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Ask HN: Learn Math the Hard Way

171 points by nsomaru 14y ago

I am looking to teach myself Mathematics, mainly to improve my thinking processes and learn more about the basis for CS based topics like algorithms and proofs, and other practical applications. Suggestions for resources? (textbooks, videos, etc)

1331 14y ago
_Concrete Mathematics_ [1] is a good book that is fun to work through. The book can be considered an introduction/prerequisite to Knuth's _The Art of Computer Programming_ series [2], which contains much mathematics. If you would like a book that is not so difficult, _Introduction to Algorithms_ [3] (aka "CLRS") is also highly recommended. [1] http://www.amazon.com/dp/0201558025/ http://www.amazon.com/dp/0201558025/ [2] http://www.amazon.com/dp/0321751043/ http://www.amazon.com/dp/0321751043/ [3] http://www.amazon.com/dp/0262033844/ http://www.amazon.com/dp/0262033844/
Arun2009 14y ago
In addition to Concrete Mathematics by Knuth, which contains a lot of Mathematical tools and techniques useful in areas of Computer Science, I'd recommend the following to learn how good problem solvers approach Mathematics: 1. The Art and Craft of Problem Solving by Paul Zeitz. 2. How to Solve It by George Polya. I'd also recommend learning any of the competition Mathematics topics like number theory, combinatorics or geometry as base material to improve your problem solving skills. I have also found the Schaum's series on Combinatorics and Graph Theory concise and useful.
verganileonardo 14y ago
I have a business background and use Project Euler to evolve my programming, math and problem solving skills. :) projecteuler.net
- big_data 14y ago
  +1 for Project Euler! It totally rocks!
madmanwalking 14y ago
The Khan academy (www.khanacademy.org) has plenty of videos and exercises, and if you're looking more at the CS side of things udacity.com runs a few online courses looking at algorithms, discrete maths and the like.
- tonetheman 14y ago
  This was going to be my suggestion. Excellent resource.
tel 14y ago
I'm going to second Arun2009's recommendation for How to Solve It by Polya. And then I'm going to recommend Haskell. I recently realized how much my understand of set theory improved and was solidified purely by programming Haskell a lot.
- vitno 14y ago
  I started messing around with haskell a few months ago. It completely inspired me to revamp my entire mathematical understanding... I've bought (and read!) like 4 math text books...
- jasomill 14y ago
  I'll third How to Solve It, and would also recommend Polya's Induction and Analogy in Mathematics.
- motxilo 14y ago
  Math + Haskell = http://homepages.cwi.nl/~jve/HR/ http://homepages.cwi.nl/~jve/HR/
nimeshneema 14y ago
The best text-book I can suggest for you is: "How to Think About Algorithms - Loop Invariants and Recursion" By Jeff Edmonds. It takes you through all the relevant mathematics required to learn about basics of CS based topics you are looking for (algorithms, proofs and other practical applications). It also acts as a good primer for learning basic Computer Algorithms. The USP of book is it's noob friendly and I think perfectly suited to your needs. Check it out here: http://www.amazon.com/Think-About-Algorithms-Jeff-Edmonds/dp/0521614104/ http://www.amazon.com/Think-About-Algorithms-Jeff-Edmonds/dp...
alex-g 14y ago
You might like "To Mock a Mockingbird" by Raymond Smullyan. It's a book of logic puzzles with a focus on combinatory logic (a model of computation related to the lambda calculus). Aside from the logic/math/CS nature of the content, the progression of the exercises should help with mathematical-style reasoning in general.
- psykotic 14y ago
  As fond as I am of the Mockingbird book and Smullyan's books in general, I'd say that most of its puzzles are pure exercises in abstract symbol manipulation and not terribly enlightening. If you have some prior exposure to the SKI combinators, one big aha-moment may come from the book's exploration of several alternative axiom systems that, while preceding the SK system chronologically, are often left out of modern CS-oriented treatments of combinatory logic. Also, while I think the exercises on symbol manipulation are not very enlightening on their own, the syntactic rewriting point of view is a useful complement if your view of combinatory logic has been mostly semantic. Proving completeness in the syntactic framework becomes a matter of showing that any combinatory rewriting rule can be expressed in terms of your base combinators. This in turn can be done if your combinators allow arbitrary permutations to be expressed in addition to duplication and dropping of symbols (cf. lengthening and shortening rules in GEB). For example, K is a dropping combinator, and rather than viewing S as a substitution operator on the semantic level (as is done when translating lambda calculus to SK calculus), it can be viewed as a hybrid permutation and duplication combinator at the syntactic level. If these dual roles are split into separate combinators, you get alternative axiom systems.
jcdreads 14y ago
The Project Euler problems might be right up your alley. They are little programming exercises that start easy and rapidly become harder. Along the way you figure out enough math to make the programming easier, or at least enough to make better algorithms. The forums are enormously helpful. http://projecteuler.net/ http://projecteuler.net/
ether 14y ago
I don't think you need to learn really hard core math to do CS. In my experience, the one most important "genre" of math was discrete mathematics. My textbook was Elements of Discrete Mathematics, and I think it was a really great intro: http://www.amazon.com/Elements-Discrete-Mathematics-Computer-Science/dp/007038133X/ http://www.amazon.com/Elements-Discrete-Mathematics-Computer...
drostie 14y ago
You may not come from the same culture that I came from, but the early significant milestones in my mathematical education were: 1. Reading the "aha!" books by Martin Gardner, as a child. 2. Reading Lockhart's "A Mathematician's Lament" [1] 3. Linear algebra, calculus, and complex analysis classes. I was taught these at Cornell; you might look for them via MIT's OpenCourseWare [2]. 4. A bunch of combinatorics, from Cornell's classes in probability. The most key thing I can tell you about mathematics is, always go back and read the definitions. Always. I have sometimes helped out math students on topics which were way out of my depth, simply because I picked up their textbooks, flipped several pages back, and said "hm, what does this word mean? what does that word mean?", building up connections all the while. You may also want to start in a completely different direction, since you're interested more in CS-based topics (I am interested more in physics-based topics) by starting with number theory and modular arithmetic, and of course MIT has a Comp Sci section [3] which also produces lectures on OpenCourseWare; you may wish to watch those lectures. [1] http://www.maa.org/devlin/LockhartsLament.pdf http://www.maa.org/devlin/LockhartsLament.pdf [2] http://ocw.mit.edu/courses/mathematics/ http://ocw.mit.edu/courses/mathematics/ [3] http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/ http://ocw.mit.edu/courses/electrical-engineering-and-comput...
- dwc 14y ago
