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Posers and Puzzles

Posers and Puzzles

  1. Standard member genius
    Wayward Soul
    21 Jun '03 16:28
    0.9 recurring (i.e. 0.999999999999999999...9) is equal to one-can anyone work out a proof for this?
  2. Standard member royalchicken
    CHAOS GHOST!!!
    21 Jun '03 19:00
    Okay. 1/9 = 0.1 rec. 2/9 = 0.2 rec. 3/9 = 1/3 =.6 rec. 6/9 = 2/3 = .6 rec.

    1/3 + 2/3 = 1
    1/3 + 2/3 =0.3 rec + 0.6 rec = 0.9 rec. Thus 1= 0.9 rec. This is not entirely perfect, but is rhetorically easier than:

    0.9 rec = lim(n->infinity) (10^n - 1)/10^n = lim(n->infinity) (1-10^-n) =1.

    That is a mathematically sound proof. Genius, you are familiar w/ limits, correct?
  3. Standard member CK
    22 Jun '03 10:37
    Another classic proof of this is:

    x=0.9 rec.
    10x=9.9 rec.
    9x=10x-x=9
    x=1

    therefore x= 1 AND x= 0.9 rec.
    so 0.9 rec. = 1
  4. Standard member genius
    Wayward Soul
    22 Jun '03 11:25 / 2 edits
    Originally posted by royalchicken
    Okay. 1/9 = 0.1 rec. 2/9 = 0.2 rec. 3/9 = 1/3 =.6 rec. 6/9 = 2/3 = .6 rec.

    1/3 + 2/3 = 1
    1/3 + 2/3 =0.3 rec + 0.6 rec = 0.9 rec. Thus 1= 0.9 rec. This is not entirely perfect, but is rhetorically easier than:

    0.9 rec = lim ...[text shortened]... tically sound proof. Genius, you are familiar w/ limits, correct?
    the third proof (i.e. from CK) was the one that i'd heard before, but i suppose yours would work too... and i have no idea about limits (although i think we did something on them in maths saying that a grath tended to infinity or summant )
  5. 22 Jun '03 16:01
    1/9 = .111 rec

    So:
    1/9*9 = .999 rec

    But also:
    1/9*9 = 1
  6. Standard member royalchicken
    CHAOS GHOST!!!
    22 Jun '03 16:55
    All of these work....and depend on different fundamental details, from the definition of a recurring decimal (the limits proof), to the multiplicative inverse property of the real numbers (zamba's). Good going all...
  7. 06 Jun '04 15:54 / 2 edits
    Originally posted by zamba
    1/9 = .111 rec

    So:
    1/9*9 = .999 rec

    This is not really a proof.

    It's just moving the goalposts.


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  8. Standard member TheMaster37
    Kupikupopo!
    06 Jun '04 16:38
    No, it isn't;

    1 = 9 * 1/9 = 9*0.11111111... = 0.9999999....
  9. 07 Jun '04 01:14
    Don't overlook this proof.

    Observe that 0.999... may be viewed as a geometric series as such:

    T = .9 + .09 + .009 + ...

    Recall the derivation of the formula for geometric series:

    S = a + ar + ar^2 + ... + ar^n + ... where |r| < 1.

    Then:

    Sr = ar + ar^2 + ar^3 + ... + ar^(n+1) + ...

    Subtracting the second series from the first cancels all terms on the right-hand side except one:

    S - Sr = a

    S(1 - r) = a

    S = a/(1 - r)

    For the number in question, a = 0.9 and r = 0.1:

    S = 0.9/(1 - 0.1)

    S = 0.9/0.9

    S = 1.0

    -Ray.

  10. 07 Jun '04 03:19
    Originally posted by rgoudie
    Don't overlook this proof.

    Observe that 0.999... may be viewed as a geometric series as such:

    T = .9 + .09 + .009 + ...

    Recall the derivation of the formula for geometric series:

    S = a + ar + ar^2 + ... + ar^n + ... where |r| < 1.

    Then:

    Sr = ar + ar^2 + ar^3 + ... + ar^(n+1) + ...

    Subtracting the second series from the first cancel ...[text shortened]... er in question, a = 0.9 and r = 0.1:

    S = 0.9/(1 - 0.1)

    S = 0.9/0.9

    S = 1.0

    -Ray.

    ==============================================

    The above is a proof because it defines what an infinitely recurring decimal IS.

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