Closed. This question needs to be more focused. It is not currently accepting answers.
Want to improve this question? Update the question so it focuses on one problem only by editing this post.
Closed 6 years ago.
Improve this question
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).
There are a few questions around that circle around this question but I feel this is different enough.
I've decided I want to improve the breadth and depth of my maths skills specifically in areas that are useful and/or interesting to programmers.
What topics should I study?
What resources do you recommend (blogs/books/online lectures...)?
I'm looking for easy to consume resources because I'll be doing this in my free time, I don't want to spend days struggling through a dense text but I want to get deeper than the surface. I've read the Yegge article on the topic (and most of the comments) which is useful but I think the voting system here will help me focus on the most useful/best resources and topics.
Edit:
I am looking to create myself a study course that I'll follow over the next few years, I'm not looking to solve a particular problem I just want to learn some new skills that will interest me and may be useful in my career in the future.
Concrete Mathematics: A Foundation for Computer Science would be my suggestion for a book that covers some advanced topics.
For an introduction to Discrete Mathematics I strongly suggest this.
I feel very lucky to have been provided this book from University
Any programmer would do well to have a solid understanding on the undergraduate level of these following math courses:
Calculus (at through multivariate calc)
Discrete Mathematics (absolutely essential)
Linear Algebra (necessary for an understanding of matrices)
Combinatorics (further development of Dicrete maths)
Introduction to Abstract Algebra (this will solidify your understanding of modulo number systems, in particular binary, octal, hex etc.). It also gives a deep understanding of set theory which is ubiquitous in practical programming and the comp sci literature.
This is the fundamentals. If your are thinking about graphics or game programming then you have a whole slew of additional courses in physics, graphic arts, and possibly fluid dynamics. Also Differential Geometry is essential for any real world modeling of motion on curved surfaces.
It's a bit off from your question, but let me suggest the Princeton Companion to Mathematics.
It gives an overview of all of mathematics, so it is more than "math useful to programmers", but it's style is as easy to understand as it gets, and the important parts are in there.
Some time ago Steve Yegge wrote a dedicated article about math for programmers. His thesis is: As a programmer you should learn math but you should do so in different way than in shool/university.
His summary is this:
Math is a lot easier to pick up after you know how to program. In fact, if you're a halfway decent programmer, you'll find it's almost a snap.
They teach math all wrong in school. Way, WAY wrong. If you teach yourself math the right way, you'll learn faster, remember it longer, and it'll be much more valuable to you as a programmer.
Knowing even a little of the right kinds of math can enable you do write some pretty interesting programs that would otherwise be too hard. In other words, math is something you can pick up a little at a time, whenever you have free time.
Nobody knows all of math, not even the best mathematicians. The field is constantly expanding, as people invent new formalisms to solve their own problems. And with any given math problem, just like in programming, there's more than one way to do it. You can pick the one you like best.
Math is... ummm, please don't tell anyone I said this; I'll never get invited to another party as long as I live. But math, well... I'd better whisper this, so listen up: (it's actually kinda fun.)
Sad note: Steve abandoned his blog because of too much aggressive feedback.
If you have any interest in game development, 3D graphics, or anything somewhat related to those, then do multivariate calculus and basic physics. This will help you understand the basic concepts much better. Also, linear algebra will help immensely with all of the matrix/vector stuff you will be doing.
If you are NOT interested in these topics, I would still say study calculus and physics. Why? Solving calculus and physics problems gives you good experience in problem solving and exercises the brain. Programmers NEED to be good problem solvers... that is our job. Concepts you pick up from these courses are things you will keep with you the rest of your life.
MIT and Stanford both have really good online courses for topics such as this. Of course you can't just jump into multivariate calculus without some more basic calc, but MIT and Stanford have resources for your basic calculus classes as well. Basic physics will be a little bit easier to pick up. Again, you can check MIT and Stanford for physics.
