Reddit reviews A Concise Introduction to Pure Mathematics (Chapman Hall/CRC Mathematics)

As someone just finishing their last year of Masters in maths undergrad, A lot of the stuff that you find in The Art of Problem Solving won't really show up until year 2 probably.

Here are the books I used in the summer before starting uni
"How to think like a Mathematician"
Bridging the Gap to University Mathematics
A Consise introduction to Pure Mathematics

Those books were interesting reads for me so I would recommend them. I'll answer any questions you have if you want.

I highly recommend this book here. Got it for my undergrad, it's concise and affordable and covers many different topics so you can flick around, with problems at the end of each section, also very affordable compared to other uni-level books. I don't know what level you are at but I think it's suitable for anyone heavily into maths, pre university/college. Very neat book

I think something like A Concise Introduction to Pure Mathematics is a good place to start. there's a fair bit of number theory in it. imo number theory is a good place to pick up the tools of proofs.

what would you like to know?

I jumped back in after a decade a little under two years ago. I had enough calc that I started in with a mathematical statistics text. There was a ton I had to backtrack on (logarithm rules, basic trig stuff, some basic algebra stuff, proof methods) but as I went, it all slowly clicked together, especially since I took notes and scheduled regular review so once I saw something again, I got to keep it.

Do you have any particular thing you're excited to head towards? 'Math' is a giant area. It helps if you have some practical reason, even if it's just an abstract question or a thing you want to understand. That's my two cents at least.

As for where to start... I like books personally. how to think about analysis is a great place to start. You can read through the whole thing in a few weeks, it's not a terrible investment, but it'll ease you into thinking about what math 'is', why you care, and how to pursue it. If you enjoy Alcock's book, a concise introduction to pure mathematics would be a great followup. It'll still be accessible, but a lot more rigorous and in depth than what you'll get in how to think about analysis. It's written for someone with just a high school level background, and builds a bridge up into thinking in terms of proofs, and goes through a number of interesting results.

Beyond that, there's a really cool thing called the infinite napkin project that you might have fun checking out as well. It's written by an Olympiad coach that struggled with talking about his research to high school students. Math is SO hierarchical, it's absolutely insane, so to get into some crazy topics you might be interested in (quantum computing algorithms) you might need a seemingly absurd amount of background knowledge first (linear algebra, complex numbers, hilbert spaces...) so... the infinite napkin project is meant to be a whirlwind tour through 'higher math' for a fairly accomplished high schooler. It's absolutely not meant to get you functional anywhere (his section on group theory is about 50 pages long. I'm currently working through a text on the topic that's 500 pages) but it DOES give you a good flavor for different topics, and his resources list is excellent, I've been really happy with the ones I picked up that he said he enjoyed. You could go through a chunk of the napkin, see what you're excited by, and then pick a resource yourself to really dig in. I've self studied my way through a number of math texts in the last two years. It can get a little lonely if you don't have any friends that share your hobby (so make some!) but it beats doing sudoku and crossword puzzles, haha. And if you do it for long enough, this weird little hobby can add some serious money to your paycheck if you're already an engineer.