, thank you for the pageviews…


  61. The proof that mathematicians use is x=0.999… and 10x=9.999… However, if you replace 10x with ANY OTHER NUMBER times x, x is still equal to 0.999… and not 1. The proof only works for that specific situation, not the entire set. x=0.999…=1 is an extraneous answer.

  62. the actual proof goes

    10x-x (9.999…-0.999….)=9, therefore 9x=9;
    Simplify 9x=9; x=1

  63. Also, in response to what you said, you can’t just separate it. The fact is that in that case, it=1, but it otherwise it=0.999. Therefore, they are equal. Just because two different methods of thinking lead to two different answers doesn’t make one wrong. It just makes them equal. For instance, 1/2 is equal to one half, but it’s also 10 tenths divided by two, which is 0.5. That doesn’t mean one is wrong. It means they’re equal.

  64. Congratulations for rediscovering hyperreal numbers! Unfortunately, they are already well-known.

    Hint: your “countable numbers” are what are commonly known as “real numbers”, and your “real numbers” are in fact the hyperreals. Hyperreals include reals, positive and negative infinity (what you call Penultimate, or P), as well as infinitesimals (which you usually write as “0.000…1” – the common notation is “0.(0);1(0)”). The full notation is a.b(c);d(e), which corresponds to a.b, followed by infinitely many repetitions of c, followed by d, followed by infinitely many repetitions of e. In your notation, that would be “a.bc…de…”.

    You are perfectly correct that 0.(9);(0) != 1.(0);(0) using hyperreals. However, 0.(9);(9) = 1.(0);(0).

  65. 0.999… is equal to the limit of 1-1/10^n as n increases withouth bound. When n is raised without bound, 1/10^n approaches 0, and thus the limit, and 0.999… are equal to 1!!!

  66. The string of nines is infinite. I take the string of nines, let’s call it string 1, and duplicate it to String 2. I can correspond the nth 9 in String 1 with the (n+1)th 9 in String 2, giving us a one-to-one correspondence, but we fill the first slot with a 9. Now we multiply String 2 by 10, subtract String 1 from String 2, and everything cancels out, giving us 9x=9, thus x=1.

  67. “a near-blaphemous use of the transitive propertey”
    When applied to numbers, a mathmatical property is always true.There is no way to use it ‘blaphemously’ , only incorrectly. However, since all the steps in this chain are true, the entire path is true.

  68. Noted Scholar wrote:

    You write, “(100 * .999…) – (1 * .999…)”. But one of the paradoxes of infinity is that you cannot subtract one from the other. For example, Hilbert’s Hotel.

    First of all, .9… is not an infinite quantity, so this objection doesn’t really apply. You don’t have a basis for arguing that .9… is a number which somehow cannot be subtracted from itself.

    Secondly, you apparently accept that the “rules” can be different when it comes to infinite sequences. Well, your original argument against the equality was inductive: 1 is greater than .9, .99, and so forth. This argument takes for granted that the same principle must apply even if the number of nines written down is infinite. But of course, the “rules” are different for infinities, in ways that are entirely self-consistent. For example, as the number of nines in the sequence increases, the difference between the written number and 1 becomes smaller and smaller. When the number of nines is infinite, the difference is “infinitely small”, which is necessarily the same thing as 0 in our number system.

  69. You know, I have a similar problem with limits and calculus. It’s one thing to toy with the theory, but in physical application, I dunno, it seems problem ridden. For example, the idea of “instantaneous velocity”. O, we just ram a limit in there, as delta-t approaches zero, call it calculus, and claim that we have an equality. But we don’t REALLY. Because OBVIOUSLY if there is no duration of time, stuff doesn’t move, an therefore, can not have a velocity. It has always really bothered me that we use limits in this way, to tease out a quantity that we can use for further calculation and experimental verification, and we just sort of ignore the fact that what we are claiming is, actually, physically impossible. I have brought this up with one of my profs, and was sort of patronized about it, so I just keep it to myself now. But I feel like, I dunno, we are sort of missing some profound, and we keep passing over the clue to the revelation because it is inconvenient.