MIT OpenCourseWare:
Single Variable Calculus
Calculus
Multivariable Calculus
Physics I: Classical Mechanics
Mathematics for Computer Science
Generally speaking, the applications of math to computer programming are pretty domain-specific - that is, you need to know whatever math the specific program you're writing requires. The only mathematical topics I can think of that are generally applicable to all kinds of programming are simple arithmetic and boolean logic, but I think if you didn't already know those you wouldn't be much of a programmer ;-)
Basically, I would just recommend learning the math as needed for whatever project you're working on. If you want to give yourself a good excuse to learn some new math, start a hobby program that does something mathematical.
As for topics, look at some of the answers here. Recommended ressources are difficult for me to give, I'm German speaking. I would recommend starting with linear algebra and geometry, which you will find in computer graphics. Look at the undergraduate math series by Springer for example.
Number theory doesn't have many direct applications to programming (though there are some neat tricks you can use for optimization), but there are several basic concepts that make cryptology much easier to study.
My number theory class used Silverman's Friendly Introduction to Number Theory, which is one of the best math textbooks I've ever seen. It's very easy to read (the title is entirely accurate about its friendliness), but covers a wide range of topics. Silverman is also an author on my cryptography textbook, An Introduction to Mathematical Cryptography. It's more technical, addresses most areas of cryptography, and provides plenty of references for where to find more detail.
Consider Knuth's Art of Computer Programming series. It can get dense, but it will ground you in the math most needed for programming. I'd suggest going for the available fascicles of Volume 4 early on. These books are not for everybody, but if you find them interesting you will learn a whole lot.
They won't teach you calculus or geometry, which are important in many aspects of programming but tend to be more specialized.
I think you should dive into whatever interests you most and in order to find out what that is you should get some books which cover the facts and offer orientation and some books which nurture your motivation and curiosity. You really have to dive into it to find out, it's a pretty individual thing imho.
Facts / Orientation:
Donald Knuth -
Bronstein, Semendjajew
The Science of Programming -
Data Structures and Algorithms
Motivation / Curiosity:
The Road to Reality -
Fermat's Last Theorem -
Godel, Escher, Bach
Also for motivation on the more practical side:
projecteuler.net
What sorts of math problems do you want to solve? 'Math' is a pretty big area!
MIT has some online courses, but that's probably a big time investment.
Wolfram has some tutorials, but again, you need to know what you're looking for.
I regrettably haven't studied mathematics since I was 16 (GCSE level), I'm now a 27 year old C# developer.
Would it be a fruitless exercise trying to work through Structure and Interpretation of Computer Programs (SICP)?
What kind of mathematics standard is expected of the reader?
Having worked through all of SICP, I can tell you with confidence that you don't need a lot of math background to understand it. SICP is (used to be?) a first or second semester course in MIT, for students with practically no college/university level math. Whenever it discusses mathematical topics, it provides sufficient background for any intelligent reader to understand.
From the little you tell about yourself, it's great time to work through SICP. Reading the book and solving (at least some of) the exercises, and playing with the code of the projects, can teach you a lot about programming. Don't worry about math - you'll handle it without any problems. What's really needed is a true, deep curiosity about programming, and some patience.
It's never too late to start SICP. And it doesn't really require any higher maths at all, except perhaps in the signal processing with infinite streams parts. That can be skipped without losing too much though.
The most important thing while reading SICP is solving the problems, IMO. Some of the tougher ones can be mind-expanding and force you to really understand the topic. If you are confident about some solution you can skip it though. And the solutions can be found at - http://eli.thegreenplace.net/category/programming/lisp/sicp/
The danger in reading SICP is that after completing it, you will not like using any programming language other than Scheme. :)
I had a ganders at this book. My maths knowledge is not great ... but there is a key:
For understanding things like this, providing you have a creative mind and a good grasp of the abstract nature of structures and mathematical principles you should be fine. My mental arithmatic is pretty poor by anyones standards, but I love reading about discrete mathematics because of it's abstract nature.
I wouldn't consider myself a very good mathematician in the numeric sense, but as a software developer I like to think I have a mathematics (or mechanical) mind.
I wouldn't worry too much about your numeric strength but more about the nature of mathematics and the personality of the concepts underpinning computer science. If you have a good programming mind, maybe try and enhance that with combinatorics/discrete/concrete mathematics (which, besides counting theory, in many cases avoids dry numbers).
