Ask HN: What are some important math topics every programmer should know?

I've used a lot of math in my work. But the point is, I have a lot of math to use. So many people say "I've never used math" or "Oh look, another day without using algebra," but really what's happening is that they don't realise that there are times when they can use mathematical thinking and techniques.

It's blub all over again.

So to be specific, I've used calculus in a bunch of places, Linear Algebra, Markov Chains, Quaternions, Topology, and Graph Theory is just everywhere. It's impossible to make recommendations - there's far too much, and what you need to do is get started on something, and keep track of things that look like they'll be interesting, useful, or the basis of something needed.

In part this is why I've started the "Topics In Maths" project I'm working on.

So good luck - keep pursuing stuff, and I hope you get more and more intrigued by what you find.

Graphs.

Depends on the industry I suppose. I work in the financial industry and I have used linear algebra quite a bit and graph theory as well. I would say linear algebra in pretty important as it is also used heavily in graphics/games.

Every programmer should know? None.

Middle school maths is "good enough" for the majority of programming jobs (85%+). There are of course niche programming jobs in: finance, statistics, AI, CS, gamedev, and so on but if you look at the programming industry as a whole those jobs are a tiny proportion overall (the majority are "boring" enterprise-style developers and web-developers: Java/Tomcat, ASP.net/MVC, PHP, RoR, and so on).

The thing is that the "boring"/average programming jobs are the silent majority. You'll likely read in this thread tons of obscure topics and how they apply to obscure jobs, but the reality is that few programmers work in those industries, but nobody [else] is going to post just to admit that for their job they don't need any maths day to day because it is boring and redundant.

This is JUST my opinion, but if you want to learn obscure topics, you're better off spending time learning CS topics (e.g. floating point number representation/bugs/limitations, encryption/hashing/etc, binary representations, assembly, kernel inner workings, CPU inner workings, and even borderline electronics). While advanced maths has a place in some jobs, advanced CS topics seem to be useful even on the web-stack/enterprise-stack to understand where bottlenecks might be materialising.

Maths is not useless. But most programmers won't use a lot of it day to day, if for no other reason that they have a library which does it for them (and they'd be foolish to reinvent the wheel).

I work in digital advertising (sorry, world), and use linear algebra often. ... I suck at it, so I usually get somebody with a math degree to help me out.

Like other commenters have said, it depends a lot on what programs you're writing. You might want to know some harmonic analysis if you're writing audio software. On the other hand, if you're making user interfaces, knowing a little bit of geometric math (vector algebra, trigonometry, calculus) can go a long way.

DeMorgan's laws. (You'll probably end up negating a boolean expression at some point. Don't screw it up!)

Bottom line: You're constructing and executing math and likely do not realize how much math you are constructing and executing on a daily basis. This includes formal and experimental advanced theoretical mathematics, probability, statistics and number theory in some cases. Some academics do not like to hear this, however, you deal(industry term) in functions, variables, minimums, maximums, rates of change, statistics, probability and overall, symbols. If you're duplicating the next facebook (as facebook did with myspace and myspace, friendster, etc etc), snapchat or yo app your likely not directly engaged in much math or innovation compared to inventing something on the algorithmic level like Google. If you are creating or inventing something new on the software engineering level including the algorithmic level, then you are certainly creating a lot of theoretical math and perhaps without even knowing it. This is key.

Here's a good book to understand how you've leap-frogged certain aspects of mathematics via true and real innovative Computer Science: (the Einstein of India. Every country and family has their very own Einstein, you might be one of them.)

The Man Who Knew Infinity: A Life of the Genius Ramanujan http://www.amazon.com/The-Man-Who-Knew-Infinity/dp/067175061...

Binary numbers (1010 it's off to work we go!)

Boolean logic.

How numbers are encoded in binary. How ints, doubles, floats, etc. work under the hood on a computer. Can be learned on the job.

set theory, equivalence relations, partial and total orders, algebra (monoids, lattices, ...), graphs

I find understanding of symbolic logic to be very important.

