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I have studied mathematics, but that was long time ago. I have been a programmer for 8 years but when I started to study concepts in AI and data mining I find it very difficult to understand the theory.
Now I have wasted 2-3 years and I have got nothing. I need to first understand the math concepts required to learn AI and data mining.
I don't know where to start. Which books and tutorials do you recommend I should start with from the AI point of view.
How should I go about obtaining the fundamental requirements to use AI and Data Mining concepts.
EDIT:
I got this list from internet
Matrix algebra: most machine learning models are represented as matrices and vectors. Concepts like eigenvectors and singular value decomposition appear all over the place.
Bayesian statistics: probability, Bayes' rule, common distributions (e.g., beta, Dirichlet, Gaussian), etc.
Multivariable calculus: most learning techniques use gradients and Hessians at their core to fit parameters. (If you want to get fancier, study numerical optimization.)
Information theory: entropy, KL divergence, etc. Just the basics here.
In limited cases, higher-level math can be useful. E.g., to understand manifold learning, you'll want to know some basic notions from geometry and topology. Occasionally abstract algebra is used (e.g., see "expectation semirings" for learning on hyper-graphs). I would learn these as-needed, but if you have a chance to learn them early it can't hurt.
Can anyone recommend some books on those
My resource for studying math : http://www.khanacademy.org/
You will be able to find A LOT on all math fields.
I agree with #Lostdreamer that KhanAcademy.org has great material for learning various math concepts.
For an excellent introductory online course on Machine Learning I highly recommend the Machine Learning course being offered on Coursera.org. It is taught by Stanford Professor Andrew Ng You can watch the videos as many times as you need to understand the concepts.
The exercises and programming assignments help drive home the concepts.
I recommend that you register for it the next time it is offered. Here's a link to the course registration page.
Here's a link to a preview of the material in the course.
The course contains a basic review of linear algebra including basic matrix concepts that help me review this material.
I highly recoment #HeatfanJohn 's course, I've already made it, without any knoledge of AI and it turned out pretty good, the teacher is amazing and the course is extremely clear, try it!
In addition I made this other AI course in the same time as the other. This one is much more general, you will learn a bit about everything in AI and there are not any previous knowledge you should have. If you are not used to do math, this one is easier than the ML one (in ML you need to make exercises in matlab, that are sometimes a little bit tricky) but I found it more interesting for a general overview.
I highly recommend you to do both
https://www.ai-class.com/
One you become addicted to AI (you will for sure if you make this two courses!) I reccomend
Udacity, an amazing computer science free online "university". The best teacher in the world teaching you awesome things for free. If this is not awesome enough I'll tell you that the AI class teachers made this web page. One is Google's research director (Peter Norvig) and other one is the guy that made the first autonomous driving car (Sebastian Thrun). Awesome people
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I would like to know in which order I should learn different areas of maths so I can have a robust overview of all the theory in case I need something for a computer programming problem.
So I've created this mind map
I do not intend to know all those small details about how to do a certain thing (e.g. "gauss-jordan reduction"), I would rather look over an example, but then do it with math software like sage-maths or mathematica.
I would like to know, for instance, how to get to a taylor series, given the analytical function (I know it already, I am merely illustrating the kind of knowledge depth I expect).
So all I all, I want to be able to read academic articles about maths which have applicability in computer science / programming, and actually understand something from those articles, so I can use that knowledge in solving actual programming problems.
The open question is:
(a) In what order do you suggest to learn about these areas, on what areas should I insiste more?
(b) Do you see any missing areas in the mind map?
I was recommended this book in a data structures and Algorithms class a few years ago. It covers a lot of relevant areas ( probability, graphs, counting, relations, etc.) and it's free! :-)
If you want to be serious, you should get a graduate level course on computer science in a university. There is no replacement for this.
You must know basic set theory, big O, basic data structures, basic real analysis. I suggest looking at Cormen et al. 'Introduction to Algorithms', and/or Manber's 'Introduction to Algorithms: A Creative Approach'. For number theory, check Victor Shoup's book - contains far too much, but it is readable at any level.
I suggest not bothering at all with topics such as: complex analysis, functional analysis, projective/inversive geometry, control theory, mathematical physics, until you know you need them.
Instead of making your graph bigger and bigger, make it as small as possible.