  FYI, Lockhart has a new book coming[1] out with the kinds of things we should be doing, which is something I (and probably everyone who read & liked his Lament) have been pining for. I'm waiting like everyone else, so I can't explicitly recommend it. But I certainly anticipate a mental treat. 1. http://www.amazon.com/Measurement-Paul-Lockhart/dp/0674057554/ref=pd_rhf_ee_p_t_1 http://www.amazon.com/Measurement-Paul-Lockhart/dp/067405755...
Someone 14y ago
For getting to know what is a proof in Mathematics, I recommend Proofs and Refutations by Imre Lakatos. It will not teach you many theorems (in fact, 80% of it only proofs one, http://en.wikipedia.org/wiki/Euler_characteristic http://en.wikipedia.org/wiki/Euler_characteristic again and again and again), and it will almost certainly not be applicable to writing programs but it will give insight into how mathematicians in the last century have made mathematics much more rigorous (and abstract).
hendzen 14y ago
Start with these topics. Work your way through them, and make sure you understand everything you have read so far before progressing. Do the exercises, or it will be a waste of time. 0) Algebra, Trigonometry, Calculus Make sure you have a decent grasp over high school level math topics. You might not need to use these topics frequently (though trig comes up a surprising amount), but they are necessary to establish a base level of mathematical maturity. 1) Linear Algebra Obviously very important if you want to do any 3d work, but it comes up in later topics like graph theory. Suggested book: Linear Algebra and Its Application - Strang 2) Discrete Math You need an understanding of proofs and logic before you can get to the real algorithms material. Counting and probability is also very important. Suggested book: Discrete Mathematics and Its Applications - Kenneth Rosen This looks like a good free option, but there are no exercises: http://courses.csail.mit.edu/6.042/fall10/mcs-ftl.pdf http://courses.csail.mit.edu/6.042/fall10/mcs-ftl.pdf 3) Algorithms Enough said. Suggested book: CLRS, but here's a free option: http://www.cs.berkeley.edu/~vazirani/algorithms.html http://www.cs.berkeley.edu/~vazirani/algorithms.html This should take you some time as each of these topics usually corresponds to a college class. When you finish with those, start reading about CS Theory, Combinatorics, or pick up a graduate Algorithms text.
- ekm2 14y ago
  Another Linear Algebra book that also has a good introduction to proofs is Sheldon Axler's Linear Algebra Done Right.
- sshrin 14y ago
  You say 'trig comes up a surprising amount'. It would be great if you could give a few examples of these instances. I have always thought of trigonometry as one of the areas I really enjoyed in school but something I have never used ever since.
  - Dn_Ab 14y ago
    I think the surprising commonality is conditional. So you won't see it unless you are focusing on physics or signal processing.
gmoes 14y ago
Here's a post I did on free math resources, including a list of resources on various topics: http://www.elegantcoding.com/2011/06/free-math-resources-you-can-use.html http://www.elegantcoding.com/2011/06/free-math-resources-you... There's some really great stuff listed there, I use it all the time.
pdelgallego 14y ago
I am in the ship, I have signed in the statistics 101 class [1] at udacity. Its very basic, but it's a good starting point. Coursera have also several math related courses [2] (calculus, autmata, logic, ...). I have also signed in a class in nearest community college in my area. [1] http://www.udacity.com/overview/Course/st101/CourseRev/1 http://www.udacity.com/overview/Course/st101/CourseRev/1 [2] https://www.coursera.org/category/math https://www.coursera.org/category/math Happy logarithmic!
- radicalbyte 14y ago
  Coursera's Automata course is really good, but it's hard work. It's a six week course, and you'll spend around 10hrs per week if you do everything (and I'd recommend doing everything). Udacity's introduction to statistics, so far, has been really easy. From what I've seen so far, Udacity are aiming to be more inclusive "community college" level and Coursera are aiming for a true university level education.
sidman 14y ago
As mentioned probabilities is something you want to get into very early. Its easy to understand at the start and its practical uses in many different computing areas along with many other things in life is priceless. Good place to start would be to go through khan academy's probability videos.
mdkess 14y ago
When I was in early university, I went through Introduction to Algorithms and chapter by chapter, I implemented nearly every algorithm presented in the book. At the time, I had no idea why I was doing it (fun? ACM practice?), but in hindsight it was incredibly important. I learn by doing, not by reading (at least doing cements what I read), so I make sure that I incorporate a lot of practice into my plans.
- viscanti 14y ago
  That's really the basis for the "Learn X the hard way" tutorials. There's a number of books that propose to teach a topic quickly/easily, with the shortcut being that you won't have to go through all the exercises. It turns out that's an ineffective teaching style for most computer/science topics. For many of these topics, you really can't short-cut that process of implementing solutions and learning from doing.
ten_fingers 14y ago
Just follow the usual path for education in relatively applied math. Here is a nutshell description: The standard high school level subjects are algebra, plane geometry, second year algebra, trigonometry, and solid geometry. The standard college level subjects are calculus, abstract algebra, linear algebra, advanced calculus, ordinary differential equations. Might also take elementary courses in probability and statistics. Standard graduate school topics are point set topology, measure theory, functional analysis. For more in applications, can do (1) optimization -- linear, network linear, linear integer, quadratic, non-linear, and dynamic programming -- and (2) probability, statistics, and stochastic processes based on measure theory. Here is an overview: The course in abstract algebra will make you familiar with a long list of topics that come up in computing including sets, Boolean algebra, logic, relations, mappings, integers, prime numbers, modular arithmetic, rational numbers, the fundamental theorem of arithmetic, the fundamental theorem of algebra, elementary number theory, and the Euclidean greatest common divisor algorithm. You will also see finite field theory which has long been important in algebraic coding theory. You may touch on the question of P versus NP. You may see some of the work of Goedel, etc. and model theory. You will get good with math based on definitions, theorems, and proofs and develop 'mathematical maturity', i.e., ability to read, understand, and do abstract mathematics. Your ability to describe logical material in writing will improve. Famous authors in abstract algebra include Birkhoff, MacLane, Herstein, Lang. More recent authors likely touch on algebraic geometry. Linear algebra, mostly about 'vectors' and linear transformations, is the core of an ocean of applications in multivariate statistics (regression, factor analysis, analysis of variance, discriminate analysis, and more and, thus, applications in ad targeting, machine learning, data mining, anomaly detection, recommendation engines, etc.), optimization, group representation theory as in molecular spectroscopy, Shannon's information theory, and more. E.g., linear algebra is the place to learn about solving systems of linear equations, e.g., by Gauss elimination, and, thus, the place to get started with matrix inversion and the simplex