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Proving 0.999... Is Equal To 1 1260

Posted by CmdrTaco
from the nerds-like-math-right dept.
eldavojohn writes "Some of the juiciest parts of mathematics are the really simple statements that cause one to immediately pause and exclaim 'that can't be right!' But a recent 28 page paper in The Montana Mathematics Enthusiast (PDF) spends a great deal of time fielding questions by researchers who have explored this in depth and this seemingly impossibility is further explored in a brief history by Dev Gualtieri who presents the digit manipulation proof: Let a = 0.999... then we can multiply both sides by ten yielding 10a = 9.999... then subtracting a (which is 0.999...) from both sides we get 10a — a = 9.999... — 0.999... which reduces to 9a = 9 and thus a = 1. Mathematicians as far back as Euler have used various means to prove 0.999... = 1."
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Proving 0.999... Is Equal To 1

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  • by Anonymous Coward on Thursday October 14, 2010 @09:26AM (#33892528)

    (0.999...)st Post!

  • by MyLongNickName (822545) on Thursday October 14, 2010 @09:27AM (#33892532) Journal

    I was able to prove that with even one less "9" after the decimal point, it STILL equaled 1. I plan on doing this for a few more iteration until I can prove that . = 1

    • by MyLongNickName (822545) on Thursday October 14, 2010 @09:29AM (#33892562) Journal

      And seriously... is this really front page material? The simplest proof is to say "express 1/9" as a decimal. Now multiply both sides by 9. I remember this in elementary school algebra.

      • by betterunixthanunix (980855) on Thursday October 14, 2010 @09:31AM (#33892606)
        Small numbers usually win; express 1/3 as a decimal, and multiply by 3. The problem with that, though, is that people have trouble accepting that there was nothing wrong with what they did -- a lot of people have this implicit assumption that if a few simple steps bring them to a result that doesn't look like it makes sense, then they did something wrong. If you get them a more complicated proof (assuming they can follow it), they are more willing to accept the result.
        • by MozeeToby (1163751) on Thursday October 14, 2010 @10:19AM (#33893478)

          a lot of people have this implicit assumption that if a few simple steps bring them to a result that doesn't look like it makes sense, then they did something wrong.

          It should be noted that this is not a bad thing. Indeed, it is one of the first things that a good math teacher will teach to the class - all answers should go through a 'does this make sense?' filter before you consider the problem done. It is only very rarely that it causes problems, and it is exceedingly common that it prevents them.

        • by radtea (464814) on Thursday October 14, 2010 @11:06AM (#33894450)

          The problem with that, though, is that people have trouble accepting that there was nothing wrong with what they did -- a lot of people have this implicit assumption that if a few simple steps bring them to a result that doesn't look like it makes sense, then they did something wrong.

          Nope, the problem is that the people who discuss this question are lousy teachers. They set it up deliberately to create a block in other people's minds that makes it unnecessarily difficult for them to understand what is being claimed and why it is true.

          If instead they said, "It is possible to represent numbers in different ways. We all know this, and it's completely uninteresting, but I'm going to bore you with it anyway. You know you can represent 1/3 as 0.3333... right? No big deal. Now curiously that also means you can represent 1/1 = (3*1/3) as 3*0.3333... or 0.99999... It's just a different representation of exactly the same value. You can of course also represent 1 as 5*1/5 1/2+1/2 and all kinds of other awkward and unintersting ways, too."

          I'm not sure why people insist on presenting this result in the most counter-intuitive way possible and then wasting vast amounts of time trying to undo the damage they've inflicted with their incompetent introduction of the problem. My guess is that they are simply not very smart, as anyone who isn't fairly dumb would see that there is an obvious pedagogical problem at play here, and correct their presentation accordingly, rather than blindly and stupidly repeating the rote "0.9999... = 1" introduction to the remarkably dull fact that you can represent the same value in different ways.

          Of course, in an insanely strictly typed language with infinite precision 0.999... would not quite be the same as 1, as the former is a real and the latter is an integer, so despite having the same value their different types would mean they could not be used identically in all circumstances.

          • by Benfea (1365845) on Thursday October 14, 2010 @11:39AM (#33895120)

            As soon as you get to "You know you can represent 1/3 as 0.333... right?", you hit a brick wall. People who believe that 0.999... does not equal one also believe that 0.333... does not equal 1/3, and for many of the same reasons. Taking your approach, you simply shift from arguing about whether or not 0.999... equals one to arguing about whether or not 0.333... equals 1/3. You have to get at the root of the problem of why they refuse to believe those numbers are equal before you can get anywhere.