There is a good book, that I think would help you to get more out of computer science research papers and dissertations. It's called "Concrete Mathematics: A foundation for Computer Science", and it's available on Amazon:
http://www.amazon.com/Concrete-Mathematics-Foundation-Computer-Science/dp/0201558025/ref=sr_1_1?s=books&ie=UTF8&qid=1341081763&sr=1-1&keywords=math+computer+science
I think this would help because it will all be relevant, and its consolidated which will help expedite the learning process.
Even if you don't have any money, just Google it and take a look at the index to get an idea of what areas you might want to learn.
And here's one more interesting book.
This is almost impossible to answer as many programming tasks require no mathematical knowledge (other than counting and basic logic) at all. If you have specific interests in an area (such as numerical linear algebra or statistics for example) then start there. Still what I would suggest is getting a good grasp of how finite precision arithmetic works. Perhaps read an introductory text on numerical analysis as this will give a good understanding of what numerical stability is. A good book on analysis of algorithms (with regards to speed and efficiency) would do no harm, and if you plan on doing any kind of mathematical programming, getting a sound knowledge of linear algebra is a very good idea. Almost everything in applied mathematics reduces to ultimately solving a linear system of equations (or doing so iteratively).
There is no right answer to be honest.
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I haven't taken any math classes above basic college calculus. However, in the course of my programming work, I've picked up a lot of math and comp sci from blogs and reading, and I genuinely believe I have a decent mathematical mind. I enjoy and have success doing Project Euler, for example.
I want to dive in and really start learning some cool math, particularly discrete mathematics, set theory, graph theory, number theory, combinatorics, category theory, lambda calculus, etc.
My impression so far is that I'm well equipped to take these on at a conceptual level, but I'm having a really hard time with the mathematical language and symbols. I just don't "speak the language" and though I'm trying to learn it, I'm the going is extremely slow. It can take me hours to work through even one formula or terminology heavy paragraph. And yeah, I can look up terms and definitions, but it's a terribly onerous process that very much obscures the theoretical simplicity of what I'm trying to learn.
I'm really afraid I'm going to have to back up to where I left off, get a mid-level math textbook, and invest some serious time in exercises to train myself in that way of thought. This sounds amazingly boring, though, so I wondered if anyone else has any ideas or experience with this.
If you don't want to attend a class, you still need to get what the class would have given you: time in the material and lots of practice.
So, grab that text book and start doing the practice problems. There really isn't any other way (unless you've figured out how osmosis can actually happen...).
There is no knowledge that can only be gained in a classroom.
Check out the MIT Courseware for Mathematics
Also their YouTube site
Project Euler is also a great way to think about math as it relates to programming
Take a class at your local community college. If you're like me you'd need the structure. There's something to be said for the pressure of being graded. I mean there's so much to learn that going solo is really impractical if you want to have more than just a passing nod-your-head-mm-hmm sort of understanding.
Sounds like you're in the same position I am. What I'm finding out about math education is that most of it is taught incorrectly. Whether a cause or result of this, I also find most math texts are written incorrectly. Exceptions are rare, but notable. For instance, anything written by Donald Knuth is a step in the right direction.
Here are a couple of articles that state the problem quite clearly:
A Gentle Introduction To Learning
Calculus
Developing Your Intuition For
Math
And here's an article on a simple study technique that aims at retaining knowledge:
Teaching linear algebra
Consider auditing classes in discrete mathematics and proofs at a local university. The discrete math class will teach you some really useful stuff (graph theory, combinatorics, etc.), and the proofs class will teach you more about the mathematical style of thinking and writing.
I'd agree with #John Kugelman, classes are the way to go to get it done properly but I'd add that if you don't want to take classes, the internet has many resources to help you, including recorded lectures which I find can be more approachable than books and papers.
I'd recommend checking out MIT Open Courseware. There's a Maths for Computer Science module there, and I'm enjoying working through Gilbert Strang's Linear Algebra course of video lectures.
Youtube and videolectures.com are also good resources for video lectures.
Finally, there's a free Maths for CS book at bookboon.
To this list I would now add The Haskel Road to Logic, Maths, and Programming, and Conceptual Mathematics: A First Introduction to Categories.
--- Nov 16 '09 answer for posterity--
Two books. Diestel's Graph Theory, and Knuth's Concrete Mathematics. Once you get the hang of those try CAGES.
Find a good mentor who is an expert in the field who is willing to spend time with you on a regular basis.