I found my love for things like set-theory studying compilers, and I wouldn't want to sit my maths A-level without alot of cramming!
Give it a go, what have you got to lose?
(im 22 and in a similar situation to you)
Good luck
PS: I also found the video lectures interesting. You can torrent them from
http://groups.csail.mit.edu/mac/classes/6.001/abelson-sussman-lectures/
It definitely wouldn't be a fruitless exercise, it's an excellent book. On the other hand, it would be kind of tough going, as they do expect you to have some mathematical sophistication, if not tons of advanced math.
You might find How to Design Programs, by Felleisen et al, a bit of an easier start while giving you much the same approach, using Scheme and all.
From what I can remember from this book, it talks about some matrix calculations, which might be hard to understand at first. But it is just list of lists, or array of arrays... so you will need to deal with that sooner or later in programming.
If there was any difficult math, I think you can skip it. This book was (and probably still is) used in Berkeley's first year computer science class (many students take it in the first semester), without any need of understanding calculus at all, so I think general understanding of math is good enough to understand the book.
The book talks about a function being a black box... and after reading the book, I think it helps a person's understanding of math in general as well.
The Numerical Programming section might require some higher math, but you should be able to digest the rest of the book with high-school math.
As it currently stands, this question is not a good fit for our Q&A format. We expect answers to be supported by facts, references, or expertise, but this question will likely solicit debate, arguments, polling, or extended discussion. If you feel that this question can be improved and possibly reopened, visit the help center for guidance.
Closed 11 years ago.
Locked. This question and its answers are locked because the question is off-topic but has historical significance. It is not currently accepting new answers or interactions.
I happened to debate with a friend during college days whether advanced mathematics is necessary for any veteran programmer. He used to argue fiercely against that. He said that programmers need only basic mathematical knowledge from high school or fresh year college math, no more no less, and that almost all of programming tasks can be achieved without even need for advanced math. He argued, however, that algorithms are fundamental & must-have asset for programmers.
My stance was that all computer science advances depended almost solely on mathematics advances, and therefore a thorough knowledge in mathematics would help programmers greatly when they're working with real-world challenging problems.
I still cannot settle on which side of the arguments is correct. Could you tell us your stance, from your own experience?
To answer your question as it was posed I would have to say, "No, mathematics is not necessary for programming". However, as other people have suggested in this thread, I believe there is a correlation between understanding mathematics and being able to "think algorithmically". That is, to be able to think abstractly about quantity, processes, relationships and proof.
I started programming when I was about 9 years old and it would be a stretch to say I had learnt much mathematics by that stage. However, with a bit of effort I was able to understand variables, for loops, goto statements (forgive me, I was Vic 20 BASIC and I hadn't read any Dijkstra yet) and basic co-ordinate geometry to put graphics on the screen.
I eventually went on to complete an honours degree in Pure Mathematics with a minor in Computer Science. Although I focused mainly on analysis, I also studied quite a bit of discrete maths, number theory, logic and computability theory. Apart from being able to apply a few ideas from statistics, probability theory, vector analysis and linear algebra to programming, there was little maths I studied that was directly applicable to my programming during my undergraduate degree and the commercial and research programming I did afterwards.
However, I strongly believe the formal methods of thinking that mathematics demands — careful reasoning, searching for counter-examples, building axiomatic foundations, spotting connections between concepts — has been a tremendous help when I have tackled large and complex programming projects.
Consider the way athletes train for their sport. For example, footballers no doubt spend much of their training time on basic football skills. However, to improve their general fitness they might also spend time at the gym on bicycle or rowing machines, doing weights, etc.
Studying mathematics can be likened to weight-training or cross-training to improve your mental strength and stamina for programming. It is absolutely essential that you practice your basic programming skills but studying mathematics is an incredible mental work-out that improves your core analytic ability.
While advanced mathematics may not be required for programming (unless you are programming advanced mathematics capability) the thought process of programming and mathematics are very similar. You begin with a base of known things (axioms, previously proven theories) and try to get to someplace new. You cannot skip steps. If you do skip steps, then you are required to fill in the blanks. It's a critical thought process that makes the two incredibly similar.
