# In what order should the following areas of mathematics be learned?

I am in a biological field (medicine) but I have genuine passion for mathematics. I want to learn it on my own , in my spare time. Mathematics , as I gather, is learned best when you have grasped the prerequisite concepts for the area you are currently interested in. Kindly suggest a sequence of study for the following areas ...

1) Algebra
2) Calculus
3) Discrete math
4) Geometry
5) Probability and statistics
6) Mathematics Software Packages (which one do you suggest for primarily educational, nonprofessional use)
6) Any other areas of fundamental importance that I may have missed

EDIT: does first order logic and set theory belong up there ?

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You forgot analysis. Probability theory is in my opinion something completely different from statistics, so I'd place them in a different category. – Jonas Teuwen Feb 13 '11 at 22:47
Fundamental importance for what? There is a lot of mathematics out there. – Qiaochu Yuan Feb 13 '11 at 23:52
Although you haven't asked for book suggestions, if you want to refresh your high-school mathematics, try Basic mathematics by Serge Lang. – lhf Feb 14 '11 at 0:34
What do you mean by "Algebra"? – Eric Feb 14 '11 at 0:49

I'm unclear what you mean by "Algebra"; if you mean stuff like working with polynomials, basic equations, symbolic manipulation, etc., then that goes first. If you mean "abstract algebra", then you can wait.

Added. Likewise: if by "geometry" you mean classical geometry, or even projective geometry, then the following applies.

Calculus, Discrete Mathematics, and Geometry, are independent enough that their order doesn't matter.

Added. However, if by "geometry" you mean analytic geometry, then it should definitely precede calculus, and the same is true if it means trigonometry. I think it unlikely that you meant "differential geometry" or "algebraic geometry", but if you did those are very advanced topics that should wait until well after calculus, abstract algebra, and real/complex analysis.

For introductory probability and statistics you'll find Discrete Mathematics very useful; for more advanced probability and statistics, Calculus is a must.

An "introduction to proofs", which would include some set theory, some basic logic, etc., can be done at the same time as Discrete Mathematics, or immediately after.

After all this, then you can hit linear algebra, abstract algebra, real or complex analysis, in pretty much any order (though complex analysis should follow real). Abstract algebra is a bit easier if you've taken linear algebra, but this is not strictly necessary.

If you happen to find probability and statistics very interesting, then you should do some measure theory after the real analysis.

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Similar to what you wrote about "algebra", the statement "calculus, discrete mathematics, and geometry, are independent enough" really depends on what the OP meant by geometry. Classical euclidean geometry? I agree. Analytical and differential geometry? I think those are rather deeply connected with calculus. Metric geometry? That often uses quite a bit of tools from discrete mathematics. – Willie Wong Feb 14 '11 at 0:22
@Willie: Good point; let me edit. – Arturo Magidin Feb 14 '11 at 0:23

This is the order I would suggest:

• Linear Algebra
• Calculus
• Discrete Math
• Probability and Statistics
• Geometry

The first two are interchangeable. According to some people(article about teaching math), discrete math should really be taught first.

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 does first order logic and set theory belong up there ? – explorest Feb 13 '11 at 22:53 @explorest: some of it may be covered under Discrete Math. – Arturo Magidin Feb 14 '11 at 0:57

I'm doing the same thing.

I find its best to interleave your study of these topics.

1   algebra / geometry / calculus          | math
2   discrete math                          | software pkgs
3   probability+stats                      | (matlab/octave/maple)


You'll find a math software package takes a while to get used to (esp if you don't have a programming background). So starting with one right away is a good idea. Octave is a free version of MATLAB. Be warned: Octave can be painful to use at times. MATLAB is the Cadillac.

Also if you're doing proof-based study, you'll find a lot of calculus and algebra (at least as it appears to me at the moment) is mostly orthogonal. That is, the Fundamental Theorem of Calculus won't really help you to prove the Triangle inequality. But how to prove and your way of thinking should be exercised by proving in either topic.

Algebra is closely related to geometry. Linear transformations, geometric intersection - to me geometry seems to boil down to simply applied algebra.

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In my opinion you want to learn the topics in this order :

• Algebra
• Geometry
• Calculus/linear algebra, (statistics if you want, not necesarry)
• real analysis
• complex analysis

Any duplicates on a line means that you can learn them simultaneously! I would assume that you would cover some calculus rigorously in analysis and also touch upon some set theory, and of course, learn the fundamentals of "proofs."

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I think I can recommend some books on algebra, such as the book An introduction to field theory by Iain.T.Adamson or the book Galois Theory by Joseph Rotman of which I think as pretty suitable for beginners.
As for further study, I am wondering if you are interested in algebraic number theory or other ones(As Gauss once said, Mathematics is the king of science and Number Theory is the queen of Mathematics). As Jurgen Neukirch said, Number theory is Geometry, you might firstly be familiar with analysis or geometry to study Numbers. Of course you might not be interested in numbers, nonetheless, if you do, get the book by Hilbert whenever you can fully understand it.
As for calculus, the books by Richard Courant is absolutely good and worth studying.

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