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Math Help - p-adic expansion

  1. #1
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    p-adic expansion

    I just started learning about p-adic numbers so I am having trouble with a relatively easy question:

    Compute the 2-adic expansion of 2/3 and then verify your answer.

    Help would be appreciated. Thanks.
    Last edited by mr fantastic; May 22nd 2009 at 03:12 PM. Reason: Restored original question deleted by OP
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  2. #2
    Senior Member TheAbstractionist's Avatar
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    Quote Originally Posted by curiousmuch View Post
    I just started learning about p-adic numbers so I am having trouble with a relatively easy question:

    Compute the 2-adic expansion of 2/3 and then verify your answer.

    Help would be appreciated. Thanks.
    Hi curiousmuch.

    Note that 1+2^2+\left(2^2\right)^2+\cdots converges to \frac1{1-2^2}=-\frac13 in the 2-adic norm.

    Hence \frac23=1-\frac13=2+2^2+2^4+\cdots so the 2-adic expansion of \frac23 is

    \frac23=\cdot01101010\ldots
    Last edited by TheAbstractionist; May 19th 2009 at 11:26 AM.
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  3. #3
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    Wow that's really helpful, but I would really appreciate it if you could explain the last step. How did you generate the 2 adic decimal expansion from the infinite series. Thanks.
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  4. #4
    Senior Member TheAbstractionist's Avatar
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    Quote Originally Posted by curiousmuch View Post
    How did you generate the 2 adic decimal expansion from the infinite series. Thanks.
    They are the coefficients of the powers of 2 in the 2-adic power series.
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