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stackexchange,

I'm recently trying to get into the p-adic numbers with some materials on the web. I understand the very basics of arithmatics with the p-adic numbers and followed some examples I found as well as calculated some of my own.

So far, however, all the examples I found dealt with primes $p < 11$. I now wonder how to write down p-adic numbers where p is a prime with multiple digits.

For example, given the prime $p = 11$, I can write the number $n = 120$ down as

$n = \sum_{i=0}^{s-1} a_i p^i$

$120 = \cdots + 0*11^2 + 10*11^1 + 10*11^0$

$120 = \cdots 0 10 10$

But if now someone were to only give me the 11-adic number $\cdots 0 10 10$ without any further explanation, how would I know that it is the number 120 and not

$\cdots 0 10 10 = 1*11^3 + 0*11^2 + 1*11^1 + 0*11^0 = 1331 + 0 + 11 + 0 = 1342$ ?

Sadly, all the examples I could find so far in the materials I have use primes $< 10$. Is there a writing convention for this issue in more advanced material on the topic of the p-adic numbers?

MaChaeHa
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    Does this answer your question? How would you count a base $> 36$ system? – mr_e_man Nov 08 '22 at 13:12
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    $\cdots+10 \times 11 + 10\times1$ would not be written $\cdots 01010$. To write in base $11$ you would need an eleventh digit sumbol to represents ten. See video https://www.youtube.com/watch?v=-nC8J6YXqdo – GEdgar Nov 08 '22 at 13:18
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    I would give the advice to not waste too much time with such notational questions. If you have to do an example with $p>10$, just write the elements out as power series and don't waste time trying to force it into a representation including one-letter symbols for $10,11, ...$. – Torsten Schoeneberg Nov 08 '22 at 15:58
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    Using further characters to represent numbers $> 10$ is definitely one way of doing it (e.g. something like using capital letters $A, \cdots, F$ akin to the hexadecimal system).

    Another method I came across was to put multi-digit numbers into brackets, i.e. $\cdots0(10)(10)$ which seems a lot easier to read, especially when it comes to really high primes.

    Ignoring it altogether and just writing it - as @TorstenSchoeneberg suggested - as the power series might be the best solution, though.

    So in the end it seems that there is no common writing convention?

     – MaChaeHa Nov 08 '22 at 20:20
  • I do it by putting line-level dots between the 11-adic “digits”, so that $-1$ becomes $…10.10.10.10;$ – Lubin Nov 11 '22 at 23:04

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