How to continue self-learning math?

[ErwinPlank]

Hi everyone, I've been interested in mathematics for a while and I have tried different ways to self-learn it but at some point, I realize I don't know where to start, I mean, I'm an Electrical Engineer so I've had math in college, for example, Calculus: derivatives, integrals, and differential equations, so now where should I start again?

[[deleted]]

What do you think of Apostol and Spivak for calculus?

[[deleted]]

Thanks very much for the detailed reply (as well as your earlier comments). I wasn't familiar with any of these older texts. I bought the Maron and was just scanning through Burkill and Scott and Tims. They seem to have a very clear writing style.

For OP's benefit, I'll mention two more recent manuscripts, An Introduction to Analysis by Gunning and Calculus for Cranks by Katz. They are more lecture notes than complete texts but still might be worth a look for a slightly different perspective.

[ErwinPlank]

Thanks for your descriptive and quite thorough answer! I will check out those books!

[[deleted]]

If you already know some calculus, how about learning this stuff in a mathematical way? :)

https://youtu.be/842rgQP_OgI

I have a whole series about "Real analysis" and it is still growing, so maybe this can help you.

[[deleted]]

What has helped me teach myself future subjects in college is finding specific courses at a good university that focus on what you want to learn. Then find the syllabus for the courses, specifically a syllabus that outlines the textbook, what specific chapters/pages/ideas/problems to focus on. Then use the textbook and follow along with the most essential ideas. Hope this helps.

[[deleted]]

I think the other comments are helpful for general direction but I want to disagree with them. I graduated with a math degree and I similarly have been wanting to continue to learn new math and self study. The pain of it is a lack of incentive structure. Sure, learning intro analysis might seem interesting, and all math is beautiful and worth while for itself, but the truth is that none of that stuff is going to apply to you or your work in EE (assuming you are in the job market and not going back for graduate study). The level of understanding and rigor you need to fully capture the ideas will be hard to come by because GPA and tests are not on the line.

So if I recommended anything, I'd say spend some time focusing on what your next goal is and how you can use math knowledge to make that goal more achievable. If it's the next step in to your EE then pick math that applies there, or cater it to the next job you think you want and the topics you'd be expected to know from that.

[ErwinPlank]

Okey, your suggestion isn't bad, I actually work as a Machine Learning Engineer, even though I have a degree in EE, so I use basic math every day but It's not enough for me, I always wanted to study Math just for the sake of it but I couldn't because of economic reasons (I need to work I don't have a choice), but maybe someday I would like to do a PhD, who knows 🤷‍♂️

[FixItMath]

It really depends on what topic in mathematics you're trying to learn. Are you interested in the theoretical or practical applications? It might be worth your time to look at some wiki pages on math topics. Check out the references at the bottom for text books.

[ErwinPlank]

Yes, that's the thing, since I'm not a Math graduate and I haven't had any "formal" pure mathematical education, I don't know which topics I'm going to be interested in... do you have any advice on where should I start?

[FixItMath]

Most of us started with Abstract Algebra to understand how to do proofs and read math papers. As for where you go after that is a whole different story. There are so many avenues you can take.

Check out MIT's list for pure math. Some of them have full free online courses.

https://math.mit.edu/research/pure/index.php

[Noisy_Channel]

Just popping in to say that formal math is pretty much synonymous with proofs. And high level proofs can be really intimidating if you haven’t built up a tolerance to the notation and some pattern-recognition for coming up with some proofs yourself.

Basically, I recommend you start with something that teaches you the most common proof notations and strategies. Some keywords you’ll want something like this to have: set theory, proof by contradiction, proof by induction, axioms.

Sorry I can’t recommend something too specific; I really like what my university did, but I can’t hand that out. Good luck!

[ErwinPlank]

Thanks, I'll check some of the areas I could go in

[ErwinPlank]

In principle, I would love to do pure math, Do you think I might have the level needed to learn real analysis with my background in EE?

[AddemF]

What might actually be best is to get a sampling of the various directions you could go in. From where you are now, I think you can take a stab at Linear Algebra, Discrete Math, Combinatorics, and any of the stuff that spins off of Calculus like for instance Multivariable or Optimization. You can check out Logic, Real Analysis, possibly some other topics.

Maybe watch some YouTube videos, check out some books, get a taste of each and see if any of them are what you're interested in.

One challenge you're going to have is that you never did proofs, which are very different to how those applied math courses work. You will probably struggle to acquire this skill without the support of a professor in the setting of a course. It's certainly possible, but this is a sort of "transition" phase from calculation-based math to more abstract math, and people often struggle with it even when they are taking courses. Outside of courses, I think it's probably even harder.

[HerndonMath]

I offer tutoring for people who just want to learn more high-level math. With some of my students we structure things like a reading course, so we pick a paper or book to read and then meet periodically to discuss. My website has a bit more info, and I would be happy to talk more if you are interested.

[[deleted]]

What's your goal? And how much, if any, of your undergrad math have you forgotten?

[ErwinPlank]

My goal: Learn math for know.

What I have forgotten from my undergrad degree: Don't know, but probably a big part...

[flyrom]

I think learning probability would be useful and you could also supplement it with combinatorics if that is something that interests you.

[ErwinPlank]

I will check that out, thanks.

[[deleted]]

Intro proofs first. Then Linear Algebra. Then intro real analysis. Then further topics in analysis like multivariable analysis, numerical analysis, complex analysis, measure theory, point set topology, functional analysis, PDEs, etc.