Ask HN: Books or courses to understand college level mathematics? | Hacker News Reader

Ask HN: Books or courses to understand college level mathematics?

93 pointsnoob_eng31 comments

With high school math knowledge of geometry and algebra and quite a bit of programming knowledge, what is the best resource to get started with studying college level mathematics?

Basically I want to learn how to read and write proofs. The main goal is to understand and work through higher math books like analysis, combinatorics, graph theory, abstract algebra, etc.

View on HN

How about interactive books? Like Immersive Linear Algebra by J. Ström, K. Åström and T. Akenine-Möller; or http://immersivemath.com/ila/index.html or Interactive Linear Algebra by Dan Margalit and Joseph Rabinoff https://textbooks.math.gatech.edu/ila/index.html

I suppose they should work well for self-study, but I never tried learning by one myself.

Honestly the best bet is to just read the best books for a given subject. You can figure out which books are used in curriculums by checking syllabi. Seems reasonable that if they are using the books at Harvard/Yale/MIT/Caltech/Stanford that they are probably decent.

I personally like to read a few books on the same subject in a row. It might be helpful to use wolfram alpha, SAGE, or chatgpt to explain concepts or help answer questions. Also buy a solutions manual and use it to check your answers.

As someone who did a math degree at a decent institution, I can assure you learning on your own (assuming you have the discipline) is a lot less stress than taking a class and you’ll likely learn more.

If you’re really motivated, hire a TA in the subject you are studying (online meets are fine). If you need to, just walk into the math department at a local university and ask where you can find tutors for a specific class. Just tell them you’re planning to enroll but you can’t right now.

Or check out a multitude of mathematics communities on discord or slack and ask around.

Book of Proof: https://www.people.vcu.edu/~rhammack/BookOfProof/

Art of Problem Solving for if you need/want to rebuild your pre-college foundations: https://artofproblemsolving.com/store

Cal Newport has great tips on studying efficiently/effectively (and you can implement some of his strategies with Anki): https://calnewport.com/case-study-how-i-got-the-highest-grad...

> Art of Problem Solving for if you need/want to rebuild your pre-college foundations

AoPS will do more than that. It will show you how to think about proving things carefully, an on ramp to the "rigor" of introductory analysis texts.

Huh. I'd heard of the AoPS stuff, but never knew much about it. Based on your comment here, I just ordered their "Basic Algebra" set (book + solution manual). I'll give it a spin and may consider trying other books in their series depending on how it goes. I like the idea of something that presents "high school level" material, but starts introducing the rigor needed for later mathematics. Sounds like something that could be very beneficial.

huh interesting links. I do appreciate the "rebuild the pre-college foundations" part too, IMO I think that's important.

I kinda undersold it, too.

They go much deeper than what you learn in high school. You'll build a foundation strong enough to flourish in any undergraduate math program in the country, including those in the Ivies.

More than enough for a CS program or a working SWE.

> what is the best resource to get started with studying college level mathematics?

> Basically I want to learn how to read and write proofs.

> The main goal is to understand and work through higher math books like analysis, combinatorics, graph theory, abstract algebra, etc.

I feel like these are three somewhat distinct goals, and may need distinct approaches.

If you only have HS geometry and algebra, you may want to fill in some additional foundational knowledge first. If the algebra is what we refer to in the US as "Algebra 1", then definitely do "Algebra 2" / "Precalculus" and preferably calculus as well. Someone else recommended Khan Academy, which is a great resource for this level. Calculus is not strictly required for most of the upper-level topics you mentioned, with the exception of analysis, for which it is a strong prerequisite.

For proofs, check out one of the dedicated "how to write proofs" books recommended by others. You can probably dive right into these, as they rarely require calculus or other prerequisites, as they focus mainly on simple proofs in geometry, number theory, set theory, combinatorics, and other fairly "entry level" topics.

A discrete math textbook is another good way to get started with proofs. These often cover basic topics in logic, set theory, and proof methods (induction, proof by contradiction, etc.). They also typically cover basic graph theory, combinatorics, and sometimes even basic abstract algebra (groups, rings, etc.), which will give you a head start on some of the advanced topics. Rosen's discrete math book is a good choice, widely used in courses at respectable universities.

Once you've covered that material, you should be good to go on the advanced topics you mentioned.