  70. Unnoticed,

    Your theory OBVIOUSLY builds in the loaded notions of “limit” and “bound.” But these notions, as any student of maths knows, are intertwined with the very notions under dispute.


  71. John Doe,

    This is completely insane. If the string of nines is infinite, then you can’t duplicate it. Let’s keep in mind that I am working in a non-Kantian framework here, so I reject the notion of multiple infinities.


  72. Thomegemcity,

    You write, “When applied to numbers, a mathmatical property is always true.”
    I can’t understand this sentences. Properties are not true or false, they are had or un-had.

    Please come again,

  73. Lenoxus,

    Thank you for your stimulating comment. It is better than many other comments here.

    I’m not sure I’ve accepted that the rules can differ for infinity, since I argue there is no such thing! There can’t be rules for, e.g., unicorns. I say there is a “paradox” of infinity, but that is a paradox for my enemy, not for me.

    As for the “infinitely small,” I think this is really just an argument for infinitesimals, which are not the same thing as zero (notwithstanding absurd formalisms to the contrary).


  74. .1 > .0X for any number X;
    We’ll take X = 9 because that’s the most we can fit into a single digit.
    .1 is more than .09 by .01; and greater than .0XX (with n X’s) by .001 (with n 0’s)
    With an infinite number of 9’s, the 1 at the end of the amount that .1 is greater by is endlessly far from the decimal; we can’t get it there by adding digits one at a time. (Because you can’t count to infinity.)

    This means that if we start with an endless number of 9’s (and therefore a difference between .1 and .09999… of 0.000…00001) Then the 1 has to be at the end of the endless decimal; such a thing simply doesn’t exist. There stops being a 1 at the end as soon as there stops being an end, which is why it becomes irretrievable. The difference between .1 and .0999… with an endless sequence of 9’s is 0.

  75. 1-0.(9)=0.(0)1?
    The parentheses indicate that there is no end. If there is no end, where does the one go?

  76. 0.999… = x
    10x = 9.999…
    10x – x = 9.999…- x
    9x = 9
    x = 1
    0.999… = 1

    Your opinion is irrelevant.

  77. 1 is a finite number. If you add 1 to a finite number, it remains finite. Therefore, by your induction, if we add infinitely many 1s, we still have a finite number. Thus, infinity is finite!

  78. 0.999… = x
    10x = 9.999…
    10x – x = 9.999…- x
    9x = 9
    ^ this is the mistake, this should be:
    9x = 8.999…1

    This Is not a valid method because the number of 9s whether finite or infinite are countable. So if they are countable they can be bijectively associated between the sets of infinities, just like Georg Cantor attempted to do between integers and rationals…

    The parentheses indicate that there is no end. If there is no end, where does the one go?”

    ^ this is mathematically incorrect. You can have an infinite distance between two points in euclidean and non euclidean geometry. In mathematics, you cannot compare infinities from one size to infinities of a different size. You could write:




    (which is a perfectly reasonable thing to do as you can write 0.(0)1 as 0.000…001), if you did this, then the algebraic “proof” becomes invalid.

    Counter proof. Supposing we have two integers:

    999…999 and 1000…000

    Factorising 999…999 we see that it is a perfect multiple of 3, since all the digits are multiples of 3. 1000…000 is a perfect exponent of 10, so has factors 2 and 5. We know that 1000…000 cannot have any factors 3. Likewise we know that 999…999 cannot have any factors of 2 or 5.

    According to the Fundamental Theorem of Arithmetic numbers with unique factorisation are unique numbers. Therefore 999…999 does not equal 1000…000.

    Since the number of digits and the relative sizes of the difference is completely identical to the 0.999…[999] and 1.000…[000] problem, it’s safe to say that 0.999… does NOT equal 1.

  79. “You are under no obligation to believe the centuries of mathematical theory that claims that .9 repeating equals 1. However, you are under a moral obligation to do so with a little humility, which ought to make you think twice about saying that those centuries of mathematical theory by some of the smartest minds in history completely missed a simple induction argument. It’s not that they missed it; it’s that it’s incorrect.”

