Posted by: danb | January 13, 2009

Interesting Proof By Contradiction

With the final coming up tomorrow and the possibility of a proof by contradiction on the exam, I remember one that I saw in high school and would like to share it, both as a review and as an interesting proof.  Does 1=.99999…?  At first I thought it didn’t, but then when I saw the proof I was convinced.  I think it went like this.

Thm: 1=.9999…

Pf: We will use contradiction to show that 1=.9999…

First, assume 1 does not equal .9999….

Let x=.9999….

Therefore, 10x=9.9999…

10x = 9.9999…

–   x =   .9999…

——————

   9x = 9

So, x=1.  But we initially said x=.9999….  This is a contradiction from out initial statement, that .9999…. does not equal 1.  Therefore, we have shown that .9999…=1.

Now, not only is this a good review for the exam on how to construct proofs by contradiction, it also can be used to impress friends (and works well on dates).

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Responses

  1. That’s not a proof by contradiction, except in the loosest sense. You’ve assumed the that what you’re trying to prove is false, then proved it, and said that what you assumed was false — which is true, but you could have just cut out everything except the middle bit. Just leaving the “then proved it” step. The important part was not a proof by contradiction.

    A better example of proof by contradiction is the proof that root two is irrational:

    Assume that a/b = root 2, with a/b integers with no common factors. (Which you can always do if root two is rational, by cancelling). Then a^2 = 2 * b^2. So a^2 is even, so a is even. So let c = a/2. So 2 * c^2 = b^2, so b^2 is even, so b is even. But then a/b share a factor of two, contradiction.

  2. Dear
    Mr. danberardo,
    Mr.Simon Woolf
    and All..

    The above discussions are very interesting. I have met another problem that until know I can’t explain my solution formarlly.

    You know, when we are tasked to calculate the root square of 2. Surely, we can calculate it easily by hand using the common method called Calandra Method. But we will need great effort to calculate the value of a^(1/n) by hand for n>2 and for arbitrary values of a included complex nuimbers.

    I have triend to calculate the above of a^(1/n) using Newton Raphson scheme. But as another numerical methods of solving the nonlinear equation f(x)=0, the Newton Raphson needs one initial guess, that always fail when I used zero (null) initial guess.

    Okay, here I give you my example below for calculating the value of 123456^(1/17) using Newton Raphson scheme that was performed using Matlab software with default script of form Newton_Raphson(a,n,x0), where x0 is the initial guess.

    The output when I used zero initial guess x0=0;
    >>Newton_Raphson(123456,17,0)
    By Rohedi, Physics Department- ITS, Surabaya.
    Indonesia.
    %%%%%%%

    iteration $$$$$ The value at each iteration
    0 $$$$$ 0
    1 $$$$$ NaN

    But, when I used x0=1, we found..
    >>Newton_Raphson(123456,17,1)
    By Rohedi, Physics Department- ITS, Surabaya.
    Indonesia.
    %%%%%%

    Iteration $$$$ The value at each iteration
    0 $$$$ 1.0000000000000000
    1 $$$$ 7263.0588235294117000
    2 $$$$ 6835.8200692041519000
    3 $$$$ 6433.7130063097902000
    4 $$$$ 6055.2593000562729000
    5 $$$$ 5699.0675765235510000
    6 $$$$ 5363.8283073162829000
    7 $$$$ 5048.3089951212078000
    8 $$$$ 4751.3496424670193000
    9 $$$$ 4471.8584870277828000
    10 $$$$ 4208.8079877908540000
    139 $$$$ 1.9929808960093915
    140 $$$$ 1.9929698003654466
    141 $$$$ 1.9929697998712288

    The tolerance of 1e-015 is satisfied at 141 iteration.

    O o o.. too many iterations, hence indeed very tiring.

    My question : can the searching of a^(1/n) above be performed by another methods by using zero initial guess and also needs least iterations?.

    Thank you for your attention,
    Sincerely,
    Rohedi.

