• Sharebar

As penance for liberally beating a dead goat in the other Brainteaser thread, here is a fresh one that should be easy for many of you:

Does .9 repeated = 1?

Please prove mathematically and explain in English. SB for the first to do both.

1

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Comments (43)

  • FreezePops's picture

    A Geometric series is probably easier.

  • Anacott_CEO's picture
  • illiniPride's picture

    You can explain this without any fancy math, using the definition of numbers alone.

    Leadership can be defined in two words: "Follow Me"

  • WSOusername's picture

    x = 0.9
    10x = 9.9
    10x - x = 9

    GBS

  • TechBanking's picture

    .9 repeated has a limit value of 1, but is never actually 1.

  • Cola Coca's picture

    x = 0.9

    10x = 9.9

    10x - x = 9

    9x = 9

    x = 1

  • In reply to Cola Coca
    illiniPride's picture

    .

    Leadership can be defined in two words: "Follow Me"

  • bcbunker1's picture

    just do a geometric series with a=.9 and r=.1

    or could do something like this:
    1/9=.1111111...
    9*(1/9)=9*.1111111....
    1=.999999...

  • In reply to Cola Coca
    youngblood90's picture

    Cola Coca:
    x = 0.9

    10x = 9.9

    10x - x = 9

    9x = 9

    x = 1

    I don't think your answer is correct.

    10*.9=9

    not 9.9, so in your equation x does not equal .9 like you said.

    the short answer is no it does not equal 1

  • In reply to TechBanking
    FreezePops's picture

    TechBanking:
    .9 repeated has a limit value of 1, but is never actually 1.

    If abs (x) <1, the infinite series sum(x^n) converges to 1/(1-x)
    let x = (1/10)

    finite sum = 1/(1-.1)

    this gives you 1 + .1 + .01 + .001 ....
    subtracting 1 and multiplying by 9 gives you .9 + .09 + .090 ....

    solution = [1/(1-.1) -1] *9 = [10/9 - 9/9] * 9 = 1

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  • JDimon's picture

    Yes - my simplest 'explain in english' is that .3 repeating is generally known to equal one third, and 3 times one third equals 1

  • In reply to Cola Coca
    jgx101's picture

    Cola Coca:
    x = 0.9

    10x = 9.9

    10x - x = 9

    9x = 9

    x = 1

    Wtf is this shit...

  • JDimon's picture

    Let me also do it with math:

    0.999 repeating equals x.
    10*x equals 9.999 repeating equals 9 + x
    Subtract x from both sides
    9*x = 9
    therefore x=1

  • JDimon's picture

    Oh Cola Coca got the answer first. Everyone who's disagreeing with him isn't taking the time to think about what it means for the decimal to be repeating indefinitely

  • In reply to JDimon
    Old Grand-Dad's picture

    JDimon:
    Yes - my simplest 'explain in english' is that .3 repeating is generally known to equal one third, and 3 times one third equals 1

    That's a good way to explain it. Also if
    1/3 = .3 repeating
    2/3= .6 repeating

    Add them together and you get

    3/3=.9 repeating
    1=.9 repeating

  • In reply to JDimon
    FreezePops's picture

    JDimon:
    Let me also do it with math:

    10*x equals 9.999

    Assuming you mean 10* .999 repeating = 9.999 repeating, this step is only true if the series converges (i.e it equals 1). In order to finish your proof you essentially assumed that what you were proving was true.

    Are you retarded?

  • bumpthethread's picture

    Here's my answer:

    Yes. When you want a repeated decimal, you just take that number and divided it by 9.
    0.7777.....=7/9
    0.5555.....=5/9

    Thus, 0.999999.....should theoretically equal 9/9, and 9/9 = 1.

    Do I get the SB?

  • In reply to illiniPride
    Cola Coca's picture

    illiniPride:
    Cola Coca:
    x = 0.9

    10x = 9.9

    10x - x = 9

    9x = 9

    x = 1


    That's the math, now why does it work?