Also, mathematicians and programmers both think critically in the abstract. Real world things are represented by objects and variables. The ability to translate from concrete to abstract also links the two fields.
There's a very good chance that if you're good at one, you will probably be good at the other.
computer science != programming
OK, seriously, I know good and bad programmers who were English and Psychology majors and some that were Computer Science majors. Some very famous guys that I admire as developers didn't have a CS background. Larry Wall(Perl), for example, was a linguist.
On the other hand, it helps to know something about the domain you are working on because then you can at least see if your data makes sense and help your customer/users drill down to what they really want.
And yes, there's the issue of computational complexity and efficient data structures and program correctness. That's stuff you learn in Computer Science and that's useful to know in almost any domain, but it's neither necessary nor sufficient.
I guess I am going to be the first person to say you do need math. As others have said math is not all that important for certain aspects of development, but the fundamentals of critical thinking and structured analysis are very important.
More so, math is important in understanding a lot of the fundamentals that go into things like schedulers, optimizations, sorting, protocol management, and a number of other aspects of computers. Though the math involved from a calculation level is not complex (its mostly High school algebra) the theories and applications can be quite complex as a solid understanding of math through calculus will be of great benefit.
Can you get by without it, absolutely, and you shouldnt let a less then thorough knowledge of math hold you back, but if you had the chance, or the inclination I would study as much math as you could, calculus, numeric theory, linear algebra, combinatorics, practical applications, all of it has both practical and theoretical applications in a wide range of computer science.
I have known people who were highly successful on both sides of the fence (those without a strong focus on math, and those who went to school for physics or math), but in both groups they enjoyed numerical problems and learning about algorithms and math theory.
I have a maths degree, but I can't remember requiring that maths a single time in my career. It was useful in terms of training my mind for logical thinking, but I've not written any code using fluid dynamics, quantum theory or Markov Chains. (The last is the most likely to come up, I suspect.)
Most line-of-business developers won't need advanced maths most of the time. Sometimes knowing trigonometry can help, and certainly being able to understand enough maths to implement algorithms described mathematically can be important - but beyond that? Nah.
Don't forget that most programmers aren't advancing computer science - they're building applications. I don't need to know advanced engineering to drive a modern car, even though that car has almost certainly been improved through advanced engineering.
I would argue that having advanced logic (discrete) math can really help. That along with set theory. When dealing with common computer programs, these disciplines can help a lot. However, a lot of the other math I took in university was calculus, which as far as I can see, had very limited usage. Since 90% (or something like that) of programming is doing business apps with very simple math, I would say that for the most part, you can get by with very little math knowledge. However, a good understanding of boolean algebra, logic, discrete math, and set theory can really put you up to that next level.
I'll go against the grain here and say "Yes"
I switch from Civil Engineering to programming (Concrete Sucks!). My math background consists of the usual first year stuff, second and third year Calculus(Diff EQ, volume integrations, Series, Fourier and Laplace transforms) and a Numerical Analysis course.
I find that my math is incredibly lacking for computer programming. There are entire areas of Discrete math and logic that I am missing, and I only survive due to an extensive library of textbooks, Wikipedia and Wolfram. Most advanced algorithms are based on advanced math, and I am unable to develop advanced algorithms without doing extensive research (Essentially the equivalent to a half-course worth of work.) I am certainly unable to come up with NEW algorithms, as I just don't have the mathematical foundations as the shoulders of giants upon which to stand.
It depends on what you are doing. If you do a lot of 3D programming, knowledge of 3D geometry is certainly necessary, don't you agree? ;-) If you want to create a new image format like JPG or a new audio format like MP3, you are also pretty lost if you can't understand a cosine or fourier transformation, as these are the basics most lossy compression are based on. Many other problems can be resolved better if you know your math rather well.
There are also many other programming tasks you will find do not need much math.
If you find the subject fascinating enough to post this, just go ahead and start learning. The rest will come naturally.
Yeah, there is no need for advanced mathematics - if you are programming commercial - off the shelf software.
However when dealing with hardcore stuff such as:
Calculating trajectories to control
a robot
Creating AI-like applications to
support uncertainty and automatic
reasoning
Playing with 3-D motion and graphics
Some advanced mathematics knowledge might come in handy. And it's not like they are "out-of-this world" problems.
