Since my last post I’ve started attending a course on$p$-adic numbers.  Initially my only real motivation for doing so was that a closely related concept had come up in my research; I had previously been of the opinion that the study of$p$-adic numbers was something of a niche pursuit that bore little relevance to other areas of mathematics.  However, having attended 2 lectures, I am finding the subject quite fascinating, and pleasing in the way it relates concepts from algebra, number theory and analysis.  So today I’m going to write highly non-rigorously about some of the interesting bits…perhaps I will even do a short series of posts on the subject.

So what are the p-adic numbers?  I think the best way to explain this is to start by talking about something a bit more familiar: the real numbers.  A space is complete if, intuitively, it “has no gaps”; this is a very desirable property from the analyst’s point of view (in fact analysis can only be done in a complete space, as the notion of a limit does not make sense if there are gaps in the space).  The formal definition of a complete space is one in which every Cauchy sequence – that is one in which the gaps between elements eventually get infinitesimally small – converges to a point in the space.  The rational numbers are not complete because, for example, we can construct a sequence$(a_i)$that converges to$\sqrt{2}$by defining:

$a_1=1$

$a_2=1.4$

$a_3=1.41$

$a_4=1.414$

…and so on.  The real numbers$\mathbb{R}$can be obtained by completing the rational numbers$\mathbb{Q}$, that is, by “filling in the gaps”.  The way we do this is to take every Cauchy sequence in$\mathbb{Q}$and let$\mathbb{R}$be the set of points that these sequences converge to (for the more technically-minded,$\mathbb{R}$ is the quotient ring$\frac{C}{M}$, where$C$is the ring of Cauchy sequences in$\mathbb{Q}$and$M$is the maximal ideal of$C$consisting of all sequences converging to zero).  A helpful way to think of this is by envisaging the decimal expansion of every number as being a convergent sequence, in the same way as we saw above for$\sqrt{2}$.  Sequences are considered to be equivalent if they converge to the same point, and so for example$0.9999...=1.000...$, because the sequences:$0,0.9, 0.99, 0.999,...$and$1, 1.0, 1.00 ,1.000,...$both converge to$1.$

Now, to say a sequence converges to a point is to say that it gets “infinitely close” to that point, and so the notion of convergence depends on what we mean by “close”.  There are many different metrics – that is, ways to measure distance – in mathematics.  In the above construction of the real numbers the metric we used was induced by the usual absolute value$|\cdot |,$so we considered the distance$|x-y|$between two numbers$x$and$y$to be$x-y$if$x\geq y,$or$y-x$if$y\geq x.$

Let$p$be some prime, and let

$x=p^{e_{0}}.q_{1}^{e_{1}}.q_{2}^{e_{2}}...q_{n}^{e_{n}}$

be the prime decomposition of some rational number$x$, where the$q_i$are distinct prime numbers different from$p,$and the$e_i$are integers.  We then define the$p$-adic absolute value$|x|_p$of $x$ to be $p^{-e}$.  So for example, the$2$-adic absolute value of$12$is$2^{-2}=\frac{1}{4}$, because the prime decomposition of$12$is$2^{2}.3.$$|12|_{3}=\frac{1}{3}$, and$|12|_{p}=1$for any other prime$p.$

You will no doubt have noticed that numbers which are divisible by higher powers of$p$have a smaller$p$-adic absolute value.  Using this notion of size to measure the space between numbers provides us with another, rather counterintuitive way to define distance, in which numbers which differ by a large power of$p$are thought of as being close together.  For example, in the$3$-adic metric the distance between$1$and$100$is$\frac{1}{9},$while the distance between$\frac{1}{9}$and$\frac{2}{9}$is$9$.  This means that Cauchy sequences in the$p$-adic metric look rather different from those we are used to.  Take the sequence:

$p,p^{2},p^{3},p^{4},...$

In the$p$-adic metric, the gap between the$i$th and$(i+1)$th element is:

$|p^{i+1}-p^i|_{p}=|p^{i}(p-1)|_{p}=p^{-i},$

so these get smaller and smaller: this is a Cauchy sequence.   In fact, this sequence converges to zero.  But just as was the case with the usual absolute value metric, there are Cauchy sequences which do not converge in the$p$-adic metric, eg:

$p,p+p^{2},p+p^{2}+p^{3},p+p^{2}+p^{3}+p^{4},...$

Again the sizes of the gaps between elements here approach zero.  But the limit of this sequence is:

$\sum_{i=1}^{\infty}p^{i},$

which is not a rational number.  By adding the limits of sequences like this one in the same way described above for the real numbers, we can complete the rationals in a different way: this time with respect to the$p$-adic metric.  This gives us the field of$p$-adic numbers,$\mathbb{Q}_{p}.$Because$\mathbb{Q}_{p}$ is complete we can do$p$adic analysis on it, in much the same way as we do real analysis on$\mathbb{R}$and complex analysis on$\mathbb{C}.$But why would we want to do that?  And really, what is the point of any of this?  Hopefully I will get to that next time….

### One Response to p-adic preliminaries

1. […] mathematics.  I don’t refer to the space-time continuum, but to the real numbers,.  In my post onadic numbers, I mentioned how completeness is an important mathematical attribute for a space of numbers to […]