            • People who believe that 0.999... does not equal one also believe that 0.333... does not equal 1/3, and for many of the same reasons.

              For once in my life I can claim someone is underestimating the average person!

              I don't believe .999... = 1. Let me qualify that a bit, I intellectually and academically know it, but on a softer, more psychological level, I don't actually believe it. When presented with it, my first reaction would be "Hell no! Stupid.", even though I know it is true.

              Why? Because your mapping two concepts that we all were taught as a kid isn't true. Does .9 = 1? Or .99? Or .999? or ... Or .999999999999(a ridiculous but non-infinite number of times)? Most grade school kids would say "no", and be correct. Then you hit the infinite jump, and suddenly it becomes true. So you run into two problems, the problem of it not being immediately obvious (common sense), and the problem of conceptualizing infinity.

              On a lower level, its like saying A = ~A. You have a proof saying basically that ~A was A all along, so the actual preposition was wrong, which makes sense, but on a surface level all you can see is A =~A.

              I have no problem whatsoever with 1/3 = 0.3333... This makes sense, its like stating A = A. 1/3 being 0.3333 is obvious. I would even get in trouble in lower level math classes for not mucking with fractions, and going straight for the decimals, since I never say fractions outside of cookbooks and socket sizes. 1/3 = 0.33333... makes sense, it is clear and obvious, and can be explained with a single phrase (not a proof); "the "/" means division". .999999... doesn't have this.

              No, I'm not stupid, or at least for this reason. I know damn well that 0.9999... = 1, and if I ever find myself in a situation where that bit of knowledge can be applied (usefully, not just for building my ego on the internet), I will do it properly. My first reaction is still "bullshit!" on a visceral level, though. I don't perceive it as true, even if I know it is.

              I suppose I can map this experience to most of the "social knowledge vs. science" debates in our culture currently. I won't.

        • by mea37 (1201159) on Thursday October 14, 2010 @11:11AM (#33894554)

          People assuming they did something wrong when the result "doesn't make sense" isn't the problem.

          People failing to distinguish between a notation and a number, creating the belief that "0.99(9)=1" doesn't make sense, is the problem.

          Consider this proof, which follows simple steps to reach a conclusion that doesn't make sense:

          i^2 = -1 (definition of i)
          i^2 * i^2 = -1 * -1
          i^4 = 1
          sqrt(i^4) = sqrt(1)
          i^2 = 1
          -1 = 1

          Then if you want you can add 1 to both sides and divide by 2, to find 0 = 1.

          Now, do you know why this proof is bogus? When I was in high school, we were introduced to imaginary numbers, and I drew up a slightly more obfuscated version of the above; it had a lot of people (including a couple relatively sharp teachers) in "I know you did something wrong because the result doesn't make sense" mode for a long time.

          The fault, of course, lies with the sqrt() step. For a=a to imply sqrt(a)=sqrt(a), we have to interpret sqrt(a) as the pricple square root function, so sqrt(x^y) = x^(y/2) doesn't necessary work when x isn't a real number.

          Without the motivation of "this result cannot be right", I wouldn't have puzzled this out. More than that, the solution comes from understanding that rules we take for granted only apply to certain types of number. Applying that to 0.99(9), it's easy for people to convince themselves that repeating decimals are a special class of number subject to "some rule I just don't know".

          But in this instance, that reasoning is flawed, because .99(9) really is just a regular real number in a weird notation.

          • by Evo (37507) on Thursday October 14, 2010 @11:37AM (#33895086)

            Surely the problem is that you're assuming sqrt(1) = 1 when actually it is +- 1? You're throwing away the sign change in that step :)

            • Re: (Score:3, Informative)

              by agrif (960591)

              Surely the problem is that you're assuming sqrt(1) = 1 when actually it is +- 1? You're throwing away the sign change in that step :)

              Yeah, that's the problem, but for those interested...