Khanacademy.org has free online courses on AP Calculus and AP Statistics, plus other (high school) math courses.

MIT Open Course Ware has a big bunch of free online math courses. ocw.mit.edu

I found Halmos' Naive Set Theory to be very helpful when first learning about proofs. Don't be offput by the title - the material is fundamental to any working mathematician. It is also quite short.

The question that I think you should be asking is which field of mathematics to learn next. You have high school knowledge, you want to learn how to read and write proofs, I personally believe the right place to go is number theory.

Within number theory, different books and different learning styles are good for different people. I personally like to get a book, see if I like the topic, and if I do then get another book so that I can compare them. You might try one of these elementary books on number theory:

https://www.amazon.com/Elementary-Number-Theory-Second-Mathe...

https://www.amazon.com/Introduction-Number-Theory-MIT-Press/...

https://artofproblemsolving.com/store/book/intro-number-theo...

Combinatorics and graph theory are reasonable too, but IMO it is too easy to learn aspects of those subjects without really biting off the "how to handle proofs" thing. Analysis and abstract algebra make more sense after number theory first - number theory is basically "pre abstract algebra".

I hear good things about this book “how to prove it”

I also like Knuth’s art of computer programming book which is more about proving correctness of algorithms

In general, stuff like combinatorics, algorithms, etc I find to have a low barrier to entry, so it’s a nice playground

https://www.cambridge.org/highereducation/books/how-to-prove...

Be aware that this a longish journey.

Acquiring selectively mathematical skills at college level is complicated by the fact that there is quite a lot of interdependency between subjects. This goes beyond proofs. You'll be stymied by notation and assumed background knowledge.

College curricula obviously solve this somehow, so check a few of them out, and work backwards from your target subjects (eg graph theory) to find all the prerequisites.

After you’ve had a chance to learn single-variable calculus and introductory linear algebra, I highly recommend two books for “second courses” in these areas:

1. “Vector Calculus” by Baxandall and Liebeck is excellent. Begins with a semi-formal review of single variable calculus before going onto a mix of linear algebra and multi variable calculus.

2. “Linear Algebra Done Right” by Axler is one of my favorite books in any field. Might be challenging to people completely new to proofs or linear algebra but starts with pretty simple foundations and builds up an excellent rigorous foundation. I enjoyed this way more than other intro LA books but needed to study more applied LA first to be able to read this. Another approach is to start with abstract algebra and then go into the Axler.

I self-studied both of these texts many years after college and had forgotten pretty much all my math except some high school stuff. Supplemented by YouTube videos, Khan Academy, I felt pretty confident and eventuallY fell in love with both of these texts.

2 books that are worthwhile: "A mathematical bridge" (an introduction to the lie of the land in higher math) and "Chapter Zero" (a review of material not taught in school, but wich is so basic that most textbooksjust assume you know, e.g. bijections and proof by induction)

Consider inquiring at your local community colleges about classes, they tend to work with the assumption that a good portion of their students already work full time, and some of them offer online math classes.

If you aren't interested in credits, you can also just audit the classes.

Someone has already mentioned Susan Rigetti's list [0]. I also recommend Mark Saroufim's list [1].

In addition, I will recommend to you "Proof: A Long Form Math Textbook" by Jay Cummings. This is a really nice book which I can recommend highly.

Also, look at "A Programmer’s Introduction to Mathematics" by Jeremy Kun [2].

[0]: https://www.susanrigetti.com/math

[1]: https://marksaroufim.medium.com/technical-books-i-%EF%B8%8F-...

[2]: https://pimbook.org

I suggest you start out with a course or book on formal logic in computer science. In particular doing proofs in intuitionistic logic. You can use a simple proof assistant to verify your proofs and it will help you really understand how proofs are done and how they are just a kind of programs in a special programming language called mathematics (given some assumptions). Proofs in higher level books are all informal, so you can’t do them programmatically in most cases, but it really helps with understanding how proofs works. After that, follow what subjects finds your fancy.

I'm also working on this. The book I bought is https://longformmath.com/proofs-home

https://github.com/TalalAlrawajfeh/mathematics-roadmap

So this is exactly the thing that I hated so much.

