Finding expansions of $p$-adic numbers (Number Theory).

Question

I am trying to get a better grasp of $p$-adic numbers by evaluating a few examples. I want to consider the $5$-adic field. What are the reduced expansions of $127$, $5/16$, and $3/5$ in the 5-adic field? (I have seen for negative integers that it is a simple infinite series, but for positive integers and rationals I am struggling to see how to find these expansions).

Answer 1

I think that the $5$-adic numbers are the best field to work out problems in. For hand computations, I recommend writing everything in $5$-ary expansion, that is base $5$. With the higher powers of $5$ to the left, of course, so that your first question is answered by writing $127_{10}=1002_5$ — that’s $2$ plus $5^3$.

Your third question concerns $3/5$, which is simply $3\times5^{-1}$, in other words, $0.3_5$.

Your second question is the most interesting, ’cause it involves an infinite repeating expansion. Remember that $5$-adically, the tiny powers of $5$ are the ones with high exponents: as $n\to+\infty$, $5^n\to0$ five-adically. So your infinite expansion is going to run off to the left, not to the right.

What you have to do to express $1/16$ as a $5$-adic expansion is find the first power of $5$, $5^n$, such that $16|(5^n-1)$. Well, the first multiple of $16$ among the numbers $4, 24, 124, 624, 3124\cdots$ is $624=16\times39$. So we get $$ \frac1{16}=\frac{39}{624}=\frac{-39}{1-625}=\frac{-39}{1-5^4}\,, $$ in which I’ve tortured the original fraction into the form of an infinite geometric series, $-39(1+5^4+5^8+5^{12}+\cdots)$. But this is of no use to us: if that had been $39$ in front of the series instead of $-39$, we would have gotten a repeating expansion for $-1/16$ of $$ -\frac1{16}=39+39\times5^4+\cdots=\>\cdots012401240124_5\,, $$ since the $5$-ary representation of $39$ is $124_5$.

What to do? Let’s think of $1/16$ as $1-\frac{15}{16}$. Using $15\times39=585$, which has the $5$-ary expansion $4320$, we get \begin{align} \frac1{16}&=1-\frac{15}{16}=1+\frac{585}{1-5^4}\\ &=1+585+585\times5^4+585\times5^8+\cdots\\ &=1+\>\cdots4320432043204320_5\\&=\cdots4320432043204321_5\,, \end{align} and there you are with the expansion of $\frac1{16}$. But you asked for $\frac5{16}$, and you get that by just shifting the previous thing over, one place leftwards: $\cdots43204320432043210_5$

And finally, I suppose I should confess that I first found the expansion by using a $p$-adic computation package that just told me the expansion of $1/16$ in a single command. Then I worked backwards to discover it. You can actually do all simple $p$-adic computations purely by hand, never straying out of the $p$-adic universe. Just as you found $\frac17=.142857142857\cdots$ in elementary school by direct division, you can do the same for $5$-ary representations of rational numbers. You need a five-by-five addition table and multiplication table, which you write out for yourself, and you arrange the division rather differently, because the process takes place right-to-left instead of the familiar way. I don’t know any place where this is published, though.