    I resent comments like that. There is no evidence whatsoever to support this point of view. There are no books and peer reviewed documents of any mathematical credibility to support either the idea that 0.999… is 1 or that 1/3 is exactly 0.333…

    In fact… the 0.999… ~= 1 algebraic proof comes from a mathematics textbook and says:

    1895 Arithmetic for Schools says, “…when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small”

    This is not the same as saying they are equal.

  80. Anon Wibble: Even if you had found a mistake, you’ve found it in the wrong place. Assuming you agree that x can be assigned the value .999…, then you’d have to agree that 9.999… – x = 9, not that it equals 8.999…1 (although actually, it sort of does, because the “1 after all the nines” is meaningless, and 8.999… is just another way of expressing 9).

    9.999… just means “9 plus .999…”, right? So if you subtract .999…, you must be “back where you started”, at 9. Otherwise, we are turning subtraction into something dependent on representations. (However, if you choose to do it your way, you still get the right answer, because 8.999… is equal to nine. One of the nice things about accepting the equality of 0.999… and 1 is that it keeps everything consistent with everything.)

    Per your logic, we ought to calculate it like this:

    0.999… = x
    10x = “9 point infinity-minus-one nines”, because we “shifted over” the nines, right?

    But you seem to agree with Cantor, and therefore you should agree that “infinity minus one”, for countable infinity, is the same as infinity. Hence, the proof is valid: multiplying 0.999… by ten is the same thing as adding nine, because the number of nines after the decimal point is unchanged.

    As for your “integers”, they’re not integers at all, but something like p-adic numbers, which have their own rules and properties. Every integer is a finite distance from zero, while your “integers” are each infinite. There’s nothing invalid about them as numbers, but they’re simply not integers. (Otherwise, the set of natural numbers would be uncountable just like the irrationals, which we know is not true. Do you disagree with Cantor about that as well?)

  81. F U CK You notedscholar who is the person in the world with IQ 1
    hi go pick your nose and die
    you are not noted
    yourre just trying to make other people not know things in the world

  82. “Aria Turns | November 12, 2008 at 11:38 am
    density Theorem in real analysis says

    if x and y are real number with x<y, then exits a rational number r such that x<r<y

    so, it’s clear there is no rational number between 1 and 0.99999999…, it’s mean 1 and 0.99999999… are the same number."

    If x = 0.999999… with no end, then it's not a real number. If you want it to be a real number, there will be an end to it and the number between that and 1 will be the one which has something after than end…. as in 0.99999995 is between 0.9999999 and 1.

  83. You can’t induce from the finites to the infinite.
    Here’s a demonstration of the fallacy that you can:
    0.9 is 0.09… less than 0.999…
    0.99 is 0.009… less than 0.999…
    0.999 is 0.0009… less than 0.999…
    and you’ll conclude that 0.999… is less than 0.999…, which of course is nonsense.

    The fact is that 0.999… is an unending string of 9’s, so
    10*0.999… = 9.9…, and despite your claim, that is exactly true and is easily provable
    Now subtract 0.999… from both sides to get
    9*0.999… = 9 => 0.999… = 1 and that is a mathematical fact.
    Also if 0.999… is less than 1, the average is (1 + 0.999…)/2 = 0.999…
    The average must be as close to 0.999… as it is to 1. But the average = 0.999… so it must also be 1.

  84. I forgot another proof: In decimal notation, the rational numbers are etiher a finite string of digits e.g. 1/4 = 0.25, or a repeating string such as 1/7 = 0.142857142857… (note the repeat length is 6. To get the fraction back simply calculate 142857/999999 where there are as many 9’s as the repeat length. For 0.999… the repeated string is 9 and the length is 1, so 0.999… = 9/9 = 1. You’ll also see 9/9 = 99/99 = 999/999 etc. Going back a step 1/7 = 142857142857/999999999999 etc.