There is a sort of trick to learning dense material, like math and mathematical CS. Learning unfamiliar abstract stuff is hard, and the most effective way to do it is to familiarize yourself with it in stages. First, you need to skim it: don't worry if you don't understand everything in the first pass. Then take a break; after you have rested, go through it again in more depth. Lather, rinse, repeat; meditate, and eventually you may become enlightened.
I'm not sure exactly where I'd start, to become familiar with the language of mathematics; I just ended up reading through lots of papers until I got better at it. You might look for introductory textbooks on formal mathematical logic, since a lot of math (especially in language theory) is based off of that; if you learn to hack the formal stuff a bit, the everyday notation might look a bit easier.
You should probably look through books on topics you're personally interested in; the inherent interest should help get you over the hump. Also, make sure you find texts that are actually introductory; I have become wary of slim, undecorated hardbacks labeled Elementary Foobar Theory, which tend to be elementary only to postdocs with a PhD in Foobar.
A word of warning: do not start out with category theory -- it is the most boring math I have ever encountered! Due to its relevance to language design and type theory, I would like to know more about it, but so far I have not been able to deal...
For a nice, scattershot intro to bits of many kinds of CS-ish math, I recommend Godel, Escher, Bach by Hofstadter (if you haven't read it already, of course). It's not a formal math book, though, so it won't help you with the familiarity problem, but it is quite inspirational.
Mathematical notation is is akin to several computer languages:
concise
exacting
based on many idioms
a fair amount of local variations and conventions
As with a computer language, you don't need to "wash the whole elephant at once": take it one part a at time.
A tentative plan for you could be
identify areas of mathematics that are interesting or important to you. (seems you already have a bit of a sense for that, CS has helped you develop quite a culture for it.)
take (or merely audit) a few formal classes in this area. I agree with several answers in this post, an in-person course, at local college is preferable, but, maybe at first, or to be sure to get the most of a particular class, first self-teaching yourself in this area with MIT OCW, similar online resources and associated books is ok/fine.
if an area of math introduces too high of a pre-requisite in terms of fluency with notation or with some underlying concept or (most often mechanical computation and transformation techniques). No problem! Just backtrack a bit, learn these foundations (and just these foundations!) and move forward again.
Find a "guru", someone that has a broad mathematical culture and exposure, not necessarily a mathematician, physics folks are good too, indeed they can often articulate math in a more practical fashion. Use this guru to guide you, as he/she can show you how the big pieces fit together.
Note: There is little gain to be had of learning mathematical notation for its own sake. Rather it should be learned in context, just like say a C# idiom is better memorized when used and when associated with a specific task, rather than learned in vacuo. A related SO posting however provides several resources to decipher and learn mathematical notation
Project Euler takes problems out of context and drops them in for people to solve them. Project Euler cannot teach you anything effectively. I think you should forget about it, if it is popular it does not mean anything. You cannot study Mathematics through Project Euler as it contains only bits and pieces(and some pretty high level pieces) that you're supposed to know in order to solve the problems. Learning mathematics means to consider a subject and a read a book about it and solving exercices or reading solutions, that's how you learn math. If it so happens that through your reading you find something that is close to some project euler thing, your luck , but otherwise Project euler is a complete waste of time. I think the time is much better invested choosing a particular branch of mathematics and studying that. Let me explain why: I solved 3 pretty advanced Projec Euler problems and they were all making appeal to knowledge from Number theory which I happened to have because i studies some part of it. I do not think Iearned anything from Project Euler, it just happened that I already knew some number theory and solved the problems.
For example, if you find out you like number theory, take H. Davenport -> Hardy & Wright -> Kenneth & Rosen's , study those.
If you like Graph Theory take Reinhard Diestel's book which is freely available and study that(or check books.google.com and find whichever is more appropriate to your taste) but don't spread your attention in 999999 directions just because Project Euler has problems ranging from dynamic programming to advanced geometry or to advanced number theory, that is clearly the wrong way to go and it will not bring you closer to your goal.
This sounds amazingly boring
Well ... Mathematics is not boring when you find some problem that you are attached to, which you like and you'd like to find the solution to, and when you have the sufficient time to reflect on it while not behind a computer screen. Mathematics is done with pen and paper mostly(yes you can use computers .. but that's not really the point).
So, if you find a real-world problem, or some programming problem that would benefit from
you knowing some advanced maths, and you know what maths you have to study , it can be motivating to learn in that way.
