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

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.

One thing that most people do seem to remember from school is solving quadratic equations.  And many find it difficult to imagine what comes after this.  So, partly in honour of my friend Ascher, who often inquires as to how my quadratic equations are going, I will attempt to give some idea of the vastness of mathematics, using the humble quadratic function as a starting point.

So first I will remind you that a quadratic function takes the form:

$f(x)=ax^2+bx+c,$

where$a,b$and$c$ are numbers.  To a high-school pupil, setting these equal to zero to form a quadratic equation and solving for$x$is part of their study of “algebra”.  However, to a mathematician, manipulating symbols in this way in known as “elementary algebra”, and is not an object of study.  When a mathematican refers to algebra, he refers to abstract algebra.  Let me try to give some idea of what this involves.

The first thing we need to ascertain about our quadratic function is where the coefficients (the numbers$a,b$and$c)$are coming from.  For this I will need to define a ring, which I am afraid I am not going to do properly now.  For the purposes of this post let us just say that a ring  is a collection of objects (in this case numbers) such that if you add two together you get another one, and if you multiply two together you get another one.  So the integers$\mathbb{Z}$form a ring, and let us assume for simplicity’s sake that our quadratic function has coefficients in$\mathbb{Z}.$Now a quadratic function is just one example of a polynomial, which are a special type of function which consist of a finite sum of multiples of powers of$x$with the general form:

$a_nx^n+a_{n-1}x^{n-1}+\ldots a_1x+a_0$

These polynomials themselves form the ring$\mathbb{Z}[x]$(if you add two together you get another one, if you multiply two together you get another one).  And then we can form the ring of polynomials with coefficients that are polynomials, or the ring of polynomials with coefficients that are polynomials with polynomial coefficients…and so on ad infinitum.  These are just a couple of examples of rings…ring theorists are interested in general properties of rings, their structure, how to represent them, how their elements interact.

But ring theory is just one branch of algebra!  Let’s go back to our quadratic function, and this time think of it as having coefficients in the field of rational numbers (a field is just a ring in which you can also divide two elements to get another one).  Now if we set the function equal to zero and solve for$x$as we did in high-school, we will get two solutions.  But these solutions might not be rational numbers…that is, they might not be contained in the field we are working in.  So we can adjoin these solutions to our field to get a new, bigger field, and then examine how we can manipulate this new bigger field whilst leaving our original one unchanged.

This leads to the beginnings of galois theory, which is where group theory first arose…and group theory is a major area of research in mathematics.  I am beginning to think that I may have been a bit over-ambitious with this blog post!  But I will attempt a very cursory explanation.  A group is just a collection of objects, a bit like a ring, but with only one operation.  The objects could be numbers, or functions, or permutations…the operation could be addition or multiplication, matrix multiplication, composition of permutations, etc.  Groups are ubiquitous in mathematics.  For a concrete example, take some 2-dimensional regular geometrical shape, and consider the ways in which it can be rotated and flipped.  Each of these manipulations constitutes an element of the symmetry group of the object, and just as with rings, composing two elements produces another element.  For example, if I flip a square from left to right, turn it 90 degrees clockwise, and then flip it again from  left to right, it is the same as if I simply rotated it 90 degrees anticlockwise.  What I am doing here is permuting – or rearranging – the sides of the square, and there is much more in mathematics that can be permuted.  In fact pretty much any collection of things can be permuted in various ways, and many sets of permutations form groups; this is partly why group theory is so prevalent in mathematics.  In the case of our quadratic equation, a galois theorist would be interested in how he could permute the roots – or solutions – of the equation (well, actually he would find the case of a quadratic equation very uninteresting, but you have to start somewhere).  I mentioned permutations in my post on music and mathematics: in that case I was talking about permuting patterns of musical notes, or the order in which church bells are rung.

A nice, simple parabola

But enough about groups, and algebra for that matter!  What would other mathematicians have to say about our quadratic function?  Well, a geometer would obviously want to draw its graph.  It was René Descartes (of “I think therefore I am” fame) who first worked out that we can represent functions geometrically using cartesian coordinates, in the process founding analytic geometry.   In this case, the graph of a quadratic function would be a parabola…the shape a stone would trace as you threw it into the air (or an inverted such shape).   But needless to say, most representations of functions are not quite this simple.  For a start, they might have many more than 2 dimensions.  Collectively, the geometric representation of polynomial equations are known as algebraic varieties, and they are the central object of study in algebraic geometry.  I am not going to go into algebraic geometry, as it is very hard, and I don’t really understand any of  it.  In fact, apart from bog-standard Euclidean geometry, which is the one that we all thought was all there was for thousands of years, I don’t really know much about geometry in general.  There is a bewildering array of theories: hyperbolic geometry, non-riemannian geometry, projective geometry, spherical geometry…