  85. You can prove this in a simple way…
    3/3 = 1

    3/3 = (2+1)/3
    =2/3 + 1/3
    =0.66… + 0.33…

    0.99… = 1 HENCE, PROVED…

    you can also prove… a lot using this trrick… lol

  86. Proof that 0.(9) = 1:
    (0.(9) means zero followed by an infinity of nines)

    Let x = 0.(9)

    Multiply both sides by n, where n is any number at all — real, imaginary, complex, it doesn’t matter:
    nx = 0.(9)n

    Subtract x from both sides and factor:
    nx – x = 0.(9)n – x
    (n – 1)x = 0.(9)n – x

    Divide both sides by x:
    n – 1 = 0.(9)n – 1

    Add 1 to both sides:
    n = 0.(9)n

    Divide both sides by n:
    1 = 0.(9)

    Commute 0.(9) and 1 if you like:
    0.(9) = 1


  87. You didn’t divide both sides by x. Because x = 1, no error occurs. But you have unintentionally made a circular argument. In a funny sort of way, that does make a proof.

    (n-1)x = n*0.(9) – x
    n – 1 = n*0.(9)/x – 1
    n = n*0.(9)/x
    1 = 0.(9)/x
    x = 0.(9)

    I cannot bring myself to write 0.(9)n, that is horrible and ambiguous.

  88. @ NotedScholar
    In the original argument, you claimed that you had “disproved” the idea that 0.999…=1 by proving the inverse, but that’s false. You must also refute all of the proofs that demonstrate that the statement is true. Otherwise, even if your argument is correct (it’s not), you’re just demonstrating that our current system of axioms and definitions are inconsistent, because we can use them to prove two mutually exclusive statements are both true.

  89. Temporarily adopting the ridiculous notation that’s been used: 1 – 0.000…1 = 0.999…9

    So 0.999… = 0.999…9 and then we must also have 0.000…1 = 0.000… by symmetry. So the 1 at the end (of something endless – ) has no real existence.

  90. Also, to be clear, no one is claiming that 1 = 0.999…999 (that is, a decimal with a finite number of nines). What many of us are talking about is 0.999…, a decimal with an infinite number of nines, which does equal one. However, based on your arguments, you follow finitism, a philosophy of math where only finite mathematical objects exist. In such a system, 0.999… does not exist; therefore, one could give it any property and be trivially correct. This is not a new concept. There are a number of people who have held such beliefs for centuries. Most mathematicians, however, will choose a different system so that they can perform more — and more meaningful — calculations, make more accurate predictions, and have a more practical usage, both in real life and in theory. It also allows us a more complete, consistent system to use in exercising their minds. So congratulations on proving something no one disagreed with in the first place.

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  92. > 0.1 is necessarily greater than 0.0X, where ‘X’ is any countable number.

    Isn’t that what you were trying to argue? Burying the conclusion in the assumption is not scientific.

  93. Dear the Magic M,

    I many times will say and have said and say on Twitter that #ScienceNeedsLogic, so I am loving your usage of a fallacy to accuse me here! It gets at the issues in a way that other comments do not get at the issues.

    What you quote is a real quote, from my post, this very post, yes. Was it what I was trying to argue? Not exactly, although this is always tricky in maths. Recall that in maths the conclusions is ALWAYS incarnated into the premise. This is because maths is value-laden with necessity. A la priori reasoning is the name of games in maths.

    So for once hand, yes: it is what I am arguing. But on twice hand, no: I am engaging in mathematical induction, where the conclusion is slowly excavated from the premise.

    I hope this helps,

  94. Dear NS,

    Could you expand further on the definition you use for the number “0.999…” ?

    I understand most people (implicitly or otherwise) seem to view it as “the limit of the sequence of finite decimal numbers 0.9, 0.99, 0.999, …”, or equivalently, “the sum of terms 0.9 + 0.09 + 0.009 + …”, which, practically by definition, leads it to be equal to 1.