              The base problem with this is that unlike the logarithm for real numbers, the logarithm for complex values is not a function (or, if you like, it's a "multi-valued function"). This comes from the interesting fact that x^1 has 1 solution, x^2 has two solutions, x^3 has 3 solutions, and so on. We kind of fudge around it in reals, because x^n will only ever have one or two solutions, but in the complex plane it has n solutions, and things are much more compl

            • Re: (Score:3, Informative)

              by SashaM (520334)

              I'm probably too late to get modded up, but since none of the existing responses gave the exactly correct explanation, I'll have to post rather than moderated.

              sqrt(1) is 1. It's not -1. By definition.

              A list of transformations of an equality like the one given in the grandparent's "proof" is shorthand for a list of "implies" statements. For example, a proof like this:

              2x-4=0
              2x=4
              x=2

              is actually shorthand for:

              A. 2x-4=0 (assumption).
              B. 2x-4=0 implies 2x=4 (by rules of arithmetic).
              C. 2x=4 implies x=2 (by ru

        • by Benfea (1365845) on Thursday October 14, 2010 @11:26AM (#33894872)

          The problem with the argument you present is that people who don't believe 0.999...=1 also don't believe that 0.333...=1/3. They can't quite wrap their heads around the concept of infinity, so in their minds 0.333... continually comes closer to 1/3, but never quite reaches it because they can only imagine a finite number of digits. They honestly think of infinity as being a really large finite number, so they believe that no matter how many digits you add to 0.333..., it never quite reaches 1/3.

          Another part of the problem is that many people simply can't wrap their heads around is that they don't separate the idea of a number and the symbols used to represent numbers, thus they cannot grasp that some numbers can be represented in more than one way by our number system.

      • by cgenman (325138) on Thursday October 14, 2010 @10:19AM (#33893460) Homepage

        Conceptually, 0.999... keeps getting closer and closer to 1, as you add more decimal places. It approaches 1. This limit is how all calculus works. Any series that approaches another number as you flesh out the series further and further, will be that number once you have taken the series to infinity.

        • by Culture20 (968837)
          The problem is that many times in calculus, "approaches" is described that the number approached is never reached.
          • Re: (Score:3, Interesting)

            by blueg3 (192743)

            Only in a few cases (and the notable case of "infinity is not a number"). Anyone familiar with the derivation of limits, derivatives, and integrals should be familiar with finite numbers that are the result of an infinite-step process.

      • Re: (Score:3, Insightful)

        by smallfries (601545)

        And how do you do multiplication of an infinite series of digits? I'm guessing that you don't start from the right-hand side.... but beyond that your approach seems to be simple because it is incomplete. Kind of the point of the article really.

      • by MobileTatsu-NJG (946591) on Thursday October 14, 2010 @11:13AM (#33894624)

        And seriously... is this really front page material?

        You'd rather argue about smartphones?

    • Re: (Score:3, Funny)

      by gmuslera (3436)
      So, if ./ = 1/, you just messed my filesystem, and the web.
  • Finally (Score:2, Funny)

    by kannibul (534777)
    Someone disproved math. Kids around the world celebrating. Accountants are lighting themselves on fire. Corporate greed accellerates. 'Office Space' now seen as a prophecy.
    • Re:Finally (Score:5, Funny)

      by Vectormatic (1759674) on Thursday October 14, 2010 @09:38AM (#33892704)

      just as long as no-one proves 0 = 1 we computerpeople are safe...

      • by alexhs (877055)

        But b00000000 = b11111111 in one's complement [wikipedia.org] system...

        So 0 == 1 as long as you're using 1 bit wide one's complement integers...

      • Re:Finally (Score:4, Insightful)

        by Jason Levine (196982) on Thursday October 14, 2010 @10:22AM (#33893546)

        Suppose you have 3 numbers, a, b and c such that c = b - a.

        Multiply each side by (b - a) to get:

        c(b - a) = (b - a)(b - a) => Or....
        cb - ca = b^2 - 2ba + a^2 => Now add (ab - a^2 - cb) to both sides
        ab - ca - a^2 = b^2 - cb - ba => Or....
        a(b - c - a) = b(b - c - a) => Divide both sides by (b - c - a) and.....
        a = b

        There you go! Proof that any two numbers (such as 0 and 1) are equal.

        (Yes, I know there's a flaw in there. Let's see who'll spot it first.)

