Welcome to this guide, which aims to give an introduction to the mathematics behind the numeric hierarchy of type classes in PureScript’s Prelude, aimed at people who haven’t (necessarily) studied mathematics beyond a high-school level.


Beta. There are still quite a few chapters I want to write. In particular, the following are currently missing:

  • General principles for formulating these mathematical structures as PureScript type classes
  • A glossary
  • A diagram with an overview of this hierarchy


Normally, algebraic structures like rings or fields are only introduced to students at undergraduate level. One unfortunate side-effect of this is that lots of the material currently available on the web which describes these concepts is sometimes a little inaccessible for people who haven’t studied mathematics past a high-school level. My aim with these posts is to help people develop intuition for what these structures are and how they can be used so that that knowledge can be applied in PureScript code. I also hope that I can help persuade you of the beauty of mathematics and convince you that it is worth studying in its own right.

I want to stress that it is not necessary to read and understand all of this in order to be able to use the PureScript type classes like Ring or Field, and to be able to write functions which work for any type which has a Ring or Field instance. However, I do hope that it will help you answer questions such as:

  • “I want to write a function which works for many different numeric types, but should I give it a Semiring constraint, or a Ring constraint, or something else entirely?”
  • “I have written a function with a Field constraint, and I want to find an appropriate concrete type which is a Field to test it with. How do I do that?”
  • “What’s the point in all of this maths mumbo-jumbo anyway — what’s wrong with plain old Haskell-style Num?”


I will try to assume as little knowledge of mathematics as I can. If I accidentally assume knowledge of something which makes you unable to understand a part of this guide, please let me know by opening an issue on GitHub or emailing me at

Although this guide is primarily aimed at PureScript users, I will only reference PureScript infrequently for the purpose of illustrating examples. This guide is really about mathematics, not PureScript.

Therefore, as far as is reasonably possible, I am also interested in making this guide accessible to programmers using other languages or libraries which make use of these same abstractions (rings, fields, etc). If you fit into this category, and you are unable to follow something I’ve written because it requires more than a very basic level of PureScript knowledge, please feel free to file an issue.

How to read this guide

I will provide exercises throughout. Whenever you encounter an exercise, I strongly recommend you attempt it before reading on! I speak from experience as a maths student: in my personal experience, it’s simply not possible to reach the same level of understanding without having worked through problems myself.

I should note that I often find it extremely tempting to skip to the solution, read through it, and tell myself “yes, I could have done that.” Be careful of this! It’s very easy for me to persuade myself that I could have solved a problem when in fact I probably wouldn’t have been able to. But also it’s okay to look at the solution if you’re really stuck; attempting the problem first is the most important thing.

If you get stuck on an exercise for more than, say, 10 minutes, it’s okay to skip it or simply look at the solution (although if you find yourself needing to skip lots of exercises, perhaps consider going back and rereading some earlier bits). Another good idea if you get stuck is to do something else and come back to the problem the following day — of course, if you’re a programmer, you might already know this.

One more thing I will say is that you shouldn’t expect to be able to read this sort of material anywhere nearly as quickly as you might read most other types of non-fiction prose. Mathematical writing is usually extremely dense — I don’t mean this as a criticism of the writing style of mathematicians, but rather to help avoid unrealistic expectations. In fact I think this density is a mostly unavoidable consequence of the nature of mathematics. Don’t be put off if it takes you a long time to get through this!


This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

This means you are free to copy and redistribute it as well as make changes, but you must give credit, link to the license, and indicate if changes were made. The license also forbids commercial use.

Note that the work is not necessarily exclusively licensed under CC BY-NC-SA 4.0. In particular, if you’re worried about whether your use of it counts as a commercial use please contact me and we’ll probably be able to sort something out.