I know a lot of universities offer mathematics as a degree, but what do mathematicians actually do all day?

I imagine that a lot of them end up working out complex formula for scientists and engineers, although I don’t know this for sure.

Is there such a thing as pure mathematics research? For example, I remember doing imaginery numbers at school (e.g. 1+i, where i is the square root of -1). Its a fairly abstract concept, but it actually came in useful when I went to university.

How are concepts like this arrived at? Does it take research, or does it just come from inspiration? Is the maths invented first, and then applications for it found later, or do we have to find a use for it first?

Well, a lot of them teach at the university level. Many, but not all, of those people also do pure mathematical research and publish papers on it. Partly they do it because it helps them in their career (it get them tenure, it gets them paid more, and it keeps them up-to-date on what’s going on in their mathematical speciality), but partly they do it because they enjoy doing it.

Being a mathematician in industry or government can mean a number of things. There are no jobs doing pure mathematical research (except for a tiny number of jobs in think tanks or attached to universities). Other mathematical jobs range from just being a computer programmer working on somewhat more mathematical tasks than most programmers to doing research on scientific problems that require very deep mathematical knowledge which can be applied to current scientific research.

The rest of your questions would take a long time to answer. I know that you’d like the answer to them right now, but I don’t think it’s possible to explain them well in a single post. Perhaps you should get a history of mathematics and read more about this subject.

Pure mathematics research is probably a bigger field than most people realize, though it’s not some enormous underground thing. In 2000, the Clay Mathematics Institute offered a prize of $1 million for each of the top seven unsolved problems in mathematics. The first person to solve them gets the prize money. These include The Reimann Hypothesis (which you might know of if you saw A Beautiful Mind) and P vs. NP (which you probably know if you’re into computer science).

And just to be clear, yes, there is pure mathematical research. There are many, many unsolved questions in various fields. Few of them can be explained to the layperson – the math that you are taught in undergraduate courses is far from the edges of discovery.

To be even clearer: these unsolved questions are NOT simply “Mary has four apples and John gives her two more.” For instance, the field of algebraic topology is concerned with imposing algebraic structures on surfaces. Chaos theory (which emerged from theories of dynamical flows on manifolds) is a fairly new field.

Mathematicians employed by universities are usually required to publish research results. There are several monthly journals that accept such publications. It’s not much different than the English Lit professors or the History professors – publish or perish, if you’re at a research university. (If you’re at a smaller university that doesn’t put pressure on research, then you’re teaching a heavier course load than you would at a research instititution.)

Interesting stuff. I imagine that most of the actual mathematical research done these days is way beyond my comprehension these days. I was fairly good at maths at school, but I remember looking at the Clay Mathematics Institute problems a few years ago, and not being able to work out what any of them meant.

Does anyone actual still research ‘real’ numbers (or is this an extremely stupid question)?

Number theory is a fairly straightforward branch of mathematics at its core, but it gets as hairy as the other ones high up. You may have heard of the Goldbach Conjecture:[ul]Every even number greater than 2 is the sum of two primes.[/ul]It is still not known whether this statement is true or false, and though I don’t know any personally, I’m sure there are mathematicians working on it. Yet it is delightful in its elegance and comprehensibility, no?

This is wonderful… I’m studying two subjects that many people don’t see the purpose of - Philosophy and now Mathematics (note that this will be a combination of Pure and Applied). Now I’m starting to understand why I get blank faces when I tell people this…

I think others have clarified what Mathematicians do. Perhaps you would be interested in my rational for picking the subject. I wanted to study Mathematics because it is a backdoor into a lot of careers. Finance and IT are the two big ones, which are pretty attractive routes in my opinion. There is also the oppertunity to branch into research. Naturally, I have also had a lasting interest in the subject and am sufficently able. I am also taking Philosophy because the subject fascinates me, I need a little creativity and I want to develop my thinking, writing, etc.

The subject combination isn’t as odd as many think - at least in the UK many top universities offer it. There are crossovers such as Logic, though in some regards the subjects are polar opposites!

Bakhesh, what I think you need to appreciate is that few university degrees are strictly vocational. Medicine and Law are the general exceptions to the rule. Never did I ever choose the course with dreams of becoming a Mathematician, in the same way that few History students aspire to becoming Historians.

Just FYI, I graduated with a Dual Degree in Mathematics/Computer Science in 2002, and we were told by our mathematics department that the biggest employer of mathematicians was actually the NSA. So it would seem that for some mathematicians, they do code breaking work all day. Personally, I’m doing programming right now, and most of the other mathematics majors I graduated with are either teaching, or also had a dual degree and are working in that field (mostly physics or engineering)

Place I work has several mathematicians (for some reason that dosen’t look right). There are two different problems they tend to work on. One group tries to find mathmatical solutions on how to predict ground contamination from different sources, and ground types. The other group works with statistics of something, but not sure what statistics they are looking at.

Another big job market for mathematicians is in the cryptography field. Encryption is heavy into math, and so is cracking encryption. I tried to read a book on it (Applied Cryptography by Bruce Schnider) but couldn’t understand the math.

So yes there are real jobs for people with math degrees where they will actually do math.

I ended up a ‘Data Analyst’ (statistician) for a market research company. I also manage my area.

I work with survey and database data. This includes segmenting markets, setting price recommendations, analyzing competitiveness of clients vs the competition, helping design what new features are good for new products etc… I find it very satisfying and challenging.

Insurance (actuarial) attracts many math types. These are the people who analyze risk and set insurance rates.

Math is a good major to have. It has a ‘natural respect’ in the business world in that if you have a math major, you are thought to be smart.

As with all degrees, it still is basically a hunting license in the business world.

Another one of the “Clay problems” is the Poincaré Conjecture, which may have been solved by a Russian mathematician just this past year. See http://mathworld.wolfram.com/news/2003-04-15/poincare/ for an overview of the hub-bub, if you’re interested.

Another recent example of a Big Problem Solved was Fermat’s Last Theorem. Fermat was very good at thinking up statements that sounded like they might be true, but not so hot at actually proving them. Most of his conjectures were proven (or disproven) by Euler, but there was a little note he scribbled in the margin of one of his books which remained unproven until the 1990s.

Fermat stated he had found a “beautiful proof” of his last theorem just before he died, yet it is unknown whether he did or didn’t. It took Andres Wiles many years [and help from a totally different branch of mathematics, which, IIRC, was not even thought of at the time of Fermat] to come up with his famous proof.