Spherical geometry makes problems for cartographers

Most of them sound rather odd, and involve impossible-sounding things like parallel lines meeting…but they are not just figments of crazed mathematicians’ minds: spherical geometry is what we experience walking around on the earth’s surface for example, and Einstein’s theory of general relativity uses pseudo-Riemannian geometry as its mathematical basis.

As I said, I don’t know much about geometry.  So I will swiftly move on to another of the big areas of mathematics: analysis.  What would we have to say about a quadratic function if analysis was our game?  Well, analysis came out of the attempt to provide a rigorous basis for calculus, so at the most basic level, we could start by using calculus to integrate (find the area under) and differentiate (find the the slope at each point of) the curve pictured above.  We might then consider the function as an element in a space of continuous functions.  By viewing this space as a vector space we could bring the machinery of linear algebra (matrix transformations and so on) to study it.  By giving the vector space a norm or an inner product, we would be able to say something about the “size” of the function and its position relative to others.  And by endowing the space with a topology, we would be able to measure the proximity of the function to other functions in the space (I wrote a bit about topology in an earlier post).  If we were a functional analyst, we would be interested not so much in the function space itself, but in functions from that space to other spaces.  And so on.

I think I’d better stop here, although I could go on for a while yet.  I haven’t even mentioned such major fields of study as number theory, combinatorics and logic.  My point is that the simple quadratic function that everyone is so familiar with occurs in a huge number of very different – and yet inextricably linked – areas of mathematics.   And in none of these areas is everything even remotely close to being “done”!** In fact, the more we discover, the more we find there is to discover, which is lucky as otherwise I’d be out of a job.

* In fact this view does not seem to have been restricted to non-mathematicians! Alexandre Grothendieck – one of the 20th century’s most influential mathematicians – said, in reference to having been advised not to pursue mathematics: “Mr. Soula [my calculus teacher] assured me that the final problems posed in mathematics had been resolved, twenty or thirty years before, by a certain Lebesgue.”
** Apart from perhaps linear algebra

PS – as an aside, I am finding it rather difficult to write in a way which will not be totally incomprehensible to non-mathematicians, and yet also be of some interest to mathematicians. I know there are quite a lot of people clicking on these pages (if not necessarily actually reading them!) and would be grateful for any feedback as to the tone/content of my posts. If you are a mathematician, is this blog too dull and basic? If not, can you even understand a word I am saying? And regardless of what you are, do you think I am trying to cater to too many people and losing coherence in the process? Thanks for any advice!

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

1. Colin Reid says:

Hi Adam! I don’t think even mathematicians can really get to grips with the vastness and interconnectedness of mathematics, but your post does a very good job of hinting at it.

Also, abstraction does seem to be the key sticking points for people who struggle to understand maths. Abstraction for me means the removal or suppression of irrelevant information about something. In principle that ought to make things conceptually easier – you’re freeing your thoughts about something of all the unnecessary baggage. Perhaps the reason people find it so difficult in practice is that in our everyday lives, we are so used to the opposite thought process, where we subconsciously fill in plausible details for things we aren’t observing directly.

Hi Colin,

Yes…interesting point. Abstract thought and expression such as art and symbolic language seems to be something (almost) uniquely human, and yet it is hard to see how it fits into an evolutionary model. As you say, most of the things that we actually do for our day-to-day survival require the exact opposite of abstract thinking.

On one hand, sitting around thinking about things does not help with hunting down something for dinner. But then on the other hand, our ability to think abstractly is surely the main – if not only – reason for our great success as a species. Very puzzling…and probably too thorny a debate for the comments section of a blog!

As for mathematics, certainly I find I need concrete examples of new concepts in order to really understand them. But I know there are some who find the opposite – that examples merely distract from the bigger picture. I’d imagine that those people find mathematics all the more easy to learn, being able to – as you say – dispense with the baggage and get straight to the root of the matter.

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6. I discovered your blog – as I was searching for a way to understand and teach abstract logic, composition of permutations, isomorphism to an 8th grader. I could use some good examples & I found your example of the quadratic equation & the rectangle very helpful. How can I find more tutorials on this topic?