If you feel you are not motivated it is hard to study properly.
There is also the question of what you actually mean when you say learn. Does the learning process stop after you solved the problems at the end of the chapter of a book ? Well you decide. You can consider you have finished learning that subject, or you can consider you have not finished and read more about it. There are entire books on just one equation and variations of it.
The amount of programming-related math that you can learn without formal training is limited, but it's more than enough. But maybe you can self-teach yourself.
It all boils down to your resources and motivation.
To know mathematics you have to do mathematics not programming(project euler).
For beginning to learn category theory I recommend David Spivak's Category Theory for the Sciences (AKA Category Theory for Scientists) because its relatively comprehensible due to many examples that enable understanding by analogy and which quickly builds a foundation for understanding more abstract concepts.
It requires the ability to reason logically and an intuitive notion of what is a set. It proceeds from sets and functions through basic category theory to adjoint functors, categories of functors, sheaves, monads and an introduction to operads. Two main threads throughout are modeling databases in terms of categories and describing categories with annotated diagrams called ologs. The bibliography provides references to more advanced and specialized topics including recent papers by Dr. Spivak.
An expected outcome from reading this book is the capability of understanding category theory texts and papers written for mathematicians such as Mac Lane's Category Theory for the Working Mathematician.
In PDF format it is available from http://math.mit.edu/~dspivak/teaching/sp13/ (the dynamic version is recommended since its the most recent). The open access HTML version is available from https://mitpress.mit.edu/books/category-theory-sciences (which is recommended since it includes additional content including answers to some exercises).
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It has been discussed on this site before about the relationship between math and programming, and whether one is a subset of the other, etc.
In my recent study of programming, I've found myself more and more wishing I was better at math. You all know the scenario when programming books start to generalize something in a math way ("Therefore, we may say that for all <some single letter>, <lots of letters>"). My eyes glaze over in such situations. I know that that is mostly due to me being stupid, but it seems that if I could just improve my higher math skills, maybe I could get more out of such things.
Major question: Is math indeed something one can "get better at," or is your brain kinda either wired for it or not?
Important follow-up question: If the answer to the above is yes, then what are some ways to go about it?
I think anyone can get better at math. You just have to be determined and practice.
Part of the problem is that math books tend to be written by mathematicians who ceased being math novices decades ago. What you want is books geared to your level and which contain material you can work with.
Some recommendations:
If you can find a copy, get Mathematica and a good book on it (the Schaum's outline is actually pretty good and cheap). I use it all the time to visualize things.
As a programmer, you probably want to aim more for discrete mathematics than calculus.
The Concrete Mathematics book mentioned elsewhere is excellent.
Most introductory discrete math texts have good coverage of the things like logic, sets, combinatorics, probability, graph theory, etc. My school used Rosen's text which I liked.
Linear algebra is useful if you are going to do 3D graphics programming. Most intro texts for engineers will teach you what you need to know. Linear Algebra Done Right is probably the best on "real" linear algebra if you want something more theoretical.
Look for books by Martin Gardner and play with his puzzles. He's an excellent writer and teacher.
Remember that math doesn't change that much. You can get used books for cheap on Amazon and in used bookstores. I always look for the n-1 version when I buy textbooks.
When you first started learning English all those "symbols" (letters) looked like gibberish to you. I'm sure that at some point you were frustrated over your lack of understanding. But slowly, and gradually you began to understand them.
Eventually you were able to construct your own words and sentences using these symbols. After being corrected on it's structure and grammar for years you now have a command of the language.
Math is just like that. Your eyes glaze over because you haven't learned the language. Maybe in school you didn't particularly enjoy math because you didn't see any practical applications for it. Certainly the way we teach math to our students is atrocious, so it is no wonder why many get through school without being well versed in it (for further reading, check out A Mathematician’s Lament which discusses how horrible our current method of teaching math is).
However, it is never too late to get up to a degree of proficiency that will allow you to read many academic Computer Science writings. Start out with Pre-Calculus at your local community college at night (to brush up on everything you have forgotten). Then move on to Calculus and after that take Discrete math. Honestly, this is all the math you will need 99.99% of the time. In less than 2-3 semesters you can be fully caught up and you'll no longer have your eyes glaze over when reading something with some mathematical roots.
I can share my experience...
I have been terrified of math since grade school. Hated it, didn't get the point, didn't pursue it.
