What books and web pages do you recommend I read, as well as what blogs and podcasts are good to follow to learn more on a constant basis.
Thanks!
I really wish they'd put a class like that right after basic symbolic algebra in normal school curriculum - it's far more useful in the modern world than trigonometry.
Otherwise these rules come out as a set of vague half-instructions that always lead to rounds of revisions in UAT. Oh, and "we only scheduled a week of UAT".
A broader knowledge of discreet theory would be much more helpful than understanding a sine or cosine at a... trigonometric level.
But anyway, the Teaching Company offers a phenomenal course on Discrete Math, taught by none other than the "Math Magician" Arthur Benjamin (seriously, this guy is magic. He can do 5 digit squares in his head!). It's easily torrent-able / available through more traditional means.
It's incredibly well-written. Very challenging, yet totally approachable and accessible to someone with even just a rather basic (even foggy) understanding of "high-school" math.
I, quite a few times, spent more than four hours with a pencil and paper making my way through 2-4 pages of material, as since my background was lacking, there were parts where I had to figure out how they got from A to B. IIRC, I spent multiple days on a few paragraphs at one point.
That said, it is written how I think math books should be written, is considered part of the start of Discrete Mathematics, and is conversational, illuminating, well-layed-out, and approachable despite it's level of difficulty.
I can't recommend it enough.
Don't view it as something where if you're not making progress you're failing. Take the time to ensure you understand every bit of what you read before moving on, and do all the exercises you can muster.
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Good books on mathematics for somebody who's only taken high school math? : http://news.ycombinator.com/item?id=299687
Recommendation for (re)learning Math Skills : http://news.ycombinator.com/item?id=1449799
Assuming that you want to learn some 'university-level mathematics', then you'll really need to be prepared to study and work through problems rather than just read. Mathematics is an area where it's hard to get breadth of knowledge without also having at least some depth because things build very much on each other.
If you really do just want an overview of areas of different areas of mathematics to whet your appetite, there are books by people like Ian Stewart, Marcus du Sautoy and Keith Devlin, all worth reading. Just be aware that reading these is a bit like reading about different programming languages without ever having written a computer program.
If instead you just want to keep your brain engaged mathematically without learning more serious mathematics, there are also plenty of recreational mathematics books out there - Martin Gardner being the name that instantly springs to mind. On a similar vein, you may also enjoy the books of Raymond Smullyan which are more focused on logic.
The only really nice non-textbook taster of university-level mathematics that I have found in Alice in Numberland by Baylis and Haggarty. However it's out of print so you might have problems getting hold of a copy. It is a lovely book though if you can get your hands on it.
Otherwise you are looking at textbooks. I'd recommend maybe 'Introductory Mathematics: Algebra and Analysis' by Geoff Smith as a gentle but rigorous intro to the basics that I'd expect every maths student to learn at the start of their degree course. There are lots of alternatives out there too though. I taught myself lots from Herstein before going to university but that's pretty heavy going and there are better books out there these days. If you look at other books, I'd probably suggest getting one on abstract algebra maybe, covering things like sets/functions and group theory rather than say analysis or linear algebra to start off with, as it's easier to get into the right mathematical mindset if you're not distracted by content which you already have intuitions about.
If you want to build up your math muscles (as a good preparation for actually studying maths), you should have a look at some discrete mathematics books (the one I had was "Discrete Mathematics" by Norman Biggs) as they teach you to think in terms of proofs.
If you want to get a thorough foundation for non-discrete maths, you should start with a good (university math) analysis textbook (No idea what's a good one in English).
Another approach you could take is to take a math book that is targeted at physicists and EE people - those usually skimp on the proofs and don't contain enough detail to understand the fundamentals behind it all, but bring you to the interesting (to physicists and EE people) stuff much quicker than a real math course would.
Oh, and if you hang out on Youtube, be sure to watch the catsters - this is category theory, presented by actual working mathematicians, at an accessible level (and with a cute UK accent too).
It was organized in a dependency graph kind of way.
I recommend The Art of Problem Solving I and II. On the one hand, they're intended for (mathletic) middle and high-schoolers. On the other hand, some of their problems are quite challenging, and much of the material therein is what my school teaches in its intro discrete math courses since very few students learned it in middle and high school.
Basic
1. Chapter Zero (Carol Schumacher) 2. Naive Set Theory (Paul Halmos)
Linear Algebra
1. Finite-Dimensional Vector Space (Paul Halmos) 2. Linear Algebra Done Right (Steven Axler)
Real Analysis
1. Real Mathematical Analysis (Charles Pugh) 2. Introduction to Analysis (Maxwell Rosenlicht)
Algebra
1. A first course in abstract algebra (John Fraleigh)
These books are better for self-directed reading compared to some of the classics like Rudin or Herstein. These should keep you busy for a while.
Are there any good texts that are more problem workbook style? One example that comes to mind is "Exercises in Probability", or some of the 3000 Problems books as published by Schaum.
As for a good broad overview of many areas, the title that springs to mind is 'the nature of mathematical modelling' by Gershenfeld, though you'd better have some decent maths experience before tackling that one - it can be tough-going, but is refreshing in its breadth and clarity.
linear algebra (strang, trefethen, golub and van loan) optimization (nocedal, bertsekas) probability (rice, casella & berger, grimmett) statistical learning (tibshirani, bishop)
A good free online book was recently an HN topic: http://news.ycombinator.com/item?id=1738670
Also, anything by Martin Gardner.