      • Re: (Score:3, Interesting)

        by Galestar (1473827)
        1 = 0.999...

        since 0.999 can also be expressed as 1 - 1/infinity,
        1 = 1 - 1/infinity
        0 = - 1/infinity
        0 * infinity = -1 / infinity * infinity
        0 = -1
        1 = 0
    • Re:Finally (Score:4, Insightful)

      by SoVeryTired (967875) on Thursday October 14, 2010 @09:48AM (#33892878)

      If pressed, many logicians will admit that the modern foundation of mathematics (ZFC) is probably inconsistent.
      See this article:
      http://www.math.princeton.edu/~nelson/papers/warn.pdf [princeton.edu]

      The author discusses an informal survey he took among loogicians on page three.

      If someone ever discovers a paradox, we can simply scale back to some other system and keep most of what we know, but still...

      • Re:Finally (Score:4, Insightful)

        by drakaan (688386) on Thursday October 14, 2010 @10:27AM (#33893668) Homepage Journal

        Okay, but this isn't a problem with the foundation of math being inconsistent, this is a problem with people not knowing how to write the number normally known as "1" in a different way. Most people would grasp "3/3" as being the same as 1, but this *looks* different because they're unused to seeing it.

        The fact that the fractions 1/3 (known in decimal notation as .3...) and 2/3 (known in decimal as .6...) have a sum that can be written funny doesn't mean that they don't still add up to 1.

        A mathematical amusement causes people confusion and consternation. It's like asking someone why they appear reversed left-to-right in a mirror, but not top-to-bottom, and saying there's an inconsistency in the foundation of physics.

        The problem is that partial understanding of a subject and an associated problem in that subject makes things *appear* inconsistent when they are not.

  • by betterunixthanunix (980855) on Thursday October 14, 2010 @09:28AM (#33892552)
    0.999... = 1 is second place to the Monty Hall Problem on the list of things that people have difficulty understanding and accepting the proof of. It is second place because the only department where I do not see graduate students giving me a confused look is the math department; with the Monty Hall problem, I will sometimes get a confused look even from people in the math department.

    The other reason I put it in second place is that most people have difficult understanding the problem at all, whereas very few people have trouble understand what the Monty Hall problem is asking.
    • Re: (Score:3, Insightful)

      by bluefoxlucid (723572)
      I'm more interested in the .5 repetand, 5/9. Besides, 8/9 is 0.8 repetand; 9/9 would be 0.9 repetand but 9/9 = 1.
    • by MyLongNickName (822545) on Thursday October 14, 2010 @09:32AM (#33892612) Journal

      It is easy to explain.

      1. 1/9 = 0.111111111111111111111111111111.....
      2. Multiply each side by 9
      3. 9/9 = 0.999999999999999999999999999999......
      4. Simplify fraction
      5. 1 = 0.999999999999999999999999999999......

      Monty Hall trips up even serious math enthusiasts.

      • Except that I was counting non-mathematicians as well. A lot of people have difficulty grasping what is going on with 0.999... and what it means for that number to equal 1 (the idea that a number could have two representations in the same base goes over a lot of people's heads).
      • by kannibul (534777) on Thursday October 14, 2010 @09:45AM (#33892816)
        This could be done with any fraction represented as a repeating decimal.
        The trip-up is that it's repeating...since we have no concept for infinity, and, that there's no method of resolving a fraction w/ repeating decimal...it's not an accurate representation of the fraction - that's the flaw.
        Therefore, Fractions are Good. Decimals are Evil!
        Good thing our banks, credit card companies, and governments don't use repeating fractions.
        • Re: (Score:3, Insightful)

          by mcgrew (92797) *

          The trip-up is that it's repeating...since we have no concept for infinity, and, that there's no method of resolving a fraction w/ repeating decimal...it's not an accurate representation of the fraction - that's the flaw.
          Therefore, Fractions are Good. Decimals are Evil!

          So, what's the exact value of PI represented as a fraction?

      • by Johnny Mnemonic (176043) <mdinsmore.gmail@com> on Thursday October 14, 2010 @10:59AM (#33894270) Homepage Journal

        Typical engineer. Here's the operations perspective:
        a reliability of 1.0 equates to never fail.
        a reliability of .999... means "sometime fail".

        The sales guy will sell 1.0, and when failure happens, explain that what was really meant was .999...

        Good luck with that.

    • The Monty Hall problem and its delinquent cousin the Tuesday Boy problem are genuinely difficult because the answer is highly dependent on the way that the question is posed.