For UI programming of any sort, an understanding of basic geometry is very important, including graphing. There have been times in my short career where I have had to implement pixel perfect renderings due to the types of problems that were being solved.

Sometimes I feel like I should have paid more attention in class, like when looking at an expression and wondering if it could be simplified or if all the parens are really necessary

Starting with the easiest:

boolean logic, boolean algebra

What does O(n) mean?

The compound interest formula

matrix/linear algebra, numerical analysis

The statistics behind A/B testing

numerical approximation of a derivative, integral, differential equation

Read "numerical recipes in C"

High school and even undergraduate college gives you a rather late-19th, early-20th century view of what "math" is. A lot of math doesn't involve "numbers", per se. So, as a programmer in your self-described situation, my advice is to work through the math in Structure and Interpretation of Computer Programs [1], commonly referred to as SICP, and to do so seriously.

Does it not feel like math? That is, alas, a deficiency in your education, but don't feel bad, it's a deficiency in most people's education. It's math, and directly applicable to what you are doing in life.

Now, these other suggestions are fine too. But I find it to be very, very practically useful to be able to take apart the very concepts of computation, and thereby learn how to reassemble them to suit my whims, and not be swept away by the fads of the day.

The SICP is a concentrated load of exactly what you missed out on by not getting a formal computer science education. I honestly would likely prefer someone with 3 or 4 years of industry experience and the ability to prove they really worked through the SICP (not just read over it once, but worked it through) over someone with a 4.0, a bachelor's degree, and nothing else.

[1]: https://mitpress.mit.edu/sicp/full-text/book/book.html

My first two jobs out of college were in FinTech -- lots of math there (partial derivatives, linear algebra, numerical analysis methods, stats and more)

Then later, I was in image processing -- a lot of linear algebra. Anything with recognition has mathematical concepts.

I'm tutoring a teen in game programming and keep having to take breaks to teach more and more trig.

Having a good grasp of probability and statistics will serve you in almost anything you do.

There's a course at MIT called Mathematics for Computer Science (6.042). It's available on Open Courseware, http://ocw.mit.edu/courses/electrical-engineering-and-comput...

You need much more math to survive in the field beyond the appcrap/webcrap level. Here's the description of a 3D SLAM library for robotics. This is what looks at an image from a moving camera and builds a map of where the camera is and of its surroundings.

"RobotVision is a library for techniques used on the intersection of robotics and vision. The current version (1.1) comprises bundle adjustment, feature initialisation pose-graph optimisation, and 2D/3D visualisation among other things.

The bundle adjustment class follows the classical approach - the first order sparseness structure is exploited using the Schur complement. However, compared to other straight-forward implementations, it has the following features: The second-order sparseness structure -- not all landmarks are visible in all frames - is exploited using sparse Cholesky factorisation (using the CSparse library). Optionally, the implementation supports the use of robust kernels in order to guard against spurious matches. Also, the implementation generalises over different transformations, landmarks and observations using template meta programming. Among others, SE3 pose transformation with 3D Euclidean points and 2D image observations are provided as a default implementation for monocular SLAM. Furthermore, the bundle adjustment class also contains an information filter for inverse depth feature points which can be used efficiently for feature initialisation within a keyframe-based monocular SLAM framework.

The pose-graph optimisation framework is using Levenberg-Marquardt, whereas the sparse Hessian is dealt with using sparse Cholesky factorisation (CSparse). Again, it generalises over different transformations. Apart from the standard rigid transformation SE3, it also supports 3D similarity transformations Sim3. In other words, it can also deal with scale drift which occurs in monocular SLAM.

Both, the bundle adjustment class as well as the pose-graph optimisation class uses Lie theory. Poses transformations (SE3, Sim3) are represented on a manifold/ as a Lie group while incremental updates are done in the tangent space around the identity/Lie algebra. In this way, we achieve a minimal representation during optimisation while ensuring that we are always far from singularities."[1]

That's only some of what it takes today to do robotics.

