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I'm struggling to get any intuition for the following example:

Consider the sequence $\{a_k\} = \{3, \ 33, \ 333, \ 3333, \ \ldots\}$

It's easy to show, using the geometric series formula, that $a_k = \frac{1}{3}(10^k-1)$.

It follows that $3a_k = 10^k-1$ and hence $3a_k+1=10^k$, and so $3a_k+1 \equiv 0 \bmod 5^k$.

By the definition of the $5$-adic metric, we have $0 \le |3a_k+1|_5 \le 5^{-k}$.

As $k \to \infty$, $5^{-k} \to 0$ and so $|3a_k+1|_5 \to 0$. We conclude that $a_k \to -\frac{1}{3}$ ($5$-adically).

How can a sequence of positive integers tend to a negative fraction?

It seems that $|a_k|_5 = 1$ for all $k$, but then $a_k \to -\frac{1}{3}$?

Fly by Night
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  • Don't think about the $5$-adic distance geometrically. For example $-1$ is $5$-adically closer to $4$ than $1$ is because the euclidean difference between $4$ and $-1$ is $5$ which is a multiple of $5$ while the euclidean distance between $4$ and $1$ is $3$ which is not a multiple of $5$. Multiples of $5$ are small in th $5$-adic metric. – Gregory Grant Jul 22 '16 at 18:38
  • @GregoryGrant Thanks for your reply. I don't understand how all distances and norms are of the form $0$ or $5^{-n}$, but then we get $-\frac{1}{3}$. – Fly by Night Jul 22 '16 at 18:40
  • to add to Gregory's comment: 'positive' means 'to the right of zero' on the real line... The $5$-adics have little to do with the real line. – peter a g Jul 22 '16 at 18:44
  • @peterag This is the reason for my post -- please explain and fill in the details. – Fly by Night Jul 22 '16 at 18:45
  • It takes a while to get used to distance concepts other than the familiar absolute value. Do observe that $$|-1/3|_5=1=|3|_5=|33|_5=|333|_5=\cdots.$$ Also, there is no division into positive and negative numbers in the $p$-adic domain. For example, there is a $5$-adic integer $i$ with the property $i^2=-1$, and that $i$ cannot be positive or negative for much the same reason its lookalike cousin in $\Bbb{C}$ can! – Jyrki Lahtonen Jul 22 '16 at 18:46
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    @FlybyNight You don't get $-\frac13$ as an absolute value, you get that $a_k$ approaches $-1/3$ not that anything's $5$-adic absolute value is $-1/3$. – Gregory Grant Jul 22 '16 at 18:49
  • @JyrkiLahtonen Thanls for the reply. Doesn't $|-1/33|_5=1$ as well? So why is $-\frac{1}{3}$ the limit and not any other? Sorry for being obtuse. – Fly by Night Jul 22 '16 at 18:50
  • Didn't you just prove that the limit must be $-1/3$? And, yes $|-1/33|_5=1$ as well. The $p$-adic absolute value is nowhere near to being a 2-1 mapping like its archimedean cousin. – Jyrki Lahtonen Jul 22 '16 at 18:52
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    In a way, wasn't it just as striking that $4$ equals $-1$ in $\mathbb F_5$ (Gregory's example), or $3$ equals $1/2$, or that $2$ 'equals' $i$ in that field? The surprise lies there in the word 'equality' - just as it does in the word 'convergence' here. As @JyrkiLahtonen writes, it's a question of getting used to it. Edit: I see Jyrki already used my $i$ example... – peter a g Jul 22 '16 at 18:53
  • @JyrkiLahtonen I went through the details but, as my post says: I'm struggling to get any intuition for the example. I've pushed the symbols, but the answers seems incredible. I was hoping someone might be able to post some explanation to help me understand. – Fly by Night Jul 22 '16 at 18:57
  • @peterag A lot of people have edited their posts to add things, so I haven't read most of the stuff you're talking about. I'll go back and read the edited posts now. – Fly by Night Jul 22 '16 at 18:59
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    We understand! Your example is not at all unlike the common 2-adic limit $$1+2+4+8+16+\cdots=\frac1{1-2}=-1.$$ Geometric series sum formula FTW! The partial sum of that series just go closer and closer to $-1$ w.r.t. the 2-adic metric. For example $|(1+2+4+8+16)-(-1)|_2=|32|_2=1/32$ is already pretty small, and $(1+2+4+8+16+32+64+128+256+512)=2^{10}-1$ is closer still. Here $2$-adically that $2^{10}$ is the small term, and $-1$ is the large one. – Jyrki Lahtonen Jul 22 '16 at 19:00
  • @JyrkiLahtonen That's interesting. If I apply the infinite series formula then I do indeed get $-\frac{1}{3}$ as the sum of $3+30+300+3000+\cdots$. I guess I'll just have to accept this and use it as a formal construction. Thanks for all of your help and patience. – Fly by Night Jul 22 '16 at 19:04
  • YW. The real fun begins (I learned this from Lubin who is the leading p-adic person on Math.SE) with limits like that of the sequence $2,2^5,2^{5^2},2^{5^3},\ldots$ (you get the next entry by raising the previous to the fifth power). That limit gives you a square root of $-1$! – Jyrki Lahtonen Jul 22 '16 at 19:09
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    Maybe thinking of the topological language as just that might help - i.e. 'just language' to represent algebraic notions: for instance, writing that the geometric sequence of @JyrkiLahtonen's converges $2$-adically to $-1$ is 'just language' for $$ 1+\cdots + 2^{n-1} \equiv -1 \pmod {2^n },$$ for all (positive!) integer $n$. Likewise, say $k$ is a field - one can use the completion $k[[x]]$ of the ring of polys $k[x]$ to express sequences of congruences $\pmod {x^n}$. hope this helps! – peter a g Jul 22 '16 at 19:38
  • Comparison to formal power series is a great suggestion @peterag! Fly by Night, the metric in power series is very similar to the p-adic one. Two formal power series are close to each other if their low degree terms agree - the highe you need to go to see the difference, the smaller the distance. – Jyrki Lahtonen Jul 22 '16 at 20:16
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    Another way to say what you proved: the set of nonnegative rationals is not closed as a subset of the rationals, according to the $5$-adic metric. Of course you know that different metrics can have different closed sets. – GEdgar Jul 24 '16 at 12:09