      0.9999...=1 is not genuinely difficult because at the end of the day it's a very informal statement about adding an infinite number of decimals, and the only real controversy about the statement exists among 4chan trolls and Wikipedia users. Most who don't understand don't care and most who do understand also don't care.

      The only people with a problem are the people who don't understand but still care, but then that's the problem with most topics these days.

      • by metamechanical (545566) on Thursday October 14, 2010 @10:37AM (#33893834)

        The Monty Hall problem and its delinquent cousin the Tuesday Boy problem are genuinely difficult because the answer is highly dependent on the way that the question is posed.

        I would argue that the Monty Hall problem is difficult because people don't take into account the fact that the result is NOT path independent.

        It would be much easier (I think) to understand intuitively if people realized that it was highly likely that they picked the wrong door to start. A more intuitive way of explaining the problem to somebody would be to increase the number of doors - to say, infinity. If there are infinity minus one closed doors with goats behind them, and a single door with a car behind it, the odds are obviously very high that you picked a goat. The probability that you picked the car is vanishingly small. Therefore, when the host opens every door except yours and one other, and they all reveal goats, the odds are very, very high that the other door hides a car, and yours hides a goat.

        Now, reduce that to 3 doors. The same logic applies.

        • Re: (Score:3, Interesting)

          by AllieA (170303)

          I am not a mathematician, but I have always considered the Monty Hall trick to be more of a word trick than any basis in mathematics. Look at it this way:

          If you pick one door out of a million and Monty Hall opens 999,998 others and it's between yours and the other door, there's a good chance Monty Hall knew where the car was since the chances of him doing that at random are so small, so of course your chance is better if you switch to the other door since there is a strong probability he didn't miss that on

      • Re: (Score:3, Insightful)

        by characterZer0 (138196)

        Tuesday Boy is a difficult problem not because of the math behind it but because of the grammar in the question.

    • by Missing.Matter (1845576) on Thursday October 14, 2010 @09:56AM (#33893018)
      I remember being told this in highschool. There was much objection, but the teacher shut us up by simply saying "give me a number in between them."
      • by Chapter80 (926879) on Thursday October 14, 2010 @11:11AM (#33894578)

        I remember being told this in highschool. There was much objection, but the teacher shut us up by simply saying "give me a number in between them."

        Duh. 0.9999... and a half!

  • I have wikipedia too... [wikipedia.org]: "When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999... equals 9.999..., which is 9 greater than the original number. To see this, consider that in subtracting 0.999... from 9.999..., each of the digits after the decimal separator the result is 9 9, which is 0. The final step uses algebra:"
  • by Anonymous Coward on Thursday October 14, 2010 @09:32AM (#33892614)

    Now I can replace my SLA with 100% uptime.

  • Or (Score:3, Insightful)

    by Anonymous Coward on Thursday October 14, 2010 @09:33AM (#33892624)

    1/3 = 0.3333...
    2/3 = 0.6666...

    0.3333.... + 0.6666.... = 0.9999....

    1/3 + 2/3 = 1 = 0.9999.....

  • a = b
    a^2 = ab
    a^2 - b^2 = ab - b^2
    (a+b)(a-b) = b(a-b)
    a + b = b
    2b = b
    2 = 1

  • This is so old... (Score:4, Interesting)

    by DiSKiLLeR (17651) on Thursday October 14, 2010 @09:35AM (#33892666) Homepage Journal

    This is so old...

    Even Blizzard issues a press release about it years ago because people kept arguing about it on the Blizzard forums.

    http://www.mbdguild.com/index.php?topic=14915.0 [mbdguild.com]

  • by elrous0 (869638) * on Thursday October 14, 2010 @09:36AM (#33892682)

    Humans are used to natural numbers because they're simple. But do natural numbers even exist in the real world? For the vast majority of practical purposes, 0.99999 can be thought of as one. But "one" itself is usually just a construct in the real world. There is no such thing as the perfect one of anything. The more precise we get, the more "one" becomes more of a mathematical ideal than a reality. So we spend our entire lives rounding off, because that's practical. We teach kids to count 1, 2, 3, 4... We can't very well teach them to count 0.000001, 0.00001, 0.0001, 0.001... (or any of the infinite variations of "counting" without resorting to natural numbers).

    Proving that 0.99999 = 1 is an interesting intellectual exercise. But in the real world, we do it every minute of every day.

