## Quadratic Equations! (or: what do mathematicians actually do?)

April 4, 2010

When I tell people that I study mathematics they tend to have one of two reactions:

1. They make some impressed-sounding noise, or mention that they were terrible at maths at school, and then quickly make it clear that they wish to change the subject.

2. They are genuinely interested, and want me to tell them exactly what it is that I study.

The second reaction is the one that I fear most!  And at this point it is usually me who tries to change the subject.  I have an ongoing competition with myself to increase the length of time which I can spend explaining my research to someone before their eyes glaze over and their body language starts to say “I want to be somewhere else now”.  I am currently up to about 15 seconds.  And the subject I am currently working on (somewhere between graph theory and galois theory) is fairly accessible compared to some of the more exotic branches of mathematics!

The main problem in explaining pure mathematics to a non-mathematician is the level of abstraction involved in the subject.  Most people’s view of mathematics is that it deals with numbers, and it is hard for people to imagine what exactly it is that mathematicians do…add and subtract really big numbers?  Many seem to find it difficult to imagine how it could be that all the mathematics that could be done hasn’t been done already.* People rarely encounter abstract mathematics before university; and for good reason, as the transition from dealing with concrete quantities in a familiar setting, to treating those quantities and that setting as merely one very special case in a vast world of abstraction, can be rather bewildering.

January 29, 2010

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$1.$