I had to create a software to try to "predict" the necessary amount of paper for an office (and it was hell just to find out the best way to approximate values).
You have to be careful, though, because it is easy to get lost when using advanced things - there is a friend of mine who resorted to using Turing to store the state of a dynamic menu just to display it correctly - humm... perhaps he wnet too far in his imagination.
What type of programming?
In my commercial experience, I have needed no advanced mathematics, but this is heavily dependent on the field you are in.
Computer graphics require a large amount of advanced mathematics. A lot of academic computer programming requires advanced mathematics.
So saying there tends to be a correlation between people who are good at mathematics and people who are good at programming.
I hope this wishy-washy answer helps.
Mathematics are needed for developers in some fields but are almost useless in others.
If you are a game developer and have to work with physics a lot - understanding of math is crucial. If you are working with advanced visual controls - you could not do much without geometry. If you're planning to do some financial calculations - it would REALLY help to have solid knowledge of statistics.
On the other hand over last 5 years I had only 2 or 3 projects where ANY amount of math was required at all. Of these there was only 1 occurrence when a Google search did not help.
At the end of the day even financial calculations are very often something your clients do for you and give you formulas to implement.
So if you're in 'applied software' business you are likely to never use your math degree. If you're in academic software maths are crucial.
I agree with Chris. I would say "Yes", also. But this depends on your market as stated above. If you are simply creating some basic "off-the-shelf" applications or writing tools to help your everyday work...then math isn't nearly as important.
Engineering custom software solutions requires lots of problem solving and critical thinking. Skills that are most definitely enhanced when a mathematics background is present. I minored in Math with my Computer Engineering degree and I give credit to all of my math-oriented background as to why I'm where I am today.
That's my 2 cents, I can tell from reading above that many would not agree. I encourage all to consider that I'm not saying you can't have those skills without a math background, I'm simply stating that the skills are side-effects of having such a background and can impact software positively.
In my experience math is required in programming, you can't get away from it. The whole of programming is based on math.
The issue is not black and white, but more colorful. The question is not whether or not you need math, but how much. The higher levels of math will give you more tools and open up your mind to different paths of though.
For example, you can program if you only known addition and subtraction. When multiplication is required, you will have to perform a lot of additions. Multiplication simplifies repetitive additions. Algebra allows one to simplify math before implementing it into programs. Linear Algebra provides tools for transforming images. Boolean Algebra provides mechanics for reducing all those if statements.
And don't forget the sibling to mathematics, Logic and Philosophy. Logic will help you make efficient use of case or switch statements. Philosophy will help you understand the thinking of the guy who wrote that code you are modifying.
Yes, you don't need much math to write programs. Some programs may require more math than others. More math knowledge will give you an advantage over those who have lesser understanding. In these times, people need every advantage they can get to obtain those jobs.
I've been programming for 8 years professionally, and since I was 12 as a hobby.
Math is not necessary, logic is. Math is horribly helpful though, to say it's not necessary is like saying that to kill a man, a gun isn't necessary, you can use a knife. Well, it is true, but that gun makes it a lot easier.
There are a couple bare minimums, which you should already meet. You need to know basic algebraic expressions and notation, and the common computer equivalents. For example, you need to know what an exponential is (3 to the 3rd is 27), and the common computer expression is 3^3. The common notations for algebra does change between languages, but many of them use a somewhat unified methodology. Others (looking at you LISP) don't. You also need to know order of operations.
You need to understand algorithmic thought. First this, then this, produces this which is used in this calculation. Chances are you understand this or you don't, and it's a fairly hard hurdle to jump if you don't understand it; I've found that this is something you 'get', and not really something you can learn. Conversely, some people don't 'get' art. They should not become painters. Also, there have been students in CS curriculum who cannot figure out why this does not work:
x = z + w;
z = 3;
y = 5;
It's not that they don't understand addition, it's that they aren't grasping the requirement of unambiguous express. If they understand it, the computer should too, right? If you can't see what's wrong with the above three lines, then don't become a programmer.
