p-adic expansion

**curiousmuch** · May 19th 2009, 10:15 AM

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.

**TheAbstractionist** · May 19th 2009, 11:06 AM

Originally Posted by curiousmuch

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

$\frac23=\cdot01101010\ldots$

**curiousmuch** · May 19th 2009, 11:22 AM

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.

**TheAbstractionist** · May 19th 2009, 11:25 AM

Originally Posted by curiousmuch

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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