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What is pure mathematics? What is statistics?
By Anonymous on Wednesday, July 5, 2000 - 08:49 pm:

I am taking my GCSE pure maths a year early and I wondered if you could tell me what pure mathematics is.

If possible could you give me a syllabus of areas included in pure mathematics.

What is statistics in mathematics?

Thank you

By Anonymous on Thursday, July 6, 2000 - 10:10 pm:

Pure maths contains the topics of equations on shapes and sizes of matter, it deals with vectors, differentition and integration.

By Dave Sheridan (Dms22) on Friday, July 7, 2000 - 11:52 am:

Much more than this is true. What you see at GCSE level is only a brief taste of pure mathematics as it is studied by professionals. Perhaps it is more useful to describe the difference between pure and applied maths. If you want to describe a real-world situation using maths, that's physics. If you want to study the effects of changing your physical model by refining it to be more accurate (adding extra effects, for example) and rigorously study the results (ie do things actually get better or do the results just appear to be good?) then this is applied maths. For example, studying the motion of individual points in a fluid, the formation of galaxies or subatomic particles.

Pure maths, on the other hand, is more the study of objects which are constrained to have particular properties. There are three main areas, broadly called analysis, geometry and algebra. Analysis is the study of functions, and amongst other things concerns exactly what can be called a function (or generalised function, etc) as well as defining properties such as continuity, measurability and convergence properly. Geometry has its basis in the sort of thing you see at school as "geometry" but goes well beyond that and touches on topology, manifold theory and many things I don't know about... Algebra is not the study of rearranging equations, but rather the type of object which is described by a couple of simple rules - and what we can say about general constructions which satisfy these. For example, groups are an algebraic construct and can be defined by four simple rules - it's quite amazing what can and can't be proven from these and what you need to add in order to see the behaviour you expect when thinking of, say, addition on integers.

Statistics and probability come under the heading "applicable" which is somewhere between pure and applied. Probability is the study of "randomness", which can loosely be described as something unpredictable. You can't say for certain whether a coin will be heads or tails, or the precise location of a quark, but you can describe what's likely to happen, and typical behaviour is often useful for deciding what to do about the situation. Statistics is the study of random data, and making sense of it. If I conduct a survey and 20% of people I ask think my product is awful, does this mean I should improve the product or would I expect to find this many people who are awkward about everything? When are the results of a scientific experiment justified (not all will be positive)?

You won't be able to learn all of pure mathematics; the topics you need will be on a syllabus provided by your school. We can help you to understand any particular thing but the subject is so immense that it's impossible to even give you a quick guided tour of the subject without any extra information.

Hope this has been useful. I'm a probabilist so excuse me if I've not described things as well as I could in the areas I don't understand so well. If you'd like more information on particular topics, just ask.

-Dave