Lastly, you need to know whatever math is under your domain of programming. Accounting software could stop at basic algebra. If you are programming physics, you'll need to know physics (loosely) and math in 3-dimensional geometry (Euclidean). If you're programming architecture software, you'll need to know trigonometry.
This goes farther then math though; whatever domain you are programming for, you need to soundly understand the basics. If you are programming language analysis software, you'll need to know probability, statistics, grammar theory (multiple languages), etc.
Often times, certain domains need, or can benefit from, knowledge you'd think is unrelated. For example, if you were programming audio software, you actually need to know trigonometry to deal with waveforms.
Magnitude changes things also. If you are sorting a financial data set of 1000 items, it's no big thing. If it was 10 million records, however, you would benefit greatly from knowing vector math actually, and having a deep understanding of sorting at the binary level (how does a system sort alphabetically? How does it know 'a' is less than 'b'?)
You are going to find that as a programmer, your general knowledge base is going to explode, because each project will necessitate more learning outside of the direct sphere of programming. If you are squeamish or lazy about self-learning, and do not like the idea of spending 10+ hours a week doing essentially 'homework', do not become a programmer.
If you like thought exercises, if you like learning, if you can think about abstract things like math without a calculator or design without a sketchpad, if you have broad tastes in life and hobbies, if you are self-critical and can throw away 'favorited' ideas, if you like perfecting things, then become a programmer. Do not base this decision on math, but rather, the ability to think logically and learn. Those are what is important; math is just the by-product.
Of course it depends on what kind of programmer you want to be, or better what kind of programmer your employers want you to be. I think calculus and algebra are essentials, statistic and linear programming is indeed a good tool to have in your briefcase, maybe analysis (derivative, integrals, functions...) could be done without. But if you want to know how things work skin-deep (electronics, for example, or some non trivial algorhytms) "advanced" math is something you'd better not go without anywhere.
Most of the programming I have done involved physics simulations for research including things like electromagnetism, quantum mechanics and structural mechanics. Since the problem domains have advanced mathematics associated with them I would be hard pressed to solve them without using advanced mathematics.
So the answer to your question is - it depends on what you are trying to do.
Advanced maths knowledge is vital if you're going to be writing a new programming language. Or you need write your own algorithms.
However, for most day-to-day programming - from websites to insurance processing applications - only basic maths are necessary.
Someone with a solid mathematical (which is not merely arithmetic) or logic background will cope well with algorithms, variable use, conditional reasoning and data structures.
Not everyone can design a UI.
Not everyone can make efficient code.
Not everyone can comment and document clearly.
Not everyone can do a good algorithm
Mathematics will help you to a point, but only to a point.
I dont think advanced mathemetics knowledge is a requirement for a good programmer, but based on personal experience I think that programmers who have a better grasp at advanced maths also make better programmers. This may simply be due to a more logical mind, or a more logical outlook due to their experiences of solving mathematical problems.
The fundamental concept of maths is the following, devising, understanding, implementation, and use of algorithms. If you cannot do maths then it is because you cannot do these things, and if you cannot do these things then you cannot be an effective programmer.
Common programming tasks might not need any specific mathematical knowledge (e.g. you probably won't need vector algebra and calculus unless you're doing tasks like 3D graphics or physics simulations, for example), but the underlying skillsets are identical, and lack of ability in one domain will be matched by a corresponding lack of ability in the other domain.
Math is a toolbox for creating programs. I recommend Cormen's Introduction to Algorithms. It touches on the more "mathy" stuff.
- Greatest lowest limit (managing resources)
- Random variables (game programming)
- Topological sort (adjusting spreadsheets)
- Matrix operations (3d graphics)
- Number theory (encryption)
- Fast fourier transforms (networks)
I don't think that higher math is a requirement for being a good programmer - as always it depends on what you are coding.
Of course if you are in 3D graphics programming, you'll need matrices and stuff. As author of business software, you'll probably need statistics math.
But being a professional programmer for almost 10 years (and another 10 years amateur) "higher math" is not something that I needed regularily. In about 99.8% of all cases it's just plus, minus, division and multiplication in some intelligent combinations - in most cases it's about algorithms, not math.
Learning higher math, for most programmers, is important simply because it bends your brain to think logically, in a step-by-step manner to get from one thing to another.
