$p$-adic representation

Question

I'm reading about $p$-adic representation. I can't seem to understand how $p$-adic representation is a sort of mirror image of the base $p$ representation of real numbers.

Given $\alpha \in \mathbb Q_{p}$ with

$$ \alpha = \sum_{k=-r}^{\infty} a_{k}p^k,$$

we can write $\alpha$ as $$ \alpha = \cdots a_3a_2a_1a_0.a_{-1}a_{-2} \cdots a_{-r}.$$

I can't seem to understand how we can add and multiply $p$-adic numbers. Could someone explain with a concrete example? Possibly ($\cdots 555.5)^2$ in $\mathbb Q_{7} $ since I read about it and can't comprehend at all ?

Answer 1

My recommendation is this: prepare for yourself an addition table and a multiplication table for integers $0,1,\dots,6,7$, which are notated $0,1,\dots,6,10$ in septary notation. Then proceed with your multiplication in exactly the same way that (I hope that) you learned in elementary school, with carries, and so forth. You’d see a pattern, since $\dots555;5$ is the rational number $\frac{-5}{42}$, but it takes a while for it to show up. To thirty places, your square of the above turns out to be $$\dots2065432065432065432065432065;44 $$

Although I’ve done lots of $p$-adic calculations of this type by hand, I assure you that for this calculation I had the help of a package that handles $p$-adic numbers.

Answer 2

$\Bbb{Z}_7$ and $\Bbb{Q}_7$ are the completion of $\Bbb{Z}$ and $\Bbb{Q}$ for the absolute value $|p^k c/d|_7= p^{-k},p\nmid c,p\nmid d$. Then it works the same way as for $\Bbb{R}$ the completion of $\Bbb{Q}$ for $|.|_\infty$. Concretely approximate your series $\sum_{k=-r}^{\infty} a_{k}p^k,\sum_{k=-r}^{\infty} b_{k}p^k$ by $\sum_{k=-r}^n a_{k}p^k,\sum_{k=-r}^n b_{k}p^k$, multiply as standard rational numbers, you get an approximation of $(\sum_{k=-r}^{\infty} a_{k}p^k)(\sum_{k=-r}^{\infty} b_{k}p^k)$