    As with most definitions involving such ambiguous concepts as infinity however, it is rather arbitrary. It stands to reason that you could define it in another way that remains just as intuitive, yet leads to a concept of 0.999… – and other numbers with similar notation – being separate entities from the real numbers, and therefore not equal to them.

    I suspect these unspoken differences in the implicit mental model people have for 0.999… is the main reason for the difficulties they have in coming to an agreement about the conclusions.

    Therefore, I am curious about the mathematical meaning you ascribe to such numbers.


  95. Okay, first off, how old are you, because if you’re younger than 25, then there’s still hope of saving you from this New-Age Old-Age nonsense!

    Second, I get it. Infinity IS a scary concept. I mean, think about it. Picture you’re in an endless expanse, a black void where you are completely alone and somehow, despite the absence of any light, you can see yourself. (This is where a lot of people, especially people on RationalWiki, would like for you to–*SMACK*–Owwwwwwwwwwww…) Now, think on this: to the rest of the space around you, you are about the size of a grain of sand.

    But no worries; you can (for some reason) call into this endless void whatever exists in our universe, so you call in the Earth, as huge a thing as it is…yet it will still only be the size of a grain of sand relative to the rest of the void. Call in the whole solar system, and it’ll still be the relative size of a grain of sand. Call in the whole GALAXY, and it’ll still be the size of a grain of sand. CALL IN THE ENTIRE OBSERVABLE UNIVERSE, AND EVERYTHING BEYOND THAT, AND IT’LL STILL BE THE RELATIVE SIZE OF A GRAIN OF SAND.

    Even if you blow up that universe to a million bajillion times its size (that’s about the square root of googol, I checked), it would still have that same relative size. Ditto is if you blew it up another million bagillion times (half of googolplex plus 2 and a half; my friend and I had a bit of an arms race in kindergarten with these).

    So yea, infinity is pretty hard to wrap our heads around, and that’s because we are ultimately finite creatures. Our time on this earth is finite, the boundaries of our bodies and minds are finite (psychic powers notwithstanding), even the number of palpable relationships you can have is finite. It’s hard to appreciate the vast concept of infinity if we can only experience a very small fraction of a what is really just a portion of it.

    However, despite that, putting a limit on how far numbers can go is, at best, wishful thinking, despite how intuitive it may seem to have it set at the largest number of general relevance. While some sciences, like chemistry, astronomy, physics, etc., do set a limit for distance and time as it approaches zero – the Planck length and the Planck time, respectively – that’s only because there is only so far that we can divide before it all becomes superfluous; why say an electron is one million micro-Noli-lengths when you could just say it’s one regular Noli length? (The Noli length is a trademarked length for the length of the single f**k I give about significant figures. I’m not a chemist, but I try.)

    So how about going the other way? What if we try and find your number P, the “penultimate number”, and let’s say, for the purposes of this thought exercise, we say that P equals 100,000, a rather small number compared to the likes of googolplex or Graham’s number, but still fairly large nonetheless. (Think of it this way: you wouldn’t want P people at your doorstep.) Okay, so let’s say we also have lowly 3, but 3 wants to be bigger, so it just keeps adding itself to itself, going from 3 to 6 to 9 and on and on. Now what happens, after 33,332 additions, when it reaches P? Does it stop at P when 3 is added again, even though the new value should be 100,002? Or does it continue upwards regardless of P as a limit, forcing a redefinition of P? And since P IS a quantifiable number, couldn’t we also square it and get a more massive number than P? Better yet, couldn’t we also find 2 to the POWER of P, creating an even MORE massive value?

    “Alright,” you say, “then we’ll set P to where it’s divisible with EVERYTHING up to it.” Okay, then, I reply, so P becomes (P-1)!, multiplying every number up to P together to create an insanely gargantuan number that can still be reached legitimately by exponential, additive, and multiplicative means. But herein lies the problem. The difference between P and (P-1) is, well, 1, not [(P-1)^2 – (P-1)] as it would have to be in order for P to function the way it should. As a result, you’d have to keep setting P higher and higher to satisfy the finite limit, but you find you just can’t do it: P WILL CONTINUE TO GROW BIGGER AND BIGGER AS YOU TRY TO MAKE SURE ITS CONDITIONS ARE SATISFIED.