1 Answers1

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Promoting some of the comments to an answer with a view of A) removing this from the list of unanswered questions, B) covering some of the features of the $p$-adic metric that surprise learners at first, and C) also aiming to draw an analogy with the ring of formal power series.

1. The $p$-adic "size" of rational numbers does not match the intuition based on the archimedean valuation better known as the absolute value. $2$-adically a high power of $2$ is tiny in comparison to a meager $-1$. Therefore, for example, the sequences $2^n\to0$ and $2^n-1\to-1$. In other words, in a series like $$1+2+4+8+16+32+\cdots$$ the first term $1$ is the dominant one. This is not atypical of converging series in all domains :-)

2. The formula for the sum of a geometric series is your friend. Whenever it converges the sum of the series is surely (by the usual argument) $$ a+aq+aq^2+aq^3+\cdots=\frac a{1-q}.\qquad(*) $$ Here it converges if and only if the ratio $q$ has $p$-adic size $<1$ (hardly a surprise!). In other words, the formula $(*)$ holds iff $|q|_p<1$. Therefore, as Fly by Night observed $$ ...33333=3+30+300+3000+30000+\cdots=\frac3{1-10} $$ whenever $|10|_p<1$. This inequality holds when $p=5$ or $p=2$, and in both cases the sum of this series is $-1/3$.

3. The mildly surprising fact about this limit is that a sequence of positive rationals has a negative limit. This will cease to amaze a learner whenever they recall that $2$-adically $1024$ is very close to $-1024$, but relatively far away from $1025$. A more formal way of phrasing this is that the $p$-adic fields do not have a total ordering - a relation that would allow us to, among other things, partition the $p$-adic numbers into negative and positive numbers. One rigorous argument for that parallels the reasoning why we don't have a total ordering in $\Bbb{C}$ either. Remember that if $i$ were either positive or negative, then its square should be positive, which it ain't. Similarly in all $p$-adic fields some negative integers have square roots. There is a $5$-adic $\sqrt{-1}$ (see here for a crude description of the process of finding a sequence of integers converging $5$-adically to a number with square $=-1$. Exactly which integers have $p$-adic square roots is number-theoretic in nature. For example when $p=2$ it turns out that $\sqrt{m}$ of an odd integer $m$ exists inside $\Bbb{Q}_2$, iff $m\equiv1\pmod8$, so $-7$ has a $2$-adic square root.

4. The same themes recur, if we want to define a $p$-adic exponential function with the usual power series $$ \exp(x):=\sum_{n=0}^\infty\frac{x^n}{n!}. $$ The problem is with convergence. Contrary to expectations from real analysis this series will not converge for all $x\in\Bbb{Q}_p$. The reason is the denominators. For large $n$, the factorial will be divisible by higher and higher powers of $p$. Therefore we are dividing by a sequence of small numbers tending towards zero, and the numerator $x^n$ needs to compensate for that. A more careful analysis of the situation reveals, that this series converges, iff $|x|_p<p^{-1/(p-1)}$.

5. Apropos series. An analogue of the $p$-adic metric you may be familiar from courses on analysis is the $x$-adic topology (can turn it into a metric if so desired) on (formal) power series (with coefficients in, say, $\Bbb{R}$!). Two power series are close to each other $x$-adically, iff their difference is divisible by a high power of $x$. Therefore we can say that $e^x$ and $1+x+x^2/2$ are already quite close to each other, but $\sin x$ and $x-x^3/6+x^5/120$ are closer still. A common feature of all non-archimedean metrics ($p$-adic, $x$-adic,...) is that adding "small" numbers together will never make a large number, no matter how many of them you add together. So w.r.t. the $2$-adic metric no matter how many numbers divisible by four you add together you will never get a large number, say an odd number. Similarly, if you add together several formal power series divisible by $x$ you never create a non-zero constant term. This has an impact on some things. For example, when defining integrals, we want to approximate something by a sum of small things. In the $p$-adic world we need to...

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Jyrki Lahtonen
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  • Not sure this post serves any useful role in the grand scheme of Math.SE. I just try to learn away from answers in comments (except with a view of inducing a learner to see the light themselves). – Jyrki Lahtonen Jul 24 '16 at 11:53
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    I think that a major part of the Grand Scheme is educational: to present gobbets of useful or interesting mathematics to people who may somehow have missed them. – Lubin Jul 24 '16 at 12:18
  • @JyrkiLahtonen Thank you very much. Good health and good luck. – Fly by Night Oct 02 '17 at 17:40