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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.

Comments (43)

  • 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"

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

    JDimon wrote:
    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 wrote:
    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 wrote:
    Cola Coca wrote:
    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 wrote:
    JDimon wrote:
    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"

  • In reply to TechBanking
    canas15's picture

    TechBanking wrote:
    .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 wrote:
    FreezePops wrote:
    JDimon wrote:
    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.
    If not, a=b

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

  • 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!