# 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 ?

• 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 Naslund 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.

• 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.

• 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

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."

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.

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.

This is what I'm currently contemplating as well, as I have an interest in self-studying mathematics, trying really hard to cut off my other interests entirely to give it the proper attention it needs for at least a good while. I've studied maths here and there from various books over the years, but I found over and over that I would run into problems requiring fundamental understandings of other branches. So lately I've been doing research and here's some of my results, hope it helps:

Books can be broken down into 3 categories: 1) School textbooks written by education experts 2) Books written by the creator(s) of the theory and 3) Books written by qualified professionals competent enough to present the material, sometimes in the most highly regarded and excellent ways. Of course sometimes a book can belong to more than one of these catergories. Now, here's what I got so far:

Basic Algebra - STG books are good, such as Practical Algebra or Quick Algebra review(which is what I used) just as a refresher. If you're not flying through these books then take some time to understand it before moving on to anything else because they present the material as painlessly as possible.

Trigonometry - Actually, I'm using "Geometry and Trigonometry For Calculus" by Selby. It presents the material you need to know in a way that prepares you for calculus and finishes off by developing the idea of limits, which you will definitely be thankful for later!

Calculus - "Quick Calculus", written by actual physicists, teaches you the fundamentals of differential and integral calculus in a way that only applied scientists can do. DO NOT try to learn calculus from pure mathematicians' treatment of it, because a lot of the intuition and physical applicability is lost through the purification when really, calculus evolved as a framework for solving physical problems so, naturally, people who apply this stuff professionally will have a good understanding of what the stuff actually means. Then you can move on to stuff by Michael Spivak, Tom Apostol, Richard Courant or GH Hardy. Basically, you'll have single-variable then multi-variable calculus treaments so you'll want a 2 volume set when you're ready. I would suggest Richard Courant's work along with Tom Apostol.

Linear Algebra - I don't know why this subject doesn't get more attention, maybe because it's the actual key to a lot of fields, like computer science, engineering, etc. Some good books on this are written by Rothenberg and Zhang.

Set Theory - This is a tough one. I don't like set theory very much beyond the basics of what's necessary for learning more advanced mathematics. I have Enderton's 'Elements of Set Theory', perhaps not the best book for a beginner but I got through the basics alive and stopped after completing the chapters that were suggested by the author. A lot of the problem solving deals with proving relationships, so the answers early on took the form of proof writing, which can get frustrating when they talk about "the power set of the power set of a subset of the union of..." Just ignore any axiomatic treatments of set theory and try to understand set builder notation because you will need it for abstract algebra and pretty much everything else.

Abstract Algebra - I'm not quite here yet but I have a book that doesn't seem too difficult to follow after learning a bit of set theory. Here you will learn about groups, rings, domains, modules, vector spaces and much more.

Analysis - I think analysis is an extension of calculus, or perhaps a precise development of it? In any case, "A Course of Mathematical Analysis", by Whittaker and Watson, is hailed not only as one of the finest analysis books but also one of the best math books to read. Now I don't know how this breaks down but you would study basic analysis then functional or real and complex analysis.

I think that covers the bases. Beyond this, I'm not sure of any real order: it depends on what you're interested in. I know number theory belongs in there somewhere though. For example, I want to study the Bessel Functions and complex numbers. I like how the simple idea of a function has produced a never ending stream of usefulness. I also found myself exploring some basic functions, experimenting with random setups and seeing what comes out so I think I want to know more.

Here's some tools I would suggest: Buy a small dry erase board from Walmart, trust me. Persistence and mental toughness are also key here. Blast every negative thing away as quickly as possible. Every person or thought that tells you you're wasting your time or you're not smart enough or you're too old or whatever you run into that might discourage you from study, ignore it, overcome it as FAST as you can; don't waste any time thinking about it or stressing over it. I'm not even 30 years old yet and already the gods like to use age to scare us away. If you can't figure something out, come to it later. If you lose interest, do something else. The fate of the world is not resting on your shoulders and you won't be rewarded for learning stuff at a prodigious rate. As long as you're not spending entire days on a single problem, you should be okay.

I have categorized everything beyond these subjects under special topics. Partial differential equations, Lebesgue integration, Riemann's zeta function, Fourier series, it all begins to open up. Part of the difficulty of maths comes from the compression of information behind all the symbols. But most of the time those impressive symbols are just instructions on what to do.