  3. Dear Prof.Kate and All,

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    Denaya informs you all that on the math blog will discuss numerical methods. Denaya will appreciate to support the math blog analytic answer especially for problems that related to differential equation of both ODE and/or PDE, of course by guiding my father Mr.Rohedi. We may share together in solving many problems of mathematics.

    Maybe all of you here are interested to the topic of numerical methods, Denaya invites you to visit to the math blog’s of Barack Obama supported, and leaving some comments for my posts.

    Finally, thank you for your attention,
    my best regards

    Denaya Lesa.

  4. Dear All.

    Please visit to this link

    http://blueollie.wordpress.com/2009/02/05/daily-kos-the-president-strikes-back/#comment-31199

    Denaya has just posted the comparison results of Newton Rhapson scheme and Rohedi Scheme for calculating the value of arbitrary root of complex numbers. Maybe useful for you.

    Please continue your discussion here to another topic.

    Sincerely,

    Denaya Lesa.

  5. I’m curious how you, and your instructors feel about the acceptance of “Therefore, 10x=9.9999…” We all know that multiplying by 10 is the same as moving the decimal point one to the right … because it seems so obvious with FINITE decimals…. I can accept this as a limit from a calculus/analysis view…but can we accept that as a DISCRETE proof??

  6. > I can accept this as a limit from a calculus/analysis
    > view…but can we accept that as a DISCRETE proof??

    A “discrete” proof? What does that even mean? The real line is continuous. If you’re iffy about multiplying by ten being the same as moving the decimal point and want the full infinite series formalisation, it’s only a Google away (e.g. http://en.wikipedia.org/wiki/0.999…#Infinite_series_and_sequences ).

  7. Simon,
    you wrote, “A “discrete” proof? What does that even mean? The real line is continuous”
    Exactly, but a usually acceptable definition of “Discrete” (isn’t that the subject area of M152?) might be:
    “Defined for a finite or countable set of values; not continuous.”

    I don’t doubt the equality of the values, but do realize that for some there is a real distinction between saying the summation of 9/10^n as n goes to infinity EQUALS one, or has a LIMIT of one.
    Pat

  8. 9.9999… is short hand for an infinite series.. And it certainly convergent on 1, if you have already established that the scalar multiple of a series is the same as the series with each term multiplied by the same scalar.

  9. Oh, it is so interesting discussion, until this math 152 weblog forgot to celebrate Pi Day at last 3/14/9. Apologise I am physicist, not cause 3/14 was the birthday of Mr.Albert Einstein who was the person for 20st century, but I ever read on the internet that until now scientific world doesn’t have the simple analytic exact formula for the Pi number that is about 2500 years after Mr.Archimedes introduced 22/7 as approximation for the Pi for the first time. Although the simple exact formula for the pi number has been available that is Pi=4*atan(1), but as we know that the infinite series of atan(1)=1-1/3+1/5-1/7+… is difficult to convergen, hence the pi number is regarded as myterious number. I don’t know whether the celebration of Pi as representing that mathematicians not capable the simple exact formula of the Pi number, so when displaying pi more than three trillion digits the super computer must still use BPPV formula based on the utility of infinite series.

    Okay, still on the above topic of proof by contradiction.

    Let we write 0.999… using approximation for the following definiton of number 1

    x=1/2 + 1/4 + 1/8 + 1/16 + …

    Let multiply both sides by 10, then…

    10x=10/2 + 10/4 + 10/8 + 10/16 + …

    x=1/2 + 1/4 + 1/8 + 1/16 + …
    ————————————————- –
    9x=4.5 + 2.25 + 1.125 + 0.5625 +…

    Refer to above result, I believe you will agree with me that the right side never equal 9. So I take a conclusion that 0.9999 never equal to 1. But I don’t know whether my above proof is correct or not, because I am not mathematician. Please continue your discussion.

    Hehehehe…Next I need your information. If I have a simple analytic for the exact Pi number, what appropriate journal or Patent Institution for publishing my Pi number.

    Thanks you..
    Best Regards.