By contrast, I have always been fascinated by computers. I study programming from a "need to know"- I can't stand not understanding computers and programming from the lowest to highest levels. I am almost completely self-educated, and have a career as a programmer/architect.
Last year, at my wife's urging, I began to go back to college. I signed up for a remedial Algebra class, knowing it was going to be a pain. It wasn't.
Somehow, through all the years of learning to develop OO software, it seemed that I had tricked myself into learning how to think mathematically. The concepts just weren't that difficult anymore. It may be that I had learned to think in terms of complex systems made up of smaller, less complex ideas.
I am now researching game development, and that is some seriously math-oriented programming. WAY more so than the business development I've been doing to this point. However, I don't find it so daunting because it's applied mathetmatics. Working to solve practical problems seems to make the study less tedious and far more interesting. I have found Wikipedia and Wolfram's Mathworld to be helpful. If you already know how to program, you're ahead of the game learning math.
I'd say it's certainly something anyone can get better at. It takes time and patience and some texts are obscenely dense as far as the notation involved, but if you're willing to put the time in, I think it shouldn't be too horrible.
I'd check out Wikipedia's list of mathematical symbols, and keep if nearby whenever you see a large blob of symbols pop up. Translate them one at a time and put them together in the way that makes the most sense to you (or ask us a few times until you get the hang of it).
It's both. You can get better at math. But you're also indeed limited/endowed by the particular wiring in your brain. What that means is that most likely you can improve your current mathematical skills. However, because of your mental hardware's limits, you may never discover a new theorem.
And when it comes to improving, I think the way as always is to practice. To read mathematical literature, to try to solve mathematical problems and eventually, develop an outlook where you are able to break, as a matter of habit, real-world conundrums you see before you down in mathematical terms.
As for programming's relation with mathematics, I think there's a pretty strong one. In fact, one could argue that a program is nothing but a proof of a theorem, the requirements document being all the inputs to the proof.
Your skills get rusty if not used and knowledge fades with time if not used.
If you don't use your math skills you soon have no math skills. Continuous new learning and practice of the skills you already have will lead to you one day being a math/programming master.
Project Euler has lot of math problems that can only be solved through programming. The problems get more difficult but build on the skills and knowledge acquired in your previous solutions.
I also buy some interesting textbooks at second hand book shops. Their cheap and slowly your skills improve. I use them in conjunction with MIT Open Course ware.
A fun way to practice math is http://projecteuler.net/.
Although it's less systematic/effective than doing a course or reading a textbook.
I know exactly how you feel. I've always wanted to learn more Math, but as I was unable to do it at college after school (not enough space) and not able to take it at university (not able to with a CS degree) I'm still yet to study Math formally since the age of 16.
Math is something that anyone can learn. Some will argue that it gets harder with age but I've met people going on for 60 that are taking Math classes with ease. There's one woman at my university that's going on for 70 and she's a few months off of graduating with a degree in a Mathematics related field. If you want to learn Math then now is the right time, although I'll be the first person to say that it is not easy. Whilst you'll find many of the problems extremely easy with programming experience you'll still find that going through a set of problems takes a lot of time out of your day. I almost finished the MIT OpenCourseWare course on Linear Algebra, then ended up getting a new part-time job, working 10 hours a day, 7 days a week, and forgetting the majority of what I had learnt.
That being said, if you have the time and true dedication I can recommend some links to video lectures that may just help you get on your way.
Pre-Calculus Algebra
Pre-Calculus
Calculus
Applied Probability
Introduction to Statistics
Differential Equations
Linear Algebra
I'm not saying that this is what you need to know. This is what I've set out to learn myself before I graduate from my CS degree, so feel free to pick and choose whatever you feel is best for you.
Studying math is like getting a classical education, one specially suited for programmers and other computer professions. And math's the sort of thing that you can appreciate more as you get older. You realize that it's not about grinding out answers so much as it's about thinking deeply and conceptually. The "answers" you might grind out make a lot more sense that way.
At one time, I would have recommended taking a geometry course, and take some time to learn how to prove theorems, see how the concepts flow together. These days, though, I'd say it may be better to take a course in discrete mathematics. It's much more practical, and there's a lot more variety, but there's still enough theory in there to make it challenging if you want.