    And here, I think I can pinpoint a flaw in your logic: infinity is NOT a number; it’s a set of numbers. While we may talk of some infinities being bigger than others, that’s because some infinities can’t be compared one to one to other infinities because there’s always going to be an intermediary value that you haven’t found yet. For example, say you have a number line going from 1 all the way through the infinity of counting numbers. For each counting number, you’ll then make a completely random infinite irrational decimal with no repeating digits, on and on forever. However, once you’re done (again, SOMEHOW), you can always do this: take the first digit from the first decimal, the second digit from the second decimal, the third from the third, and so on, and add one to each digit, looping back around to 0 if the digit was 9. Now you have a number that doesn’t correspond with anything on your number line, meaning the decimals’ infinity doesn’t have a one-to-one correspondence with the counting numbers’ infinity.

    So, you ask, why DOES it matter? None of these numbers will ever be generally relevant, so why not just set a limit to avoid taxing our brains. While chemistry and biology and astronomy may have to comply to set limits based on the limits of our own perception of the universe, mathematics is all about the theory. It’s the same reason all those math problems in elementary school have someone buying upwards of twenty carts worth of pineapples; it’s all about concepts, no matter how much it can be applied to in real life. It exists in a nebulous space where we can have grids pop in and out of existence with meaningless lines (meaningless to our reality at least) represented by meaningless numbers (” “) within a meaningless formula (^w^); employing infinity in this nebulous space isn’t just possible to do, but it’s par for the course in mathematics, ESPECIALLY when dealing with limits. (Fun fact: a LOT of limits go THROUGH infinity, like x+2, 3x, x^4 [that one does it TWICE!], etc.)

    All in all, yeaaaaaaaa, you may be a BIT wrong on this. Just saying. I’d say try not to stroke your confirmation bias so much and try to search for some actual fact supported by loads of people who not only do this stuff for a living, but do it for the love of discovery about the world around us, not because they want to introduce some illusion to the general public (or if you are a satirist, stop yelling, let go of the girl, come out of the building, and put your AK on the ground).

  96. A proof that 0.99… = 1. The series a(n) = ∑(k=1, n) [9(10^-k)] represents the number, 0.99…99 for some finite number n of nines, and so the limit as a(n) becomes large is represented by 0.99… So, therefore, b(n) = 1 – a(n) represents a number 0.00…01 with a finite number n – 1 of zeroes. For all n, we can clearly see that b(n+1)/b(n) = (10^-1). Since b(1) = 0.1, we find that b(n) = 0.1(10^-n-1). As n grows arbitrarily large, b(n) grows arbitrarily small, and 1 – a(n) grows arbitrarily small. Therefore, the limit of the series a(n) as n grows large (i.e. 0.99..) is one. Q.E.D. I dare you to find a hole.

  97. Actually, Mutedahideen, this video shows that the believe that you are pushing is completely subjective. Mathematicians admit (behind the doors of Youtube) that 0.999… is just “what we want it to be.”

  98. Pingback: 0.999…≠1, confirmed – Science and Math Defeated

  99. rampantchaos702724, 0.999… has countably infinitely many 9s means that there is a bijection between the 9s and the natural numbers. Just like the last 9 of 0.999.., your understanding is non-existent.

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  101. Oof.
    Hi I’m from the future. I tried to read through all the comments but quit after a whole lot of nonsense.

    This ‘argument’ boils down to one misconception. It’s the difference between a decimal with a discrete number of numbers after the decimal and a decimal with INFINITE numbers after the decimal. OF COURSE, any decimal with a discrete number of repetitions doesn’t equal an integer. Nobody in all of space and time disagrees. I suspect this is the problem because notedscholar’s repeated use of ‘penultimate’. If the series after the decimal is INFINITE, it’s a different story.

    Try to fit ‘infinite series’ and ‘convergence’ into the search terms in your apparently voluminous research. You can even find out something about induction.

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