    Rohedi.
    Physics Department,
    Faculty of Mathematics and Natural Sciences,
    Sepuluh Nopember Institute of Technology (ITS) Surabaya, Indonesia.

  10. @pat ballew

    > for some there is a real distinction between saying
    > the summation of 9/10^n as n goes to infinity
    > EQUALS one, or has a LIMIT of one.

    What is a real number? Oh yeah, the limit of a Cauchy sequence of rationals. All numbers are limits in that sense. That doesn’t stop them from being a single, well-defined point on a number line.

    @Rohedi:

    Oh, dear. Where do I start?

    > until now scientific world doesn’t have the simple
    > analytic exact formula for the Pi number that is
    > about 2500 years after Mr.Archimedes introduced
    > 22/7 as approximation for the Pi for the first time.

    Nonsense. The method that Archimedes used to get 22/7, inscribed polygons, is capable of producing arbitrarily accurate approximations to pi. The first infinite series of pi was discovered by 14th century Indian mathematician Madhava.

    > as we know that the infinite series of atan
    > (1)=1-1/3+1/5-1/7+… is difficult to convergen,
    > hence the pi number is regarded as myterious
    > number.

    It’s mysterious because one (of many) of the infinite series which converge to it happens to converge quite slowly? Hardly!

    > I don’t know whether the celebration of Pi as
    > representing that mathematicians not capable the
    > simple exact formula of the Pi number

    As you said yourself, there are many simple exact series which give pi. They’re not finite, but that’s just a trivial consequence of pi being irrational.

    > 9x=4.5 + 2.25 + 1.125 + 0.5625 +…
    > Refer to above result, I believe you will agree with
    > me that the right side never equal 9. So I take a
    > conclusion that 0.9999 never equal to 1.

    ….!!!

    You may be unsurprised to learn that the right hand side of your equation *does* equal 9 (plug it into the formula for the convergence of a geometric progression, it’s not hard), and that, as a consequence, you have *not* just shown that Mathematics is fundamentally inconsistent.

    > Hehehehe…Next I need your information. If I have
    > a simple analytic for the exact Pi number, what
    > appropriate journal or Patent Institution for
    > publishing my Pi number.

    As said above, there are many (infinite, in fact) infinite series which converge to pi. If you are claiming to have found a finite, rational series which equals pi, I would direct you to Underwood Dudley…

  11. Oh thanks Mr@Simon Woolf,

    I am so very happy find quickly respon for my post.

    % my statement
    > 9x=4.5 + 2.25 + 1.125 + 0.5625 +…
    > Refer to above result, I believe you will agree
    > with me that the right side never equal 9. So I > take a conclusion that 0.9999 never equal to 1.

    %your respon

    >….!!!

    >You may be unsurprised to learn that the right %>hand side of your equation *does* equal 9
    >(plug it into the formula for the convergence of
    >a geometric progression, it’s not hard), and >that, as a consequence, you have *not* just >shown that Mathematics is fundamentally >inconsistent

    Of course, I must agree with you that the infinite series on the right side 4.5 + 2.25 + 1.125 + 0.5625 +… to be equal 1 because it forms geometric series. Again I must say “of course” this is true mathematically. Next question, whether the equality of the above result 9x=4.5 + 2.25 + 1.125 + 0.5625 +…will justify you and all that 0.99999999…=1 as posted by Mr.@danberardo. If so of course I will agree also, but if not so, please continue your discussion. Apologise, here I don’t mean to show that Mathematics is fundamentally inconsistent. But, I only wait respon especially that related to the infinite number. How big of a number by computer has assumed as infinite number.

    % About the Pi number

    Of course as shown at this link :

    http://www.contestcen.com/pi.htm

    there are so many formulas of pi created by each contestant. But important to be remember, that all formulas give approximation only, and so many of that formulas also that obtained by trial and error depend on his creator. My exact pi formula not in ratioal form, it is in simple form and building by simple way. Hehehehe…of course my exact pi formula is imposible to be posted here, so as I said previously, Rohedi needs your information about appropriate Journal or Patent Institution to prove “the term of my claim” as you introduced above.