algorithm of linear programming (optimization) and network linear programming. A good first text in linear algebra is by Ben Noble. The crown jewel is by Paul Halmos, 'Finite Dimensional Vector Spaces', written when Halmos was an assistant to von Neumann; the book approaches linear algebra much like it was functional analysis and, thus, is a baby version of Hilbert space theory and, thus, sometimes used to teach Hilbert space theory to physicists for quantum mechanics. Richard Bellman wrote tons. See also Roger Horn's books. Evar Nering is a good first book (although his treatment of linear programming is not good); so is Hoffman and Kunze (apparently now available on the Internet in PDF for free). For numerical linear algebra, see Forsythe and Moler. The standard advanced calculus text to teach the theorems and proofs of calculus is Walter Rudin's 'Principles of Mathematical Analysis'. After that book, notation such as O( n ln(n) ) as in Knuth's TACP will be child's play. Note: At one time the Halmos 'Finite Dimensional Vector Spaces' and Rudin's 'Principles' were used in Harvard's Math 55 with a colorful description in http://www.american.com/archive/2008/march-april-magazine-contents/why-can2019t-a-woman-be-more-like-a-man/?searchterm=Sommers A course in trigonometry, the use of trigonometry in calculus, and Rudin's treatment of Fourier series will help in data compression (e.g., JPG), time-invariant linear systems, digital filtering, and Shannon's information theory. Also good coverage of the binomial theorem in high school or abstract algebra and an elementary course in probability will be a help in following Knuth's TACP. Measure theory will take a huge load off your back: Calculus, that is, Riemann integration theory as taught through Rudin's 'Principles', has some rough edges, understood back to at least 1900, and Lebesgue and Borel and measure theory (heavily due to Lebesgue) provide a clean solution. As from Kolmogorov, the solution also makes a clean foundation for probability, stochastic processes, and statistics. Likely the nicest first text in measure theory is Royden's 'Real Analysis'. Also good is the first, real, half of Rudin's 'Real and Complex Analysis' where he also gives nice introductions to both Hilbert and Banach spaces and a nice treatment of the Fourier transform. My favorite authors in probability based on measure theory are J. Neveu (long in Paris) and L. Breiman (long at Berkeley). Or just read from their teacher, M. Loeve (long at Berkeley but with a writing style that looks like some cross between English and French). This background will also let you do much more in both pure and applied math in many directions and for many applications. For learning math, it is "not a spectator sport". Most of the work is between your ears as you think about the material. A good teacher in abstract algebra is likely necessary to get you started. Also without at least occasional contact with a solid university department it is too easy to get off track. Still, nearly all the work is to be done alone in a quiet room, and there a book is fine. In principle videos could help, but so far I've never seen even one that I would recommend for any utility at all in learning math.
dkc 14y ago
How to Prove It. It's cheap (paperback), highly rated, and insightful.
phektus 14y ago
http://www.reddit.com/r/math/comments/eohrr/to_everyone_who_posts_about_learning_more_math/ http://www.reddit.com/r/math/comments/eohrr/to_everyone_who_...
- opminion 14y ago
  The fist item in that list is learn algebra, and rightly so. It then follows something similar to a university-level curriculum in Europe (continental, perhaps not UK), which might lead all the way to research maths, but perhaps it is not the most appropriate path for computer science because it entirely misses the mathematical concept of computability.
keva161 14y ago
Khan Academy
jasomill 14y ago
The freely available [1] is a nice supplement to some of the material in Concrete Mathematics, and this [2] is an approachable introduction to graph theory with side notes on both proof techniques and algorithms. More generally, if you're not already comfortable with linear algebra (a couple college semesters "or equivalent experience"), I'd recommend both [3] and [4] for two entirely different perspectives. For modern algebra more generally, I'm a huge fan of [5]. If I could only take one introductory mathematics book to a desert island, I'd cheat a bit and take [6], [7], and [8]. While never directly involved in CS, Courant was very interested in the practical applications of theoretical mathematics, see, e.g., [9] and, well, most everything else he wrote. I could go on and on, so I'll stop here. [1] http://www.math.upenn.edu/~wilf/DownldGF.html http://www.math.upenn.edu/~wilf/DownldGF.html [2] http://www.amazon.com/Graph-Theory-Graduate-Texts-Mathematics/dp/1846289696 http://www.amazon.com/Graph-Theory-Graduate-Texts-Mathematic... [3] http://www.amazon.com/Linear-Algebra-Applications-Gilbert-Strang/dp/0155510053 http://www.amazon.com/Linear-Algebra-Applications-Gilbert-St... [4] http://www.amazon.com/Finite-Dimensional-Vector-Spaces-P-R-Halmos/dp/0387900934 http://www.amazon.com/Finite-Dimensional-Vector-Spaces-P-R-H... [5] http://www.amazon.com/Algebra-Chelsea-Publishing-Saunders-Lane/dp/0821816462 http://www.amazon.com/Algebra-Chelsea-Publishing-Saunders-La... [6] http://www.amazon.com/Introduction-Calculus-Analysis-Classics-Mathematics/dp/354065058X http://www.amazon.com/Introduction-Calculus-Analysis-Classic... [7] http://www.amazon.com/Introduction-Calculus-Analysis-Classics-Mathematics/dp/3540665692 http://www.amazon.com/Introduction-Calculus-Analysis-Classic... [8] http://www.amazon.com/Introduction-Calculus-Analysis-Classics-Mathematics/dp/3540665706 http://www.amazon.com/Introduction-Calculus-Analysis-Classic... [9] http://www.ams.org/journals/bull/1943-49-01/S0002-9904-1943-07818-4/S0002-9904-1943-07818-4.pdf http://www.ams.org/journals/bull/1943-49-01/S0002-9904-1943-...
elssar 14y ago
Well you haven't mentioned how much math you're familiar with. But I'll assume that you know school level mathematics and are looking to learn more about the math mostly used in computer science. Get Concrete Mathematics( http://www.amazon.com/Concrete-Mathematics-Foundation-Computer-Science/dp/0201558025 http://www.amazon.com/Concrete-Mathematics-Foundation-Comput... ) by Graham, Knuth & Patashnik - It's based on the course taught at Stanford since 1970 & it's really really good. Another book I think you should get is Meta Math: The quest for Omega( http://www.amazon.com/Meta-Math-Quest-Gregory-Chaitin/dp/1400077974 http://www.amazon.com/Meta-Math-Quest-Gregory-Chaitin/dp/140... ) by Gregory Chaitin. It's not really a math course book, and is mostly the authors take on his favorite topics in math, but if you want to improve your thinking process then this book will definitely give you a different perspective.
akulbe 14y ago
So what if you really suck at algebra, and need to start over? Geometry always made much more sense to me, as I could see it, touch it, it was real!
- wingerlang 14y ago
  Although not algebra per se, but with Linear Algebra I did that. I only read the uber-basic Vector math and such but then I skipped the rest of the class. Then I started applying it into programming step by step which made it possible for me to look at every little calculation. Then I made a tiny physics simulation: http://www.youtube.com/watch?v=ud8NirjyLAA http://www.youtube.com/watch?v=ud8NirjyLAA
srkiranraj 14y ago