Very few programming jobs, though, require anything above high school math. I've used linear algebra once. I've never used calculus. I use algebra every day.
Mathematical knowledge is often useful to a programmer, as are graphic design skill, puzzle-solving ability, work ethic and a host of other skills and traits. Very few programmers are good at everything that a programmer can possibly be good at. I wouldn't agree with any statement of the form "you're not a real programmer unless you can {insert favorite programming ability here}".
But I would be wary of a programmer who couldn't do Math. More so than of one who couldn't draw.
I think it really depends on what you're trying to do, but IMHO, the CS and OS theory are more important than math here, and you really need only the math that they involve.
For example, there's a lot of CS background of scheduling theory and optimization that stands behind many schedulers in modern OSs. That is an example of something that would require some math, though not something super complicated.
But honestly, for most stuff, you don't need math. What you need is to learn the ability to think in base 2 and 16, such as the ability to mentally OR/AND. For example, if you have a byte and within that byte there are two 3-bit fields and 2 wasted bits, knowing which bits are in which fields are active when the byte value is something like 11 will make things slightly faster than having to use pen and paper.
I began programming about the same time I entered my pre-algebra class.. So I wouldn't say math is all that important, though it can help in certain types of programming, especially functional.
I haven't taken Discrete Math yet, but I see a lot of theory stuff with programming written in a math-notation that is taught in this class.
Also, make sure you know how to calculate anything in any base, especially base 2, 8, and 16.
Also, one class that really brought home some concepts for me was this pre-programming class. We got taught unions, intersections, and all that happy stuff and It almost exactly parallels bitwise math. And we covered boolean logic very heavily. What I considered the most useful was when we learned how to reduce complex boolean statements. This was very handy:
(x|y) & (x|z) & (x|foo)
can be simplified to
x | (y & z & foo)
Which I previously did not quite grasp.
Well you generated a number of responses, and no I did not read them all. I am in the middle on this, no you certainly do not need math in order to be a programmer. Assembler vs device drivers in linux are no more or less complicated than the other and neither require math.
In no way shape or form do you need to take or pass a math class for any of this.
I will agree that the problem solving mindset for programming is quite similar to that of math solutions, and as a result math probably comes easily. or the contrary if math is hard then programming may be hard. A class or a degree or any pieces of paper or trophies are not required, going off and learning stuff, sure.
Now if you cannot convert from hex to binary to decimal quickly either in your head, on paper, or using a calculator you are going to struggle. If you want to get into networking and other things that involve timing, which kernel drivers often do but dont have to. You are going to struggle. I know of a very long list of people with math degrees and/or computer science, and/or engineering degrees that struggle with the rate calculations, bits per second, bytes per second, how much memory you need to do something, etc. To some extent it may be considered some sort of knack that some have and others have to work toward.
My bottom line is I believe in will power, if you want to learn this stuff you can and will, it is as simple as that. You dont need to take a class or spend a lot of money, linux and qemu for example can keep you busy for quite some time, different asm langauges, etc. crashable environments for kernel development, embedded, etc. You are not limited to that, but I dont believe that you have to run off and take any classes if you dont want to. If you want to then sure take some ee classes, some cs classes and some math classes..
You need math. Programming is nothing more than math. Any findings of theoretical physics does not become a practical(applicable) implication, unless they are explained in terms of mathematical solutions. None of those can be solved computationally if they cannot be interpreted on computers, and more specifically on programming languages. Different languages are thus designed to solve specific problems. But for the general purpose and wide spread programming languages like java, c, c++ much of our programming tasks involve repetitive(continuous) solution to same problems like extracting values from database, text files, putting them on windows(desktop, web), manipulating same values, sometimes accessing some data from similar devices(but given different brand names, different port and a headache) etc which does not involve more than unitary method, and algebra(counter, some logic), geometry(graphics) etc. So it depends what you are trying to solve.
IMO, you probably need an aptitude for mathematics, without necessarily having much knowledge in the field. So the things you require to be good at maths are similar to the things you require to be good at programming.
But in general, I can't remember the last time I used any sort of advanced maths in day-to-day programming, so no.