Discrete math also provides you with programming challenges you might not have thought of before. Maybe you can hack up a good heuristic to solve an NP-complete problem, like an N-city Traveling Salesman problem. Maybe even come up with a couple solutions, and test which ones work best in which circumstances.
(I never took CompSci classes in college. You can probably tell.)
Go to the local community college and sign up for Calculus 1. This covers functions in the mathematical sense, and has a rigorous refresher course on algebra, and will use just enough of the symbols to get you ahead.
First of all, I would recommend Steve Yegge's Math For Programmers. It pretty much sums up your struggle.
And now I would like to tell a personal story. I was a double major in Math and CS. I learned a lot in the Math classes, but I honestly didn't appreciate it as much as I should have. I will tell you that a lot of things that I did have helped me in my programming career. And it's not about some formula or knowing calculus, or any of that stuff. It's that a solid Math background teaches you how to think in order to solve a problem. To me, that's the Math that you need.
1) Yes.
2) Explore mathematical questions which sound interesting. Buy/read books that give you the information needed. Repeat.
It can absolutely be learned. I personally had the most benefit from the math (especially proof) courses I took in college.
Recommended courses:
Discrete Math
Mathematical Thought
Abstract Algebra
any other proof courses
Recommended book:
The Nuts and Bolts of Proofs, by Antonella Cupillari
I strongly recommend trying to take one or more of these courses at a school of some sort. Find a local college and audit a course.
A 2nd vote for Lockhart's "A Mathematician’s Lament", which recommends that math be taught like painting, poetry, or music -- not for it's practical usefulness but for simple pleasure:
There’s no
ulterior practical purpose here. I’m just playing. That’s what math is— wondering, playing,
amusing yourself with your imagination.
Look at the diagrams in a recent Knuth paper, Dancing Links, and tell me he wasn't having fun making those.
Part of the problem is that a few mathematical symbols convey a heck of a lot of information. If you are reading a normal programming book, it is full of words and code. Neither of these is super verbose (although I often have to slow down much more on code than normal words). However one complicated mathematical equation can easily be a screen full of programming code or words. We have all sorts of simple notations that convey complex processes.
Another issue is that notation is standardized but not exactly. Different books use slightly different notation so it takes a while to get used to it. Also many textbooks leave out key steps in mathematical proofs or even examples. Sometimes even college professors puzzle themselves with the missing steps in a given proof in a textbook and then give their own proof, or give a slightly modified one over the one in the book because they learned it differently or can't recreate the missing step exactly which takes the proof in a different direction.
So anyway just because your eyes glaze over doesn't mean you have to give up. The first time you see the equations you will probably be in read the english text mode and have to pause to consider them. Going over them slowly and paying attention to what all the symbols mean one step at a time may yield the answer for you. If there is some notation you haven't seen before, there is probably an intro chapter or appendix explaining the notation, so check there. Finally, look for other sources. Use google/wikipedia to look up equations for the concept and you may find a derivation and/or proof that you can follow. Additionally the other one may help you to understand the current proofs/derivations better. Even if your understanding of the proof/derivation does not improve, your additional research will probably aid in your understanding of the equation.
I think there are two things to learning math:
1. Learning the general techniques. Ie how to add two fractions, how to differentiate, integrate.
2. Learning to problem solve and apply math to the real world.
I think by picking up math textbooks yo will learn 1. Many math textbooks are organized by section where there will be a few pages showing you a technique and then a bunch of problems. The problems tend to be related to the technique that you just learned and very similar. Ie a section on logarithms will have all problems on logarithms and probably won't include any polynomials. By doing the problems in the section you will learn the techniques. The more problems you do the faster you will get and the more you will understand the concepts. Many times you will find if you work through the problems without explicitly memorizing the formulas, you will find that after you do enough the required formulas will be implicitly memorized. Ultimately if you are having trouble looking at probability formulas you will want to read a probability book. If you are having trouble with sum notation you will want to consult that section of an algebra book, etc...
To learn 2 I think math textbooks don't help as much because each section tends to have problems related to that section. Occasionally there are a few "mixed review" problems or a "chapter review" which mixes problems, but they are typically far in between. Science textbooks like Physics, Biology, Chemistry, etc. tend to be better for this. There you often read the problem, lay it out, and end up using a variety of mathematical tools to to solve it. Sometimes calculus, linear algebra, and geometry all within the same problem. The value here is that it teaches you to problem solve. Generally the SAT/GRE do not test if you know how to do Algebra, they test if you know how to apply it to the real world, and the science problems really help you here. Also programming in general is about problem solving and the better you get at problem solving the better you'll be at programming. Basically in programming you take problems, create a mental model, design a solution, and then model it in your programming language of choice. This is similar to say Physics. You look at the problem, extract a mathematical model, design a solution, right down some equations with the model of the solution, then plug numbers in. I highly recommend physics because after my college physics class word problems became simple for me and they used to be quite difficult (though not impossible).