    Okay, thanks for your respon sir.

    Best Regards
    Rohedi.

  12. Apologise Mr.@Simon Woolf and All

    There is correction my reply comment for Mr.@Simon Woolf

    >Of course, I must agree with you that the >infinite series on the right side 4.5 + 2.25 + >1.125 + 0.5625 +… to be equal 1 because it >forms geometric series.

    becomes :

    Of course, I must agree with you that the infinite series on the right side 4.5 + 2.25 + 1.125 + 0.5625 +… to be equal 9 because it forms geometric series.

  13. Dear All

    Until now I am still not believe that the following expression is correct

    1=.9999….

    I am still believe that the proof used by Mr.@danberardo above is not appropriate for this case. Please continue this discussion. Surely you will find a new breakthrough to overcome another problem.Thx.

    Best Regards.
    Rohedi.

  14. Apologise All, Why I hope this discussion to be continued until we have final conclusion?Yeah because I worry the above of danberardo’s post will be read by someone whose face less beauty then visit to math152 asks to us how to more beauty his face, because this math blog can change 0.999… becomes 1. Sorry, if mr@danberardo applied adding (+) operator to the both sides he can’t not obtain x=1. I believe you can find x=1 by another way but it is still wrong. because of inconsistency results as shown when applying + and – operators. Hopefully this will be useful.

  15. > Why I hope this discussion to be continued until we have final conclusion?

    The discussion will never reach a final conclusion, because it was over literally as soon as it begun, when the first post contained a proof of the result. Everything since has been meaningless sophistry.

    That’s the nice thing about a proof. If it has no logical flaws, and assumes nothing not derivable from the axioms, then it is — must be — correct. This is only not the case if the axioms you start with are inconsistent.

    So at this point, you either point out a logical flaw; or you accept the proof; or you claim the axioms are inconsistent. Any claims of the latter kind will be subjected to derision unless accompanied by — wait for it! — a proof.

    So if you can’t point to a flaw in the logic (and no, adding or subtracting something from both sides of an equality is not a flaw in the logic), but you still claim the result is false, then you have contradicted yourself, and there can be no discussion. There will be no “new breakthrough to overcome another problem”, as there is no other problem.

    Best regards,
    Simon Woolf.

  16. Mr.Simon Woolf, I feel this math152 blog as my home, so I am always enjoy to share with you and other visitors. Of course my purpose to point out to the logic flaw. Why I said that the proove used by Mr.danberardo in orde to obtain x=1 above is not appropriate to this case. Okay I will show you another derivation that still yeilding x=0.999…sorry I post this reply using my handphone because I am in travelling go to my campus of ITS. Let ‘s write exp(i4x)=exp(i4*0.999…) where i=sqrt(-1). If we subtract the both sides with 1, we will obtain exp(i2x)sin(2x)=exp(i2*0.999…)sin(2*0.999…), otherwise when applying add operator we still obtain similar equality but the sine changes to cosine function. As appear on the above equality we never find x=1. Next time maybe you can meet another way to prove 1 not equal 0.999…, where that way can be also used to overcome another problem. Okay.thanks for your reply, surely I am very happy find discussion friend here.

  17. > Of course my purpose to point out to the logic flaw.
    > Why I said that the proove used by Mr.danberardo in
    > orde to obtain x=1 above is not appropriate to this
    > case.

    If you think the proof in the original post has a flaw, then *point it out*. Claiming, whilst giving no reason, that the proof is somehow “not appropriate” is meaningless.

    > Let ’s write exp(i4x)=exp(i4*0.999…) … If we
    > subtract the both sides with 1, we will obtain exp
    > (i2x)sin(2x)=exp(i2*0.999…)sin(2*0.999…), … As
    > appear on the above equality we never find x=1.