Try the "MATH FOR CS" course in MIT Opencourseware :)
- bob_hancock 14y ago
  http://courses.csail.mit.edu/6.042/spring12/mcs.pdf http://courses.csail.mit.edu/6.042/spring12/mcs.pdf This is an excellent place to start.
zedshaw 14y ago
Most math books suck. The one exception, and the first book that made me realize you can teach complex topics to regular folks, is Calculus Made Easy by Silvanus P. Thompson and Martin Gardner. It's been around since the late 1800's and it's what I finally learned calculus from. I haven't found many other books that explain calculus as well as that one. If you get that and then a book of exercises from Schaum's outlines, you'll be able to get pretty good at it. Also, the book is a really good read for a math book.
- chris_wot 14y ago
  Calculus: An Intuitive Approach by the late Morris Klein is pretty good too. I found it refreshingly clear!
deleted 14y ago
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tempire 14y ago
Math: http://khanacademy.org http://khanacademy.org Computer Science: http://aduni.org http://aduni.org Another vote for khanacademy. I did this myself, and worked out a plan to put every piece of information into long term memory. When you're motivated and not limited by century-old university bureaucracy, you can move very quickly, and run circles around people with a "real" degree, most of whom have forgotten everything but the general concepts. Udacity is also coming up as an excellent resource, but the important thing to realize is that every resource adds a different perspective on the matter that is extremely valuable. For CS-specific information, aduni.org is a glorious resource. It's from 2001, but the CS basics never change. I'd recommend getting a handle on all the math from khanacademy so that you understand everything they're referencing.
dionysiac 14y ago
Have a look at http://betterexplained.com/ http://betterexplained.com/ - Kalid's got some of the most intuitive explanations I've seen.
kmfrk 14y ago
I think Mathematics for Computer Science Eric Lehman and Tom Leighton (http://www.cs.princeton.edu/courses/archive/spr10/cos433/mathcs.pdf http://www.cs.princeton.edu/courses/archive/spr10/cos433/mat...) is a great resource. It has the best explanation of induction I have seen - discrete maths books are normally completely awful in this regard. Use something lke Khan Academy to fill in the blanks.
crasshopper 14y ago
I may have answered your question here: http://isomorphismes.tumblr.com/post/15431506533/i-never-did-mathematics-where-should-i-start http://isomorphismes.tumblr.com/post/15431506533/i-never-did.... The benefit of DIY maths is that you get to focus on what your interest is, whether that be algebra or geometry or stats/data analysis or discrete maths or foundations or whatever. (Contrary to some answers here, I don't think trig, measure theory, or planar geometry are necessary, unless you read something and it catches your fancy.) If your goal is just to do more maths, then just throw away anything that bores you and dive into anything that interests you. If you're interested in C.S. applications then I think discrete maths is the thing to focus on. Eg, "Concrete Mathematics" by Knuth. But you can go an even more direct route and if algorithms is your goal, there are some free courses available online that do just that. MIT OCW and AcademicEarth are excellent for in-depth (course-length) treatments of material by seasoned teachers. Wikipedia and math.StackExchange are good for broad overviews and quick Q's, respectively. For "proofs, and other practical applications" -- that's quite broad. OCW has a course on geophysics and a course on signal processing -- both are "practical applications" but I don't know if that's what you have in mind. Most of the OCW stuff -- whether it's chemistry, materials sci, DSP, whatever -- is aimed at engineers (other than the upper level pure maths), and therefore practical. As for proofs, you will see them in almost any maths textbook. Spivak makes some challenging ones and if you haven't taken calculus or linear algebra then Flanigan & Kazdan might be worth a skim. But if you're interested in CS applications then you can find those kinds of argumentation in an algorithms course. BTW, I do not recommend the Khan Academy. It gets a lot of press but I have not found it so great. (It is fine but so is just surfing Youtube -- a lot of people make videos besides SK.) Also, if you can still edit this question, you might give us some idea of where you're starting from and how much time you expect to spend, to elicit better answers. HTH. PS By "the hard way" do you mean doing exercises? Because pretty much all mathematics texts expect you to do it "the hard way" if that's what you mean.
- anatoly 14y ago
  It's bizarre that you start your list of recommended books on mathematics for "people who never got far with it at school" with Penrose's _The Road to Reality_. No one should follow that recommendation. The book is not really readable without a background of a Ph.D.-level knowledge in math.
econner 14y ago
Check out the forums at http://artofproblemsolving.com/ http://artofproblemsolving.com/
azsxdc 14y ago
I was in a similar position 3 years ago. I jumped into an open university maths degree at the second year, have done half of the final year and have moved onto a masters degree. All part time, for me this was the best way, as another commentator has said, it is easy to get lost among the vastness of the topic - I am still pulling together the threads of how all these things fit together. I would recommend the journey for the reasons you list, also it has proven useful in pursuing my interest in machine learning and general artificial intelligence. Take the plunge, you will probably not be disappointed.
impendia 14y ago
I recommend Susanna Epp's Discrete Mathematics with Applications, which I recently taught a university course out of quite successfully. It offers a gentle but rigorous introduction to proofs, logic, and combinatorics, with applications to CS. Useful exercises too. It is squarely targeted at beginners, if you know the stuff already then many other commenters have recommended more advanced books.
zxcvvcxz 14y ago
Question: is anyone working through Numerical Linear Algebra, or has done so in the past? http://www.amazon.com/Numerical-Linear-Algebra-Lloyd-Trefethen/dp/0898713617 http://www.amazon.com/Numerical-Linear-Algebra-Lloyd-Trefeth... I'm having a really tough time with the exercises even though I should have the prereqs... What can I say, we don't do much rigorous math in engineering. Anyways this book came highly recommended to me from a friend in quantitative finance.
elrzn 14y ago
Take a look a Precalculus Mathematics in a Nutshell for a quick review. You'll be surprised how much stuff you forgot already! Then you can move to a discrete math book like Discrete Mathematics with Applications or Mathematics for Computer Science (free pdf courses.csail.mit.edu/6.042/fall10/mcs-ftl.pdf) and any Calculus book of your liking. There are also great resources to learn Math using Haskell. I'm working through The Haskell Road to Logic, Maths and Programming and it's proving to be great http://www.haskell.org/haskellwiki/Books_and_tutorials/Mathematics http://www.haskell.org/haskellwiki/Books_and_tutorials/Mathe...
HelpWFractions 14y ago
Khanacademy.org is one of my personal favorite resources.
umeshkacha 14y ago
http://www.udemy.com/math-is-everywhere-applications-of-finite-math/ http://www.udemy.com/math-is-everywhere-applications-of-fini...

