• May 19th 2009, 10:15 AM
curiousmuch
I just started learning about p-adic numbers so I am having trouble with a relatively easy question:

Help would be appreciated. Thanks.
• May 19th 2009, 11:06 AM
TheAbstractionist
Quote:

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

Help would be appreciated. Thanks.

Hi curiousmuch.

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

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

$\displaystyle \frac23=\cdot01101010\ldots$
• May 19th 2009, 11:22 AM
curiousmuch
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.
• May 19th 2009, 11:25 AM
TheAbstractionist
Quote:

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.