    Your first equation is correct, as is your second. Neither of them have any bearing whatsoever on the question at hand, since both are basically “f(x) = f(0.999…)”, which is trivially true if x = 0.999… = 1. You have disproved nothing. Incidentally, the claimed relationship between the equations (that you subtract 1 from both sides of the first to get the second) is nonsense, even if you did mean to have pi’s in all the exponents.

    > Next time maybe you can meet another way to prove
    >1 not equal 0.999…, where that way can be also
    > used to overcome another problem.

    You’re not getting this. We will never find a way to prove 1 != 0.999… in the standard reals, because it is not true, something that was proven in the first post. If a proof that 1 != 0.999… were to be discovered, it would, far from “overcom[ing] another problem”, show that Mathematics is fundamentally inconsistent.

    Best regards,
    Simon WOolf

  18. Well. I would remember you again that from early discussion, like you, I don’t believe that 1=0.999…as presenting at my latest derivation that we never obtain x=1. Okay the following I present another way that also gives x=1 like obtained the original post. Let’s write (10sqrt(x))^2=(sqrt(99.999…)^2, then subract of each by (sqrt(x))^2 and (sqrt(0.999…))^2 respectively. Continuing this step until to the result 99x=99, so we find x=1. Mr.Simon Woolf, let’s you consider two different ways in getting x=1. I don’t know whether from the ways we can’t take a conclution that 1 not equal 0.999…when comparing my way that yield x=0.999… Finding of pi? Why not,of course you must take a special trick. Okay my thought about this here, but I have been following this interesting discussion. My best regards.Rohedi.

  19. > like you, I don’t believe that 1=0.999…

    I’m sorry, WHAT?!

    13 posts and numerous proofs of everyone trying to convince you that 1=0.999…, and you are apparently under the impression that I share your misconception?

    I suggest you read through the above posts again.

    > Okay the following I present another way that
    > also gives x=1

    Basic logic: If you are shown a proof of something, and you want to assert that that something is wrong; you must point out a flaw in the proof. Or present your own proof that the something is false. You are doing neither of these. I have no idea what you are trying to achieve.

  20. O o o where is mr.danberardo now? Because further discussion between me and you about this more suitable is done via personal email. Mr.Simon Woolf the primary messages of all my reply comments, I only would say for mathematiciants be carefull in proving an equality because you will get an establish equation to become wrong only through by applying incorrect procedure. I hope discussion of this topic still continue, while me stop here because I am preparing the publication of my simple pi exact formula. Special thanks for you for the nice discussion.

  21. Its the almost hilarious and mildly ridiculous on how you have gone on to prove something which you know in your heart of hearts is incorrect.

    The basic flaw is that infinity minus infinity is non zero, 0.999… has infinite 9’s and so does 9.999…, but who the hell ever said they are the same in number !!!

    I mean its sad to see such mathematics from a harvard undergrad, and i’m an Indian though.

    Cheers and just be careful before you make such “Public Blunders”

  22. > Its the almost hilarious and mildly ridiculous on how you have gone on to prove something which you know in your heart of hearts is incorrect.

    You may be surprised to know that mathematics isn’t done by feelings that come from your “heart of hearts”.

    It’s done by proofs which come from your brain.

    Now stop embarrassing yourself and do a bare minimum of research before declaring something a “blunder” on the basis of a feeling from your heart. (Even Wikipedia will do: http://en.wikipedia.org/wiki/0.999… ).

    Oh, and by the way: just because something has “an infinite number [of digits]” doesn’t mean it’s infinity. Because the digits come after something called a DECIMAL POINT. Look it up.

  23. the only problem is that we were introduced to this method way before they explained us what infinite numbers really meant.
    Gratitude to you for now i am a more enlightened man than i was about 2 hrs back.
    But i just hope mathematics was as much of an intuition as physics to me 😛

  24. > But i just hope mathematics was as much of an intuition as physics to me

    I fear it’ll rather be the other way round. Physics rapidly stops being amenable to intuition at around the time you start doing relativity and quantum mechanics. By the time you get to your third year of Uni Physics, “physical intuition” will be a mostly nostalgic term.