171 points by nsomaru 14y ago

1331 14y ago

_Concrete Mathematics_ [1] is a good book that is fun to work through. The book can be considered an introduction/prerequisite to Knuth's _The Art of Computer Programming_ series [2], which contains much mathematics. If you would like a book that is not so difficult, _Introduction to Algorithms_ [3] (aka "CLRS") is also highly recommended. [1] http://www.amazon.com/dp/0201558025/ http://www.amazon.com/dp/0201558025/ [2] http://www.amazon.com/dp/0321751043/ http://www.amazon.com/dp/0321751043/ [3] http://www.amazon.com/dp/0262033844/ http://www.amazon.com/dp/0262033844/

Arun2009 14y ago

In addition to Concrete Mathematics by Knuth, which contains a lot of Mathematical tools and techniques useful in areas of Computer Science, I'd recommend the following to learn how good problem solvers approach Mathematics: 1. The Art and Craft of Problem Solving by Paul Zeitz. 2. How to Solve It by George Polya. I'd also recommend learning any of the competition Mathematics topics like number theory, combinatorics or geometry as base material to improve your problem solving skills. I have also found the Schaum's series on Combinatorics and Graph Theory concise and useful.

verganileonardo 14y ago

I have a business background and use Project Euler to evolve my programming, math and problem solving skills. :) projecteuler.net

big_data 14y ago
+1 for Project Euler! It totally rocks!

madmanwalking 14y ago

The Khan academy (www.khanacademy.org) has plenty of videos and exercises, and if you're looking more at the CS side of things udacity.com runs a few online courses looking at algorithms, discrete maths and the like.

tonetheman 14y ago
This was going to be my suggestion. Excellent resource.

tel 14y ago

I'm going to second Arun2009's recommendation for How to Solve It by Polya. And then I'm going to recommend Haskell. I recently realized how much my understand of set theory improved and was solidified purely by programming Haskell a lot.

vitno 14y ago
I started messing around with haskell a few months ago. It completely inspired me to revamp my entire mathematical understanding... I've bought (and read!) like 4 math text books...
jasomill 14y ago
I'll third How to Solve It, and would also recommend Polya's Induction and Analogy in Mathematics.
motxilo 14y ago
Math + Haskell = http://homepages.cwi.nl/~jve/HR/ http://homepages.cwi.nl/~jve/HR/

nimeshneema 14y ago

The best text-book I can suggest for you is: "How to Think About Algorithms - Loop Invariants and Recursion" By Jeff Edmonds. It takes you through all the relevant mathematics required to learn about basics of CS based topics you are looking for (algorithms, proofs and other practical applications). It also acts as a good primer for learning basic Computer Algorithms. The USP of book is it's noob friendly and I think perfectly suited to your needs. Check it out here: http://www.amazon.com/Think-About-Algorithms-Jeff-Edmonds/dp/0521614104/ http://www.amazon.com/Think-About-Algorithms-Jeff-Edmonds/dp...

alex-g 14y ago

You might like "To Mock a Mockingbird" by Raymond Smullyan. It's a book of logic puzzles with a focus on combinatory logic (a model of computation related to the lambda calculus). Aside from the logic/math/CS nature of the content, the progression of the exercises should help with mathematical-style reasoning in general.

psykotic 14y ago
As fond as I am of the Mockingbird book and Smullyan's books in general, I'd say that most of its puzzles are pure exercises in abstract symbol manipulation and not terribly enlightening. If you have some prior exposure to the SKI combinators, one big aha-moment may come from the book's exploration of several alternative axiom systems that, while preceding the SK system chronologically, are often left out of modern CS-oriented treatments of combinatory logic. Also, while I think the exercises on symbol manipulation are not very enlightening on their own, the syntactic rewriting point of view is a useful complement if your view of combinatory logic has been mostly semantic. Proving completeness in the syntactic framework becomes a matter of showing that any combinatory rewriting rule can be expressed in terms of your base combinators. This in turn can be done if your combinators allow arbitrary permutations to be expressed in addition to duplication and dropping of symbols (cf. lengthening and shortening rules in GEB). For example, K is a dropping combinator, and rather than viewing S as a substitution operator on the semantic level (as is done when translating lambda calculus to SK calculus), it can be viewed as a hybrid permutation and duplication combinator at the syntactic level. If these dual roles are split into separate combinators, you get alternative axiom systems.

jcdreads 14y ago

The Project Euler problems might be right up your alley. They are little programming exercises that start easy and rapidly become harder. Along the way you figure out enough math to make the programming easier, or at least enough to make better algorithms. The forums are enormously helpful. http://projecteuler.net/ http://projecteuler.net/

ether 14y ago

I don't think you need to learn really hard core math to do CS. In my experience, the one most important "genre" of math was discrete mathematics. My textbook was Elements of Discrete Mathematics, and I think it was a really great intro: http://www.amazon.com/Elements-Discrete-Mathematics-Computer-Science/dp/007038133X/ http://www.amazon.com/Elements-Discrete-Mathematics-Computer...

drostie 14y ago

You may not come from the same culture that I came from, but the early significant milestones in my mathematical education were: 1. Reading the "aha!" books by Martin Gardner, as a child. 2. Reading Lockhart's "A Mathematician's Lament" [1] 3. Linear algebra, calculus, and complex analysis classes. I was taught these at Cornell; you might look for them via MIT's OpenCourseWare [2]. 4. A bunch of combinatorics, from Cornell's classes in probability. The most key thing I can tell you about mathematics is, always go back and read the definitions. Always. I have sometimes helped out math students on topics which were way out of my depth, simply because I picked up their textbooks, flipped several pages back, and said "hm, what does this word mean? what does that word mean?", building up connections all the while. You may also want to start in a completely different direction, since you're interested more in CS-based topics (I am interested more in physics-based topics) by starting with number theory and modular arithmetic, and of course MIT has a Comp Sci section [3] which also produces lectures on OpenCourseWare; you may wish to watch those lectures. [1] http://www.maa.org/devlin/LockhartsLament.pdf http://www.maa.org/devlin/LockhartsLament.pdf [2] http://ocw.mit.edu/courses/mathematics/ http://ocw.mit.edu/courses/mathematics/ [3] http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/ http://ocw.mit.edu/courses/electrical-engineering-and-comput...

dwc 14y ago
FYI, Lockhart has a new book coming[1] out with the kinds of things we should be doing, which is something I (and probably everyone who read & liked his Lament) have been pining for. I'm waiting like everyone else, so I can't explicitly recommend it. But I certainly anticipate a mental treat. 1. http://www.amazon.com/Measurement-Paul-Lockhart/dp/0674057554/ref=pd_rhf_ee_p_t_1 http://www.amazon.com/Measurement-Paul-Lockhart/dp/067405755...

