Basic Mathematics

What Hacker News says

1. seanhunterOct 2025
   I would strongly recommend getting, and working through Serge Lang's book "Basic Mathematics" for people who want to self-study what is normally considered "basic maths" (ie the stuff you might have covered in high school plus some of what in the US is called "college algebra" (in the UK and Europe that is just covered in high school and "algebra" at university generally means abstract algebra.

   I did it to get my very rusty high-school maths back up to snuff before starting to self-study for a maths degree and it helped a lot. The problems are really excellent and since it's Serge Lang, he treats you like a mathematician right from the beginning even though he really is doing basic stuff.

   Show morethread
2. rramadassOct 2025
   There is no one book which can give you the overall sweep of Mathematics. However, you might find the following (though not textbooks per se) useful.

   1) The Princeton Companion to Mathematics by Timothy Gowers et al. and The Princeton Companion to Applied Mathematics by Nicholas Higham et al. - The closest you have to a Modern Encyclopedia of Mathematics. You get unmatched breadth after which you can move on to dedicated books as needed. Well worth the money.

   2) Mathematics: Its Content, Methods and Meaning by Aleksandrov, Kolmogorov et al. - Absolutely brilliant overview of Basic Mathematics. Published by Dover and hence very affordable.

   3) Elements of Mathematics: From Euclid to Gödel by John Stillwell - Written as sort of an update to the great Felix Klein's Elementary Mathematics from an Advanced Standpoint books. The Topics are particularly well chosen given modern advances; they include Arithmetic, Computation, Algebra, Geometry, Calculus, Combinatorics, Probability, Logic.

   Show morethread
3. nextosJul 2024
   You can start simple. Read Basic Mathematics by Serge Lang and do all exercises. Solutions are included. That book basically covers all mathematics up to junior high in a rigorous but approachable fashion. Serge Lang was a great mathematician. Then you move to logic, calculus, linear algebra and probability. Afterwards, focus on more specific areas that interest you.

   Springer Undergraduate Texts in Mathematics and Dover have lots of elegant and concise textbooks that can help you. At the beginning, the key is to move slowly and build some solid foundations.

   Show morethread
4. nextosJan 2023
   These are good recommendations, but I think beginners tend to burn out due to the lack of a structured program and/or exercise solutions if they are trying to study on their own. The simplest structured program I can think of that satisfies both is:

   * Basic Mathematics by Lang. Covers basic algebra and geometry at high school level.

   Then one of these two, depending on your interests, or both:

   * Vector Calculus, Linear Algebra and Differential Forms by Hubbard and Hubbard. Takes you through linear algebra, single-variable calculus and multiple variable calculus. Analysis is discussed in an appendix. All proofs have a constructive bias, so it's very algorithmic and natural for a CS-minded student. Solutions are in a separate volume.

   * Program = Proof by Mimram. Discusses logic and computation, and takes you from the basics to depedent type theory and beyond. Uses OCaml and Agda. Freely available at: https://www.lix.polytechnique.fr/Labo/Samuel.Mimram/teaching...

   Show morethread
5. dr-neptuneSep 2022
   A book I always recommend to folks looking to learn mathematics with lots of beauty and from first principles is Serge Lang's Basic Mathematics.

   The book starts from axiomatic arithmetic and works all the way up to what you would need as a mathematics major with a focus on pure mathematics. He also manages to touch some beautiful areas like geometry, abstract algebra, symmetry, linear algebra, set theory and more.

   There are only a handful of exercises at the end of each section and they are very good at locking in the concepts. I finished a mathematics degree and realized afterwards that there just wasn't enough of a focus on the concepts and beauty. Even with 4 years of math experience, this book still managed to open my eyes

   Show morethread
6. nextosJun 2022
   Yeah, read Basic Mathematics by Lang. Covers all precalculus you need in a rigorous way.
   thread