  25. Haha, i’m in the final yr of colg.
    physics took a silent exit after 1st yr coz i had taken up computer science. Till then i enjoyed every bit of intuition that went into solving the trivial or rather inconsequential problems.But i couldn’t agree more on the point you just made.

  26. @Writer

    Let x=.9999….
    then
    10x=9.9999…
    the problem is here, accuracy

    when you multiply x with 10 means the right side you need to add the accuracy to 5 digit.
    so we get:
    10x = 9.99999…
    x = .99999….
    then 9x = 9.99999
    x = .99999…
    since the accuracy the first time you use at x = .9999 is 4 digit then we have to conssistenly use it
    so x with 4 digit accuracy
    x=.99999…= 1

  27. @parhobass:

    No.

    The ellipses (the three dots: “…”) mean that the 9s continue on even after you stop quoting them. For ever. So it doesn’t matter how many of them you actually quote. 0.9… = 0.99… = 0.999999… (= 1).

  28. @Woolf
    i agree with you..
    x1=.9999…
    x2=.99999…
    for me these are not the same
    x1 with accuracy 4 digits
    x2 with accuracy 5 digits

    when you want to measure 1.1 then you need at least scale of 1-2 “tools” so you get the best measurement
    if you use scale of 1-100 you get good measurement, but not the best

  29. @parhobass:

    > x1 = 0.9999… with accuracy 4 digits
    > x2 = 0.99999… with accuracy 5 digits
    > for me these are not the same

    With respect, mathematics is not subjective, and saying “for me they are not the same” is irrelevent. If you want to show they’re not the same, don’t give an opinion, give a proof.

    But, as I’ve already said, they are the same. x1 and x2 are *numbers* — not ranges of numbers like 5 +/- 2, but single numbers, points on a number line. You can *quote* them to whatever accuracy you like, but that doesn’t change the original. For instance, if y = 5.284, I could say “y to 2 s.f. is 5.3”; but that won’t change the fact that what y *is* is 5.284.

  30. @Woolf
    I agree in some points…

    let me proof.

    open microsoft paint, draw a line and save as *.JPG
    then open it with any JPG viewer.

    line is a line
    but once you make a zoom you will get “a non line”, perhaps only dots
    why??
    because accuracy. you need accuracy to get “non line” to a line we have.

    i also had given you a proof.
    if you need to measure 1, then the best tools is a tools with a scale 0-2 (i said before 1-2)
    but if you use 0-100 you’ll get non best value.

    .99999… with 4 digit accuracy is equal to 1

  31. @simon : why are you even wasting your time on this guy ?

  32. @Amit Gupta: Masochism? But even I’m not sure I feel up to arguing with his, er, Proof by MS Paint…

  33. @All
    sorry for the inconvenient

    cao

  34. proof that 1 not equal to 0.9999…
    but only accuracy can make it equal

    as written above:

    Let x=.9999….
    we have to set our mind that the accuracy is 4 digit, even actualy “…” means there are still many 9 follow

    Therefore,
    10x=9.99999
    here we need to add 5 digit after comma, as my explanation about MS Paint and tools with scale 0-2

    10x = 9.99999

    – x = .9999…

    —————————
    9x = ???????
    why we cant substract these equation?
    because:
    we have two different accuracies
    we have two different value from 2 different “tools” ( i dont have an exact pharase for this)

    but if we assume that accuracy is acceptable then
    we get:
    10x = 9.99999

    – x = .99999

    —————————
    9x = 9

    or
    10x = 9.9999…

    – x = .9999…

    —————————
    9x = 9

    both lead us to x = 1

    regards
    Parhobass

  35. @All
    2nd proof:

    let
    x = .9999…
    +x=.9999…
    —————————
    2x = 1.999…8

    let say there will be somewhere there if we stop “9” we will get “8” as addition of 9 with 9