Click into the book list and you realize that there is at least 6 book on single topic and 90% of their coverage and explanation is literally the same. Then you have to dig every corner of those books to find out what you are missing.

You can never get a definitive edition of a math textbook that cover all topics and achieving teaching & show the formal proof simultaneously.

I can recognize 10~books on the list, and they are all not good. They just hide the most importance topics and proofs in exercises and of course no solution or guide provided. Good luck if you can learn any thing from math texts.

What are those not so good books?

This looks awesome! I've been wanting to make something that like myself for a while.

Personally I can't self study math. I've always had to have help and feedback on my blind spots. Find a study group or partner.

I like this guide

https://www.susanrigetti.com/math

>Basically I want to learn how to read and write proofs.

Introduction to Mathematical Think by Dr. Keith Devlin on Coursera:

https://www.coursera.org/learn/mathematical-thinking

Paul's Notes https://tutorial.math.lamar.edu/

Farleigh Algebra Mazur Combinatorics West Graph Theory

see also https://news.ycombinator.com/item?id=35866068 - asked a couple hours after you asked this

Nothing beats mathacademy.com

"the Haskell road to logic and mathematics"

Bill Shillito | Introduction to Higher Mathematics (YouTube lecture course)

Richard Hammack | Book of Proof (pdf book) - https://www.people.vcu.edu/~rhammack/BookOfProof/

Short book: Groups and their Graphs

Taylor Dupuy | Fundamentals of Mathematics (YouTube lecture course)

Silvanus P Thompson | Calculus Made Easy (html book) - https://calculusmadeeasy.org/ (This shouldn't be your only exposure to Calculus. It is more for building intuition.)

Gilbert Strang | Highlights of Calculus (YouTube lecture course)

Josh Starmer | StatQuest (Short various statistics videos) - https://www.youtube.com/c/joshstarmer/playlists

Bob Franzosa | Introduction to Topology (single public lecture) - https://www.youtube.com/watch?v=zsN_guq__Ac

Socratica | Abstract Algebra (short videos)

MIT Calculus Revisited (Single Variable Calculus): https://www.youtube.com/playlist?list=PL3B08AE665AB9002A

MIT Calculus Revisited (Multivariable Calculus): https://www.youtube.com/playlist?list=PL1C22D4DED943EF7B

MIT Calculus Revisited (Complex Variables, Differential Equations, Linear Algebra): https://www.youtube.com/playlist?list=PLD971E94905A70448

Matthew Macauley | Visual Group Theory, Differential Equations, Discrete Mathematical Structures, Advanced Linear Algebra, and Advanced Engineering Mathematics (YouTube lecture courses)

Open University (BBC) | Geometric Topology (YouTube lecture course)

Joel David Hamkins | Philosophy of Mathematics (YouTube lecture course)

Marco Taboga | Probability and Statistics & Matrix Algebra (html book, need calculus) - https://www.statlect.com/

On YouTube you can literally watch a good lecture course for just about any typical undergraduate course. You just need to know where to look. Also there are even some really good master's degree courses out there.

Of course the only way to really learn the mathematics deeply is to "learn by doing", aka problems and proofs.

Other than the usual big American universities another good source from India is NPTEL (https://nptel.ac.in/course.html).

For somewhat more entertaining short lectures try:

Grant Sanderson | 3Blue1Brown - https://www.youtube.com/c/3blue1brown

Brady Haran | Numberphile - https://www.youtube.com/c/numberphile/

Tai-Danae Bradley, Gabe Perez-Giz, and Kelsey Houston-Edwards | PBS Infinite Series - https://www.youtube.com/c/pbsinfiniteseries/

Raymond Flood (YouTube public lectures at Gresham College) | History of Mathematics

There are a ton of channels starting to pop up like Grant's 3B1B (I find like a new one every week). He had a contest recently so maybe look at some of the winners.

This is pretty useful if you get into higher mathematics:

Math Vault | The Definitive Glossary of Higher Mathematical Jargon - https://mathvault.ca/math-glossary/

Some books:

There is Eugenia Cheng's book: The Joy of Abstraction

See this list its pretty good: https://abakcus.com/30-best-math-books-to-learn-advanced-mat...

P.S. I created some Youtube lecture aggregation channels:

https://arisbe.carrd.co/#section03

Reply on news.ycombinator.com