The world is about to change. Tomorrow, everyone will discover that powerful humanoid robots are here.[2]

[1] https://openslam.org/robotvision.html [2] http://www.theroboticschallenge.org/

I work in quantitative finance and find that basic probability and statistics help a lot in this field. In fact, they are almost like prerequisites for hiring.

That said, I think everyone would benefit by reading a bit about these subjects, whether or not they’re developers. We seem to have a built-in tendency to underestimate the effect of randomness in life (see Fooled by Randomness and How to Lie with Statistics). To counteract that, I find statistical methods (e.g., hypothesis testing, confidence intervals, etc.) to be very useful tools to keep in mind. Having basic familiarity with such tools may not necessarily make us better programmers but it'll probably make us more rational decision makers in the general sense.

Every programmer should be able to convert between decimal, binary, and hexadecimal notation, and be able to understand the bit-shift and bitwise operations. That's it. Programming is too diverse a field for any one branch of mathematics to be required for all of them.

Many fields require linear algebra, geometry, trigonometry, graph theory, and discrete math.

Graphics and 3D-modeling programmers will need to know matrices (possibly also quaternions) and vector math. You may also need screw theory and geometric algebra.

Engineering support will need to know calculus.

AI will need probability.

Scientific support will need statistics.

Cryptography will require all the maths that exist, plus some that haven't been invented yet.

Business support will need to know sexagesimal math--for the clock-watching. You will really need to know when 5:00 rolls around, so you can stop working on your stupid CRUD app.

In short, the more math you know, the more likely it is that you will able to apply it to any given program. And the more you know, the easier it will be to pick up something you need and don't have already. Practically speaking, you are better off seeking a shallow knowledge of a broad array of disciplines, and deepen your knowledge of specific topics as needed.

If you know imaginary numbers, that makes it easier to learn about duals, quaternions, null vectors, spacetime manifolds, and such.

Ai/data sciences uses a lot of partial derivatives, linear algebra, etc

The jobs I've held have not been particularly 'mathy' but knowing linear algebra, numerical analysis and statistics have come in very handy over the years.

Knowing how to 'read' math has been helpful digesting Computer Science papers too.

Best math class I took in college was discrete math, it is extremely applicable to computer programming.

http://en.wikipedia.org/wiki/Discrete_mathematics

Highlights include Combinatorics, Set Theory, Graph Theory, etc. Great stuff, I liked it so much I kept the book even though the book store offered me $80 for it.

In the first year of my CS degree we studied some rudimentary CS-related mathematics.

Some of it I barely used beyond the exam, but there are two topics which rise high above the rest in terms usefulness when working on real-world web apps.

Set Theory: http://en.wikipedia.org/wiki/Set_theory

and

Combinatorics: http://en.wikipedia.org/wiki/Combinatorics

I can't tell you how useful these branches are in terms of the indirect and direct positive influence they have your thinking about how you organise and process data. Which at the end of the day is the core work of most web apps.

Even the very basics will help you greatly, remember I'm talking about first year CS courses here!

The tough part here is "every programmer should know" as there are very few universal "Must Knows". Much of CS theory is math so there is a lot to be said for "You're learning Math whether you think so or not."

Some general comments on math topics:

Partial Differential Equations - Useful for financial technology and engineering applications.

Statistics - Extremely useful in financial technology and marketing applications. (A/B testing, etc) Also good for having an informed worldview.

Regression Analysis - Same as Statistics, though less important for having an informed worldview.

Logic - Whether you call it Philosophy or Math, it's useful in hardware, formal verification, and general thinking.

Linear Algebra - Good for financial technology and machine learning.

Set Theory - Good for understanding databases.

Number Theory - Useful for cryptography.

Graph Theory - Great for networking, and algorithm thinking in general.

Theory of Functions - Good for understanding recursion (LISP, Scheme, etc) and functional programming.

Does anyone need to know all of this? No, but learning the math behind whatever niche you choose will make your intuition and thinking stronger. I think Stats is the only general purpose math that everyone should understand as part of being a critical thinker.

Set theory & basic boolean algebra (De Morgan's laws, etc).