In day to day programming you probably won't use more than algebra and logic (for if statements and loop conditions). There are some places that use high math like computer games, cryptology, data mining, etc. but for a typical business application you probably won't use more than algebra and logic and maybe a bit of set theory (the stuff so basic you already internalized it). Even in places that use high math (like financial companies) often the business users (or some industry literature) will have done the higher math and you will just need to implement the equations (with some algebra). I only mention this because most programming books don't have more than algebra and logic either, unless you are reading textbooks on Algorithm Analysis (Introduction to Algorithms), Artificial Intelligence, or some other research area. General application books on how to do things are usually short on math.
But depending upon what you are reading math can help. For most computer science algebra + discrete math should be enough. Couple that with some physics practice and you should be good to go. It may still be a slow go but you should have the proper background.
I like combinatorics and algorithms - having fun you learn faster.
study study study!
wikipedia is actually a fairly good math reference. start with something you're interested in learning and follow links until you understand all the building blocks for that initial thing.
practice practice practice!
Schaum's outlines are good for this. If you're interested in probability (which touches on combinatorics), see 50 Challenging Problems in Probability.
Short answer:
There may be people who are to stupid to get good a maths. But those people generally are to stupid to program, too.
So if you have some skills in programming, you might consider yourself smart enough to learn maths as well.
Note: I know there are smart people with a serious math learning disability, but I think thats more like an exception.
I would also recommend project euler While it does not exactly teach math, it gives you problems that you can then look up how to solve. I've always preferred solving actual problems instead of just learning theory.
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I'm trying to reteach myself some long forgotten math skills. This is part of a much larger project to effectively "teach myself software development" from the ground up (the details are here if you're interested in helping out).
My biggest stumbling block so far has been math - how can I learn about algorithms and asymptotic notation without it??
What I'm looking for is some sort of "dependency tree" showing what I need to know. Is calculus required before discrete? What do I need to know before calculus (read: components to the general "pre-calculus" topic)? What can I cut out to fast track the project ("what can I go back for later")?
Thank!
Here's how my school did it:
base:
algebra
trigonometry
analytic geometry
track 1 track 2 track 3
calc 1 linear algebra statistics
calc 2 discrete math 1
calc 3 (multivariable) discrete math 2
differential equations
The base courses were a prerequisite for everything, the tracks were independent and taken in order.
So to answer your specific question, only algebra is needed for discrete. If you want to fast track, do one of these:
algebra, discrete
algebra, linear algebra, discrete (if you want to cover matrices first)
HTH... It about killed me when I returned to school and took these, but I'm a much better programmer for it. Good Luck!
My advice is to lazily evaluate your own dependency tree. Study something you think is interesting -- when you hit something you don't know, go learn about it.
I always find it easier to learn something new when I already have a context in which I want to use it.
This is a particularly cool site for visualizing how everything in the math world fits together:
http://www.math.niu.edu/Papers/Rusin/known-math/index/mathmap.html
It's also got short summaries of many subfields you've probably never heard of, which is fun.
Usually, an overview of each field is a good thing to have when looking at any topic, but it's rare to have a genuine dependence the way we'd think of it. Algebra is always needed. I can't think of a time I've needed any trigonometry. (except to expand it with new things from calculus) I'm even quite sure people wouldn't agree on what a dependency graph would look like, or even in which field each topic belongs.
I think the right way to approach it is to just collect a wide range of topics from all of branches and read them in whatever order you feel like, recording dependencies between topics as you go. (respecting them, or not, as you please.) This should have the far more important property of keeping the student interested.
It's also my experience that if something just has you stumped, just mark it and set it aside for later.
As for my school, well, it was similar to Harrison's:
cominatorics,
linear algebra,
calculus,
numerical analysis (error analysis in particular.)
logic,
statistics, (with operations research / queueing therory.)
Take a look at MathWorld. Browse topics or search for one, you'll get your position in the overall tree.