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 $\displaystyle 1+2^2+\left(2^2\right)^2+\cdots$ 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
$\displaystyle \frac23=\cdot01101010\ldots$