    Because an infinite string of 0.9 will always equal 1? There is no number between 0.9 repeating and 1; they are the same number, represented differently.

    0.25 = 0.24(9) repeating

  • In reply to FreezePops
    JDimon's picture

    FreezePops:
    JDimon:
    Let me also do it with math:

    10*x equals 9.999

    Assuming you mean 10* .999 repeating = 9.999 repeating, this step is only true if the series converges (i.e it equals 1). In order to finish your proof you essentially assumed that what you were proving was true.

    Are you retarded?

    All I'm doing is using that fact that the 9s go on infinitely. If they go on forever, then multiplying everything by 10 doesn't reduce the amount of 9s after the decimal point....they still go on forever (not forever - 1, because that value still equals 'forever')

  • illiniPride's picture

    Gave it to Dimon, wasnt thinking of it that way but it makes sense.

    You can prove this another way using negation

    Leadership can be defined in two words: "Follow Me"

  • Febreeze's picture

    how can one number equal another number? even if a limit, it still doesn't technically equal it...

    plz splain

  • In reply to JDimon
    BTbanker's picture

    JDimon:
    Yes - my simplest 'explain in english' is that .3 repeating is generally known to equal one third, and 3 times one third equals 1

    Give it to this guy.
  • In reply to TechBanking
    canas15's picture

    TechBanking:
    .9 repeated has a limit value of 1, but is never actually 1.

    This statement makes no sense at it stands. If you mean to say constructing a sequence where each additional term is made by appending another 9, then yes, this sequence has a limit of 1. If you mean the number .9 repeating, well, it's a constant, so its "limit value" must be itself. By the way, that is most definitely 1.

  • In reply to JDimon
    canas15's picture

    JDimon:
    FreezePops:
    JDimon:
    Let me also do it with math:

    10*x equals 9.999

    Assuming you mean 10* .999 repeating = 9.999 repeating, this step is only true if the series converges (i.e it equals 1). In order to finish your proof you essentially assumed that what you were proving was true.

    Are you retarded?

    All I'm doing is using that fact that the 9s go on infinitely. If they go on forever, then multiplying everything by 10 doesn't reduce the amount of 9s after the decimal point....they still go on forever (not forever - 1, because that value still equals 'forever')

    JDimon is right. This is a legitimate proof. He's not assuming anything about the series convergence- all he assumes is the presence of infinite 9s (follows by the assumption of repeated 9s). Multiplication by 10 is a well defined operation- even for infinite decimal places- as a "place shifter".

  • illiniPride's picture

    This was how I thought of it:

    If a <> b,
    x exists where a < x < b or a > x > b.
    If not, a=b

    Leadership can be defined in two words: "Follow Me"

  • RubiksCubeMath's picture

    The answer is yes. There is a long mathematical proof that you learn in calc 2.

    But the easy straightforward answer is...

    1/3 = .3 Repeated
    2/3 = .6 Repeated

    Add the two previous

    3/3 = .9 Repeated

  • canas15's picture

    Wikipedia has a pretty cool proof with (Dedekind) cuts too! I didn't know that one since my analysis class skipped the construction of real numbers, but it was nice. I liked it!

  • gobucks44's picture

    Assume that 9.999.../=10

    Then you can list a number that is greater than 9.999... and less than 10.

    Since you can't, 9.999...=10

  • jesus of nazareth's picture

    1/3 + 1/3 + 1/3 = 3/3 = 1 and 1/3 is 0.33... so 0.3333..... + 0.3333..... + 0.3333.... = 0.9999 = 1

  • juked07's picture

    Weird corollary imo: every terminating number has 2 equivalent decimal representations

  • In reply to juked07
    canas15's picture

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  • manbearpig's picture

    -MBP

  • BlackHat's picture

    I hate victims who respect their executioners

  • In reply to BlackHat
    manbearpig's picture

    -MBP

  • In reply to manbearpig
    canas15's picture
  • Febreeze's picture