Someone 14y ago

For getting to know what is a proof in Mathematics, I recommend Proofs and Refutations by Imre Lakatos. It will not teach you many theorems (in fact, 80% of it only proofs one, http://en.wikipedia.org/wiki/Euler_characteristic http://en.wikipedia.org/wiki/Euler_characteristic again and again and again), and it will almost certainly not be applicable to writing programs but it will give insight into how mathematicians in the last century have made mathematics much more rigorous (and abstract).

hendzen 14y ago

Start with these topics. Work your way through them, and make sure you understand everything you have read so far before progressing. Do the exercises, or it will be a waste of time. 0) Algebra, Trigonometry, Calculus Make sure you have a decent grasp over high school level math topics. You might not need to use these topics frequently (though trig comes up a surprising amount), but they are necessary to establish a base level of mathematical maturity. 1) Linear Algebra Obviously very important if you want to do any 3d work, but it comes up in later topics like graph theory. Suggested book: Linear Algebra and Its Application - Strang 2) Discrete Math You need an understanding of proofs and logic before you can get to the real algorithms material. Counting and probability is also very important. Suggested book: Discrete Mathematics and Its Applications - Kenneth Rosen This looks like a good free option, but there are no exercises: http://courses.csail.mit.edu/6.042/fall10/mcs-ftl.pdf http://courses.csail.mit.edu/6.042/fall10/mcs-ftl.pdf 3) Algorithms Enough said. Suggested book: CLRS, but here's a free option: http://www.cs.berkeley.edu/~vazirani/algorithms.html http://www.cs.berkeley.edu/~vazirani/algorithms.html This should take you some time as each of these topics usually corresponds to a college class. When you finish with those, start reading about CS Theory, Combinatorics, or pick up a graduate Algorithms text.

ekm2 14y ago
Another Linear Algebra book that also has a good introduction to proofs is Sheldon Axler's Linear Algebra Done Right.
sshrin 14y ago
You say 'trig comes up a surprising amount'. It would be great if you could give a few examples of these instances. I have always thought of trigonometry as one of the areas I really enjoyed in school but something I have never used ever since.
- Dn_Ab 14y ago
  I think the surprising commonality is conditional. So you won't see it unless you are focusing on physics or signal processing.

gmoes 14y ago

Here's a post I did on free math resources, including a list of resources on various topics: http://www.elegantcoding.com/2011/06/free-math-resources-you-can-use.html http://www.elegantcoding.com/2011/06/free-math-resources-you... There's some really great stuff listed there, I use it all the time.

pdelgallego 14y ago

I am in the ship, I have signed in the statistics 101 class [1] at udacity. Its very basic, but it's a good starting point. Coursera have also several math related courses [2] (calculus, autmata, logic, ...). I have also signed in a class in nearest community college in my area. [1] http://www.udacity.com/overview/Course/st101/CourseRev/1 http://www.udacity.com/overview/Course/st101/CourseRev/1 [2] https://www.coursera.org/category/math https://www.coursera.org/category/math Happy logarithmic!

radicalbyte 14y ago
Coursera's Automata course is really good, but it's hard work. It's a six week course, and you'll spend around 10hrs per week if you do everything (and I'd recommend doing everything). Udacity's introduction to statistics, so far, has been really easy. From what I've seen so far, Udacity are aiming to be more inclusive "community college" level and Coursera are aiming for a true university level education.

sidman 14y ago

As mentioned probabilities is something you want to get into very early. Its easy to understand at the start and its practical uses in many different computing areas along with many other things in life is priceless. Good place to start would be to go through khan academy's probability videos.

mdkess 14y ago

When I was in early university, I went through Introduction to Algorithms and chapter by chapter, I implemented nearly every algorithm presented in the book. At the time, I had no idea why I was doing it (fun? ACM practice?), but in hindsight it was incredibly important. I learn by doing, not by reading (at least doing cements what I read), so I make sure that I incorporate a lot of practice into my plans.

viscanti 14y ago
That's really the basis for the "Learn X the hard way" tutorials. There's a number of books that propose to teach a topic quickly/easily, with the shortcut being that you won't have to go through all the exercises. It turns out that's an ineffective teaching style for most computer/science topics. For many of these topics, you really can't short-cut that process of implementing solutions and learning from doing.