    if we continue it ten times…
    then we will get

    3x = 2.999…7
    4x = 3.999…6
    .
    .
    .
    10x = 9.999…0

    then
    10x = 9.999…0
    -x = .9999…9
    —————————
    9x = 8.9999..1

    x=0.9999…9

    so 1 /=.9999…

    shalom alenu
    Parhobass

  36. @All
    3rd proof

    let
    x =.9999…
    assume we have 4 digit accuracy
    then

    x = .9999…
    +x = .9999…
    —————————
    2x = .9998…

    continue ten times

    10x=9.999…

    then
    10x=9.999…
    x =.9999…
    hence we need to be conssistent, whether continue with 4 accuracy or reduce it to 3, because somewhere there at 10x=9.999… 0(zero) existed

    if we continue with 4 then
    follow the 2nd proof

    if we choose 4 or 3, then follow 1st proof

    Shalom alenu, Horas
    Parhobass

  37. @All
    4th Proof

    written x = .9999…
    lets make it
    x = lim(n–>∞) .9999…9(1)9(2)9(3)…9(n)
    then
    10x = lim(n–>∞) 9.9999…9(0)9(1)9(2)…9(n-1)

    all 9 are shifted
    here we get

    9(n) and 9(n-1)

    if we assume this is equal then
    10x = lim(n–>∞) 9.9999…9(0)9(1)9(2)…9(n-1)
    -x = lim(n–>∞) .9999…9(0)9(1)9(2)…9(n-1)
    ——————————————————————————
    9x = 9

    or

    10x = lim(n–>∞) 9.9999…9(1)9(2)9(3)…9(n)
    -x = lim(n–>∞) .9999…9(1)9(2)9(3)…9(n)
    ——————————————————————————
    9x = 9

    both lead us to x = 1 = .9999…

    but actualy
    9(n) is not the same with 9(n-1)
    9(n-1) = 0

    since 10 x 9 = 90 (so by n–>∞, at n-1, 0 existed)

    10x = lim(n–>∞) 9.9999…9(0)9(1)9(2)…9(n-1)
    -x = lim(n–>∞) .9999…9(1)9(2)9(3)…9(n)
    ——————————————————————————
    9x = 8.9999…9(-1)

    x = 8.9999…9(-1)/9
    x = .9999….

  38. alo woolf/writer how bout my proof?

  39. Excelent idea and nice explanation of you @parhobass.

  40. @Rohedi

    matursuwun…, sepurane,
    sampean dosen nang ITS ta?

  41. mas @parhobass aku sih di instrumen , cuma pas liat temen posting disini takjub juga liat cara anda ” proof contradiction ” , dan lebih kaget lagi , lha kok iso ngomong jowo.
    Sekedar tahu ” rohedi ” salah satu pengajar di Fisika ITS dan dikatagorikan rodho gendheng.
    Salam kenal dari temen – temen instrumen fisika ITS, Suwun

  42. ben mumet londo ne wess
    @Writer and Wolf

    proof 5th

    assume 1 = 0.9999…

    then

    0.9999… = 1
    ______________x9
    8.9999..1…. = 9

    9×9 = 81, so somewhre there at … 1 existed

    then

    8.9999..1…. = 9
    ______________: 9
    0.9999… /= 1

    horas;
    parhobass

  43. @writer and Woolf

    prrof 6th

    let say
    1 = .9999…

    then both side we add with 0.1111….

    1 = 0.9999…
    ____________+0.1111…..
    1.1111….. /= 1.0

    so
    .9999 /= 1

  44. proof 7th

    let say
    1 = .9999…

    then both side we substract with 0.1111….

    1 = 0.9999…
    ____________-0.1111…..
    0.9999….. /= 0.8888…

    so
    .9999 /= 1

  45. proof 8th

    let say
    u1 = .9999…
    u2 = 1

    hence
    u2-u1 = .1111…(1)

    then
    10u1 = 9.9999…
    10u1 = 9 + u1
    9u1 = 9 …..(2)

    then
    10u2 = 10 …..(3)

    then substract (3) with (2)
    10u2 = 10
    9u1 = 9
    10u2-9u1 = 1….(4)

    here if we assume u2 = u1 then
    (4) becomes

    10u2-9u2 = 1
    u2 = 1

    or

    10u1-9u1 = 1
    u1 = 1

    but this assumtion will be inconsisten with (1)

  46. @parhobass. All your “proofs” are based on the blatantly, obviously wrong assumption that 0.999… + 0.111… = 1.000… .