    In other words--eh, close enough.

    • Re: (Score:3, Insightful)

      by Garble Snarky (715674)
      You're missing the point. Unless that whole post was just a set up for the punchline, what you're talking about is almost entirely unrelated to the issue in the article.

      It's not about rounding, or counting, or any real-world interpretation of any math concepts. It's about the trouble that people have understanding that 0.999... is simply notation that refers to the limit of an infinite sequence of numbers.
    • by blueg3 (192743) on Thursday October 14, 2010 @10:41AM (#33893922)

      They're not proving "0.99999 = 1" at all. That's not true. They're proving that "0.999... = 1". One is an infinite sequence of digits, and the other isn't. The distinction is important. The proof of "0.999... = 1" has nothing to do with rounding, and to suggest so indicates a (common) gross misunderstanding of the problem.

      First, you only measure things with such poor precision because you're working well above the quantum level.

      Second, natural numbers are certainly important. For one, they're critical to our understanding of the rest of mathematics, which is important for fancy things like being able to take measurements and manipulate them at all. For another, we work with whole numbers of objects all the time -- two apples, ten antelope, four huts, etc. It's not "10 +/- 0.01 antelope".

  • So after all these years, has Intel been vindicated [wikipedia.org]?

  • by Anne_Nonymous (313852) on Thursday October 14, 2010 @09:40AM (#33892738) Homepage Journal

    In the high school gym, all the girls in the class were lined up against one wall, and all the boys against the opposite wall. Then, every ten seconds, they walked toward each other until they were half the previous distance apart. A mathematician, a physicist, and an engineer were asked, "When will the girls and boys meet?"

    The mathematician said: "Never."
    The physicist said: "In an infinite amount of time."
    The engineer said: "Well... in about two minutes, they'll be close enough for all practical purposes."

    • by Daniel_Staal (609844) <DStaal@usa.net> on Thursday October 14, 2010 @10:07AM (#33893226)

      Actually, a good physicist should have been able to give an answer (or something close to it) as well...

      Eventually, they will come to a point where they would be required to move less than 1.616252(81)×1035 meters closer together. From the uncertainty principle, we know we cannot measure position more accurately than that. So either they will not move at all, or they will superimpose at that point.

      • by Anne_Nonymous (313852) on Thursday October 14, 2010 @10:28AM (#33893678) Homepage Journal

        In the high school gym, all the girls in the class were lined up against one wall, and all the boys against the opposite wall. Then, every ten seconds, they walked toward each other until they were half the previous distance apart. A mathematician, a physicist, and an engineer were asked, "When will the girls and boys meet?"

        The mathematician said: "Never."
        The physicist said: "Eventually, they will come to a point where they would be required to move less than 1.616252(81)×1035 meters closer together. From the uncertainty principle, we know we cannot measure position more accurately than that. So either they will not move at all, or they will superimpose at that point."
        The engineer said: "Well... in about two minutes, they'll be close enough for all practical purposes."

  • And if (Score:4, Funny)

    by Dunbal (464142) * on Thursday October 14, 2010 @09:40AM (#33892740)

    0.99999... is equal to 1, then 0.999999...8 is equal to 0.99999... and 0.9999999...7 is equal to 0.999999...6 etc etc etc until 1 = 0! Holy shit!

    Or we could just admit that using a tool incorrectly produces idiotic results.

  • by qmaqdk (522323) on Thursday October 14, 2010 @10:07AM (#33893222)

    This just goes to show that people don't really know what numbers are, at least when they are infinite decimal numbers. A finite decimal number corresponds to a rational number, e.g. 9.99 corresponds to 9 + 9/10 + 9/100. The way you describe infinite decimal numbers of by denoting a sequence of finite decimal numbers that goes towards this infinite decimal, in our case: 0.9, 0.99, 0.999, etc. This, by the way, is how you construct the real numbers (pi is described in such a way).

    In doing so, however, there are multiply ways of describing the same number; the sequences 0.9, 0.99, 0.999, etc. and 1, 1, 1, etc. describe the same number, and this apparent non-uniqueness is probably what bugs people.

  • More fun... (Score:4, Informative)

    by digitig (1056110) on Thursday October 14, 2010 @10:56AM (#33894224)
    It's more fun to work out why this proof fails when using non-standard analysis (in which 0.999... != 1).

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