ten_fingers 14y ago

Just follow the usual path for education in relatively applied math. Here is a nutshell description: The standard high school level subjects are algebra, plane geometry, second year algebra, trigonometry, and solid geometry. The standard college level subjects are calculus, abstract algebra, linear algebra, advanced calculus, ordinary differential equations. Might also take elementary courses in probability and statistics. Standard graduate school topics are point set topology, measure theory, functional analysis. For more in applications, can do (1) optimization -- linear, network linear, linear integer, quadratic, non-linear, and dynamic programming -- and (2) probability, statistics, and stochastic processes based on measure theory. Here is an overview: The course in abstract algebra will make you familiar with a long list of topics that come up in computing including sets, Boolean algebra, logic, relations, mappings, integers, prime numbers, modular arithmetic, rational numbers, the fundamental theorem of arithmetic, the fundamental theorem of algebra, elementary number theory, and the Euclidean greatest common divisor algorithm. You will also see finite field theory which has long been important in algebraic coding theory. You may touch on the question of P versus NP. You may see some of the work of Goedel, etc. and model theory. You will get good with math based on definitions, theorems, and proofs and develop 'mathematical maturity', i.e., ability to read, understand, and do abstract mathematics. Your ability to describe logical material in writing will improve. Famous authors in abstract algebra include Birkhoff, MacLane, Herstein, Lang. More recent authors likely touch on algebraic geometry. Linear algebra, mostly about 'vectors' and linear transformations, is the core of an ocean of applications in multivariate statistics (regression, factor analysis, analysis of variance, discriminate analysis, and more and, thus, applications in ad targeting, machine learning, data mining, anomaly detection, recommendation engines, etc.), optimization, group representation theory as in molecular spectroscopy, Shannon's information theory, and more. E.g., linear algebra is the place to learn about solving systems of linear equations, e.g., by Gauss elimination, and, thus, the place to get started with matrix inversion and the simplex algorithm of linear programming (optimization) and network linear programming. A good first text in linear algebra is by Ben Noble. The crown jewel is by Paul Halmos, 'Finite Dimensional Vector Spaces', written when Halmos was an assistant to von Neumann; the book approaches linear algebra much like it was functional analysis and, thus, is a baby version of Hilbert space theory and, thus, sometimes used to teach Hilbert space theory to physicists for quantum mechanics. Richard Bellman wrote tons. See also Roger Horn's books. Evar Nering is a good first book (although his treatment of linear programming is not good); so is Hoffman and Kunze (apparently now available on the Internet in PDF for free). For numerical linear algebra, see Forsythe and Moler. The standard advanced calculus text to teach the theorems and proofs of calculus is Walter Rudin's 'Principles of Mathematical Analysis'. After that book, notation such as O( n ln(n) ) as in Knuth's TACP will be child's play. Note: At one time the Halmos 'Finite Dimensional Vector Spaces' and Rudin's 'Principles' were used in Harvard's Math 55 with a colorful description in http://www.american.com/archive/2008/march-april-magazine-contents/why-can2019t-a-woman-be-more-like-a-man/?searchterm=Sommers A course in trigonometry, the use of trigonometry in calculus, and Rudin's treatment of Fourier series will help in data compression (e.g., JPG), time-invariant linear systems, digital filtering, and Shannon's information theory. Also good coverage of the binomial theorem in high school or abstract algebra and an elementary course in probability will be a help in following Knuth's TACP. Measure theory will take a huge load off your back: Calculus, that is, Riemann integration theory as taught through Rudin's 'Principles', has some rough edges, understood back to at least 1900, and Lebesgue and Borel and measure theory (heavily due to Lebesgue) provide a clean solution. As from Kolmogorov, the solution also makes a clean foundation for probability, stochastic processes, and statistics. Likely the nicest first text in measure theory is Royden's 'Real Analysis'. Also good is the first, real, half of Rudin's 'Real and Complex Analysis' where he also gives nice introductions to both Hilbert and Banach spaces and a nice treatment of the Fourier transform. My favorite authors in probability based on measure theory are J. Neveu (long in Paris) and L. Breiman (long at Berkeley). Or just read from their teacher, M. Loeve (long at Berkeley but with a writing style that looks like some cross between English and French). This background will also let you do much more in both pure and applied math in many directions and for many applications. For learning math, it is "not a spectator sport". Most of the work is between your ears as you think about the material. A good teacher in abstract algebra is likely necessary to get you started. Also without at least occasional contact with a solid university department it is too easy to get off track. Still, nearly all the work is to be done alone in a quiet room, and there a book is fine. In principle videos could help, but so far I've never seen even one that I would recommend for any utility at all in learning math.

dkc 14y ago

How to Prove It. It's cheap (paperback), highly rated, and insightful.

phektus 14y ago

http://www.reddit.com/r/math/comments/eohrr/to_everyone_who_posts_about_learning_more_math/ http://www.reddit.com/r/math/comments/eohrr/to_everyone_who_...

opminion 14y ago
The fist item in that list is learn algebra, and rightly so. It then follows something similar to a university-level curriculum in Europe (continental, perhaps not UK), which might lead all the way to research maths, but perhaps it is not the most appropriate path for computer science because it entirely misses the mathematical concept of computability.

keva161 14y ago

Khan Academy

jasomill 14y ago

The freely available [1] is a nice supplement to some of the material in Concrete Mathematics, and this [2] is an approachable introduction to graph theory with side notes on both proof techniques and algorithms. More generally, if you're not already comfortable with linear algebra (a couple college semesters "or equivalent experience"), I'd recommend both [3] and [4] for two entirely different perspectives. For modern algebra more generally, I'm a huge fan of [5]. If I could only take one introductory mathematics book to a desert island, I'd cheat a bit and take [6], [7], and [8]. While never directly involved in CS, Courant was very interested in the practical applications of theoretical mathematics, see, e.g., [9] and, well, most everything else he wrote. I could go on and on, so I'll stop here. [1] http://www.math.upenn.edu/~wilf/DownldGF.html http://www.math.upenn.edu/~wilf/DownldGF.html [2] http://www.amazon.com/Graph-Theory-Graduate-Texts-Mathematics/dp/1846289696 http://www.amazon.com/Graph-Theory-Graduate-Texts-Mathematic... [3] http://www.amazon.com/Linear-Algebra-Applications-Gilbert-Strang/dp/0155510053 http://www.amazon.com/Linear-Algebra-Applications-Gilbert-St... [4] http://www.amazon.com/Finite-Dimensional-Vector-Spaces-P-R-Halmos/dp/0387900934 http://www.amazon.com/Finite-Dimensional-Vector-Spaces-P-R-H... [5] http://www.amazon.com/Algebra-Chelsea-Publishing-Saunders-Lane/dp/0821816462 http://www.amazon.com/Algebra-Chelsea-Publishing-Saunders-La... [6] http://www.amazon.com/Introduction-Calculus-Analysis-Classics-Mathematics/dp/354065058X http://www.amazon.com/Introduction-Calculus-Analysis-Classic... [7] http://www.amazon.com/Introduction-Calculus-Analysis-Classics-Mathematics/dp/3540665692 http://www.amazon.com/Introduction-Calculus-Analysis-Classic... [8] http://www.amazon.com/Introduction-Calculus-Analysis-Classics-Mathematics/dp/3540665706 http://www.amazon.com/Introduction-Calculus-Analysis-Classic... [9] http://www.ams.org/journals/bull/1943-49-01/S0002-9904-1943-07818-4/S0002-9904-1943-07818-4.pdf http://www.ams.org/journals/bull/1943-49-01/S0002-9904-1943-...

elssar 14y ago

Well you haven't mentioned how much math you're familiar with. But I'll assume that you know school level mathematics and are looking to learn more about the math mostly used in computer science. Get Concrete Mathematics( http://www.amazon.com/Concrete-Mathematics-Foundation-Computer-Science/dp/0201558025 http://www.amazon.com/Concrete-Mathematics-Foundation-Comput... ) by Graham, Knuth & Patashnik - It's based on the course taught at Stanford since 1970 & it's really really good. Another book I think you should get is Meta Math: The quest for Omega( http://www.amazon.com/Meta-Math-Quest-Gregory-Chaitin/dp/1400077974 http://www.amazon.com/Meta-Math-Quest-Gregory-Chaitin/dp/140... ) by Gregory Chaitin. It's not really a math course book, and is mostly the authors take on his favorite topics in math, but if you want to improve your thinking process then this book will definitely give you a different perspective.

akulbe 14y ago

So what if you really suck at algebra, and need to start over? Geometry always made much more sense to me, as I could see it, touch it, it was real!

