What specific branches of Mathematics are useful for a programmer to learn?

Question

I have less than 6 months to learn mathematics for my Master course in CS. (With no prior formal education in any computer/mathematics related area.) So I need to prepare my self to be tested on math as it relates to computer science.

As I found in Wikipedia, mathematics has many branches. Which ones are the most important for a programmer?

• Arithmetic
• Algebra (elementary, linear, multilinear, abstract)
• Geometry (discrete, algebraic, differential)
• Calculus/Analysis
• Set theory
• Logic
• Category theory
• Number theory
• Combinatorics
• Graph theory
• Topology
• Lie theory
• Differential equations/Dynamical systems
• Mathematical physics
• Numerical analysis
• Computation
• Information theory
• Probability
• Statistics
• Optimization
• Control theory
• Game theory

Answer 1

In my Computer Science degree many moons ago, Discrete Mathematics was a required course taught out of the Computer Science department itself. The resulting math minor was just an artifact of the origins of the Computer Science department at my University.

Answer 2

I recommend you review the prerequisites for the CS course you are going to take. They should give you a good idea of what you need to know.

As far as the CS program at my university you need to know:

• Discrete Math
• Calculus I/II
• Linear Algebra
• Physics
• Probability and Statistics

And the prerequisites thereof. (mainly arithmetic, algebra, trigonometry)

Answer 3

Personally I think math is good brain training, and I wouldn't really discourage any of these. But also I think some of these are particular to a given field, where it would be helpful to understand the math if you were programming for that field, but not for programming, in general.

Topics that are good in general for CS:

• Abstract Algebra
• Discrete Mathematics which generally includes:
  • set theory
  • logic
  • combinatorics
• Graph theory
• Control theory

For that, you'll need:

• Arithmetic
• Algebra (elementary)

For some UI work it's useful to know

• Geometry (Algebraic geometry)
• Game Theory - tells you a lot about ways humans may make decisions.

Many of the others are useful in particular fields but are not one size fits all in terms of usefulness.

Answer 4

To write non-mathematical software the amount of university level mathematics that you need is fairly minimal, to do a degree in CS, you probably want need CS style maths rather than maths maths; of the list you give:

• Arithmetic [being able to do basic sums and solve basic equations is very useful; modular arithmetic is also very useful]
• Algebra [linear algebra, things like simultaneous equations is useful, as are finding minima and maxima, but in general you would use a library for this, so just having an understanding of its existence is enough]
• Geometry [basic euclidean geometry is useful because we live in the real world, everything else is a waste of time]
• Calculus [again basic calculus is useful, but knowing how to do analysis is a total waste of time]
• Set theory [the basics again are useful, knowing what a set is, an intersection, a union... but anything more advanced is a waste of time]
• Logic [again understanding the basics like how to read and write and and or and not statements in compounds and how to manipulate those statements, not much else (knowing the proof to Goedel's third incompleteness theorem is useless)]
• Category theory [totally useless]
• Number theory [elementary number theory is useful, but again, just the basics]
• Combinatorics [very useful, many real world problems are combinatorical - I think of combinatorics as "raw problem solving skills"]
• Graph theory [quite useful - a lot of theoretical computer science problems are graph theoretical]
• Topology [useless]
• Lie theory [beyond useless]
• Differential equations/Dynamical systems [basically a branch of calculus, quite specialist]
• Mathematical physics [useful if you want to write a physics engine]
• Numerical analysis [yep, a bit useful, but you don't need to know the theory, just how to apply it]
• Computation [don't know what this is]
• Information theory [snore, snore, snore - but for a CS degree pretty useful, it's a branch of probability and algebra]
• Probability [useful]
• Statistics [snore, snore, snore, but it is very useful]
• Optimization [just a branch of calculus]
• Control theory [?]
• Game theory [might not be totally useless]

I'd say that if you have 6 months to prepare get this book for engineers and work through that. The real thing that you will get from doing mathematics is the problem solving skills.

Answer 5

All of them, some are just more dependent of what your tasks are. Logic, Algebra, Set Theory, Graph Theory, Computation and Statistics are unavoidable in my opinion, but all the others applicable depending on what you're working on.

For example, you'll need linear algebra, geometry and maybe calculus if you do some heavy physics/3d stuff. Or Probability and Numerical analysis if you do some heavy data analysis stuff.

Answer 6

Analytics and/or Statistics

In my experience, I've found that a programmer who can understand a company's business analytics is far more valuable to an employer than a programming genius.

And like it or not, you'll probably be building reports for someone at one point or another in your career :)

Answer 7

Concrete Mathematics: A Foundation for Computer Science would be my suggestion for a book that has a few branches of Mathematics to it wrapped within a CS core.

I'd also suggest looking into which textbooks does that class use and what parts of Math are used as there are numerous components where Math and CS intersect like Boolean Algebra, Big-O notation, Symbolic Computation, and Numerical Analysis to name just a few such areas.

Answer 8

I know that many computer science degrees require calculus, but from what I've heard, you can get by without it with other similar math courses. Friends with CS degrees have taken courses like Algebra (a requirement from pretty much ALL degrees), Statistics, Discrete Math (which is a huge requirement for CS majors), and Graph Theory.

Here are the requirements for a CS degree from the University of Texas @ El Paso. I cannot comment as to how their program compares to average CS programs, this just what came up on the ole' Google when searching for CS degree requirements:

http://www.cs.utep.edu/DeptCS/ugrad/requirements.html