    Think about it for just a second, and try to remember how addition works. If you add even 0.1 — let alone 0.111… — to 0.999…, you get 1.0999… (=1.1), which is definitely greater than 1.

    I recommend you spend less time trying to come up with proofs of wrong things and more time revising first grade arithmetic.

    Forgive me, but I’m not going to waste my time refuting individual pieces of nonsense. If I assume 1+1=3, I can prove wrong things too.

  47. @simon
    proof 6th right?

    thx for your comment,
    thats how i try writer and woolf to make any counter for all proof i’ve done…

    but thats fine, let me fix it for you:

    let say
    1.0000 = .9999…

    then both side we add with .0010

    1.0000 = 0.9999…
    ____________+.0010
    1.0010 /= 1.0109…

    so

    1 /= .9999…

  48. Once again, you have succeeded in demonstrating the fact that you can’t do basic arithmetic, and little else.

    Corrected version:

    1.0000 = 0.9999…
    ____________+.0010
    1.0010 = 1.000999…

  49. ignore my last comment,

    let say
    1.0000 = .9999…

    then both side we add with .0010

    1.0000 = 0.9999…
    ____________+.0010
    1.0010 /= 1.0009…

    so

    1 /= .9999…

  50. @simon
    Once again, you have succeeded in demonstrating the fact that you can’t do basic arithmetic, and little else.

    Corrected version:

    1.0000 = 0.9999…
    ____________+.0010
    1.0010 = 1.000999…

    i fix it for you…

    anyway l write it directly in the screen, and i have silindric problem,…

    but from the proof hope you can find proof my philosophy

  51. @simon,

    then show me your proof rather than ad hominem fallacy you have done!!!

  52. “1.0000 = 0.9999…
    ____________+.0010
    1.0010 /= 1.0009…

    so 1 /= .9999…”

    You’ve now successfully added the same number to both sides, and then asserted inequality. For no reason. Do you seriously think that’s a proof?

    > then show me your proof

    There’s a proof in the original blog post. That’s the one you were trying and failing to poke holes in originally, remember? There have also been several others in comments along the way, as well as e.g. links to the relevant Wikipedia article ( http://en.wikipedia.org/wiki/0.999… ).

  53. wiki says:

    Misinterpreting the meaning of the use of the “…” (ellipsis) in 0.999… accounts for some of the misunderstanding about its equality to 1. The use here is different from the usage in language or in 0.99…9, in which the ellipsis specifies that some finite portion is left unstated or otherwise omitted
    . When used to specify a recurring decimal, “…” means that some infinite portion is left unstated, which can only be interpreted as a number by using the mathematical concept of limits. As a result, in conventional mathematical usage, the value assigned to the notation “0.999…” is defined to be the real number which is the limit of the convergent sequence (0.9, 0.99, 0.999, 0.9999, …).

    oks, oks, so i miss the “intrepretation”..
    thx….

  54. @simon

    .9999… div 10 = ??

    could you help me?

  55. You know how to divide by 10. Shift the decimal point to the left.

    0.999… = 1
    Divide both sides by 10:
    0.0999… = 0.1

  56. Thanks for the post!

  57. bayou fin is shool wafiq im 14 my favourit shool is 4 èm ameè

  58. .9999… = 1 for the following reason:

    .9999…. = 9*(1/10 + 1/100 + 1/ 1000….)
    = 9 (sum(k = 1 to infinity) (1/10)^k)
    = 9 (1/(1-1/10) -1)
    = 9*(10/9 – 1) = 1

    Or put another way, .999…. can be thought of as a limit of the sequence:
    (.9, .99, .999, .9999….) which can be shown by “delta-epsilon” proof techniques to equal 1.


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