wingerlang 14y ago
Although not algebra per se, but with Linear Algebra I did that. I only read the uber-basic Vector math and such but then I skipped the rest of the class. Then I started applying it into programming step by step which made it possible for me to look at every little calculation. Then I made a tiny physics simulation: http://www.youtube.com/watch?v=ud8NirjyLAA http://www.youtube.com/watch?v=ud8NirjyLAA

srkiranraj 14y ago

Try the "MATH FOR CS" course in MIT Opencourseware :)

bob_hancock 14y ago
http://courses.csail.mit.edu/6.042/spring12/mcs.pdf http://courses.csail.mit.edu/6.042/spring12/mcs.pdf This is an excellent place to start.

zedshaw 14y ago

Most math books suck. The one exception, and the first book that made me realize you can teach complex topics to regular folks, is Calculus Made Easy by Silvanus P. Thompson and Martin Gardner. It's been around since the late 1800's and it's what I finally learned calculus from. I haven't found many other books that explain calculus as well as that one. If you get that and then a book of exercises from Schaum's outlines, you'll be able to get pretty good at it. Also, the book is a really good read for a math book.

chris_wot 14y ago
Calculus: An Intuitive Approach by the late Morris Klein is pretty good too. I found it refreshingly clear!

deleted 14y ago

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tempire 14y ago

Math: http://khanacademy.org http://khanacademy.org Computer Science: http://aduni.org http://aduni.org Another vote for khanacademy. I did this myself, and worked out a plan to put every piece of information into long term memory. When you're motivated and not limited by century-old university bureaucracy, you can move very quickly, and run circles around people with a "real" degree, most of whom have forgotten everything but the general concepts. Udacity is also coming up as an excellent resource, but the important thing to realize is that every resource adds a different perspective on the matter that is extremely valuable. For CS-specific information, aduni.org is a glorious resource. It's from 2001, but the CS basics never change. I'd recommend getting a handle on all the math from khanacademy so that you understand everything they're referencing.

dionysiac 14y ago

Have a look at http://betterexplained.com/ http://betterexplained.com/ - Kalid's got some of the most intuitive explanations I've seen.

kmfrk 14y ago

I think Mathematics for Computer Science Eric Lehman and Tom Leighton (http://www.cs.princeton.edu/courses/archive/spr10/cos433/mathcs.pdf http://www.cs.princeton.edu/courses/archive/spr10/cos433/mat...) is a great resource. It has the best explanation of induction I have seen - discrete maths books are normally completely awful in this regard. Use something lke Khan Academy to fill in the blanks.

crasshopper 14y ago

I may have answered your question here: http://isomorphismes.tumblr.com/post/15431506533/i-never-did-mathematics-where-should-i-start http://isomorphismes.tumblr.com/post/15431506533/i-never-did.... The benefit of DIY maths is that you get to focus on what your interest is, whether that be algebra or geometry or stats/data analysis or discrete maths or foundations or whatever. (Contrary to some answers here, I don't think trig, measure theory, or planar geometry are necessary, unless you read something and it catches your fancy.) If your goal is just to do more maths, then just throw away anything that bores you and dive into anything that interests you. If you're interested in C.S. applications then I think discrete maths is the thing to focus on. Eg, "Concrete Mathematics" by Knuth. But you can go an even more direct route and if algorithms is your goal, there are some free courses available online that do just that. MIT OCW and AcademicEarth are excellent for in-depth (course-length) treatments of material by seasoned teachers. Wikipedia and math.StackExchange are good for broad overviews and quick Q's, respectively. For "proofs, and other practical applications" -- that's quite broad. OCW has a course on geophysics and a course on signal processing -- both are "practical applications" but I don't know if that's what you have in mind. Most of the OCW stuff -- whether it's chemistry, materials sci, DSP, whatever -- is aimed at engineers (other than the upper level pure maths), and therefore practical. As for proofs, you will see them in almost any maths textbook. Spivak makes some challenging ones and if you haven't taken calculus or linear algebra then Flanigan & Kazdan might be worth a skim. But if you're interested in CS applications then you can find those kinds of argumentation in an algorithms course. BTW, I do not recommend the Khan Academy. It gets a lot of press but I have not found it so great. (It is fine but so is just surfing Youtube -- a lot of people make videos besides SK.) Also, if you can still edit this question, you might give us some idea of where you're starting from and how much time you expect to spend, to elicit better answers. HTH. PS By "the hard way" do you mean doing exercises? Because pretty much all mathematics texts expect you to do it "the hard way" if that's what you mean.

anatoly 14y ago
It's bizarre that you start your list of recommended books on mathematics for "people who never got far with it at school" with Penrose's _The Road to Reality_. No one should follow that recommendation. The book is not really readable without a background of a Ph.D.-level knowledge in math.

econner 14y ago

Check out the forums at http://artofproblemsolving.com/ http://artofproblemsolving.com/

azsxdc 14y ago

I was in a similar position 3 years ago. I jumped into an open university maths degree at the second year, have done half of the final year and have moved onto a masters degree. All part time, for me this was the best way, as another commentator has said, it is easy to get lost among the vastness of the topic - I am still pulling together the threads of how all these things fit together. I would recommend the journey for the reasons you list, also it has proven useful in pursuing my interest in machine learning and general artificial intelligence. Take the plunge, you will probably not be disappointed.

impendia 14y ago

I recommend Susanna Epp's Discrete Mathematics with Applications, which I recently taught a university course out of quite successfully. It offers a gentle but rigorous introduction to proofs, logic, and combinatorics, with applications to CS. Useful exercises too. It is squarely targeted at beginners, if you know the stuff already then many other commenters have recommended more advanced books.

zxcvvcxz 14y ago

Question: is anyone working through Numerical Linear Algebra, or has done so in the past? http://www.amazon.com/Numerical-Linear-Algebra-Lloyd-Trefethen/dp/0898713617 http://www.amazon.com/Numerical-Linear-Algebra-Lloyd-Trefeth... I'm having a really tough time with the exercises even though I should have the prereqs... What can I say, we don't do much rigorous math in engineering. Anyways this book came highly recommended to me from a friend in quantitative finance.

elrzn 14y ago

Take a look a Precalculus Mathematics in a Nutshell for a quick review. You'll be surprised how much stuff you forgot already! Then you can move to a discrete math book like Discrete Mathematics with Applications or Mathematics for Computer Science (free pdf courses.csail.mit.edu/6.042/fall10/mcs-ftl.pdf) and any Calculus book of your liking. There are also great resources to learn Math using Haskell. I'm working through The Haskell Road to Logic, Maths and Programming and it's proving to be great http://www.haskell.org/haskellwiki/Books_and_tutorials/Mathematics http://www.haskell.org/haskellwiki/Books_and_tutorials/Mathe...

HelpWFractions 14y ago

Khanacademy.org is one of my personal favorite resources.

umeshkacha 14y ago

http://www.udemy.com/math-is-everywhere-applications-of-finite-math/ http://www.udemy.com/math-is-everywhere-applications-of-fini...