Books for Learning Mathematics

Hi everyone I recently did a video about books for learning physics and as a follow-up to that I guess I wanted to do a video about books that I’d recommend for learning mathematics. Now there are so many math books out there, textbooks and otherwise, and so the ones that I’m going to recommend to you are just the ones that I have encountered and that I’ve read and that I didn’t think were terrible but there are going to be so many others out there that might be just as good or even better. It’s really hard to have a grasp of all math literature that’s out there, but it’s sort of my opinion that, especially if you’re just looking to learn mathematics, then it doesn’t really matter too much which introductory book you pick up, or, on any of these topics I’m going to mention, pretty much the best book for you to get is going to be the one that’s available at your local library, or that you can get access to. Don’t go too far out of your way to try and get a particular book because I think concepts like calculus or linear algebra, you know, they’re so broad that any book that claims to cover them will probably do a decent job Now as a place to start I want to recommend a reading list put out by Cambridge University – and it’s their mathematical reading list. This is a document that’s intended for students wanting to go to Cambridge and study mathematics, and it’s, I think, a really good list that contains books about maths and it really contains everything. It includes books about how to study maths well; books about the history of mathematics to give you a lot of context for the idea as you’re going to learn; there are some books about theoretical physics and the maths behind that; and there are what they class as readable textbooks. So, all the books on this list – which I’m going to link first in the description – I think are really good for someone who’s trying to learn math and wants a well-rounded idea of what math is and how you study it and how you get into the mind set of learning math. So I think that’s really good, check that out. And I do have a list here of some of my more specific recommendations And yeah, I’ll start off again with some general or ‘just for fun’ math themed books which are for… in case you’re looking for maybe a few books to get inspired about the idea of maths or you just want some fun reading. So… One of my fun reading math books would be Fermat’s Last Theorem by Simon Singh and I think that Simon Singh is a really good science communicator He also wrote a book about the mathematics behind some Simpsons episodes and he’s got some astronomy and physics books too. It’s a really good book into a little bit of a history of mathematics, but also some of the fun of math, and about proving things and about numbers themselves Also, I’m going to mention a book called Flatland by *Edwin Abbott and this is like a real old classic book and it’s not really about maths apart from the fact that it’s set in a mathematical universe on a ‘flatland’ which is like a 2D landscape where the main character is a square and it’s talking about mathematical ideas like lines and polygons and shapes, and those are all the characters, but it’s actually I guess commenting on politics and social constructs all through the lens of like a very nerdy sort of mathematical view. I thought that’s just… I’ll put it on there as a fun book. Another one that I’ll mention in this category would be ‘A Mathematicians Apology’ by G.H. Hardy and this isn’t really a book as much of it as an essay and It’s just something that I encountered multiple times during my math studies because a lecturer would recommend you read it or something, and it’s like an essay from this mathematician talking about what it’s like to be a mathematician – the mindset of being creative in that context – and I guess how to be in the mindset. It’s like his reflections on his career as a mathematician. I think it’s just maybe a useful thing if you’re hoping to head in that direction. Okay, so moving on to my list here, starting with calculus. So again, like I said I think pretty much any calculus textbook is going to be fine if you’re just wanting to learn calculus. The specific textbook I use during university like for second year studies was ‘Early Transcendentals’ by James Stewart, and this was just like the standard textbook for us at my university, and I thought it was fine. It covered like all the concepts that I really needed to know from my course. It was bearable to understand. So I guess I’d recommend it. I actually remember buying my copy of this book from one of those like textbook exchange sites and I met up with this random stranger at the stairs of my university library. He pulled out the textbook from his bag I pulled out some cash and we exchanged in I guess what was like a math majors drug deal. Another book often recommended in this category is ‘Calculus’ by Michael Spivak. So moving on to the category of linear algebra. The one I read and I guess I recommend because I thought it was good was ‘Elementary Linear Algebra’ by Howard Anton. Maybe I have a very biased liking of this book because I remember using it to study for a linear algebra test that I did really well in. But I also thought that of all the linear algebra books I encountered, this one seemed to do things reasonably intuitively, and I know it covered ideas by first giving clear definitions of them so you were never too confused. OK Differential equations is a really big category in learning math. You’ve gotta start with learning ordinary differential equations, then partial differential equations But what I’ve found in all these differential equation courses is the lecturer it tends to sort of write up their own course notes. Which you can probably learn from, completely self-sufficiently without needing a textbook. It just seems to be that since differential equations are such a big area that if you’re doing a course on them, often you will have some sort of at least recommended reading from your lecturer or they’ve written their own set of notes to go along with it. But some books that I’d say are alright are ‘Partial Differential Equations – An Introduction’ by Walter Strauss. And also another online resource, which is called ‘Mathematical Tools for Physics’ and it’s by James Nearing and I’ll give the link to this but it’s actually a completely free online PDF of this guy who’s written up a bunch of notes on math used for physics and because differential equations are so often used in physics that’s a big part of this like online textbook. I’d say check that out if you’re looking to learn not only differential equations, but a lot of these more physics or application based ideas in maths. One more book I specifically want to mention is for complex analysis, and it’s called ‘Visual Complex Analysis’ by Tristan Needham, and this book claims to give a very intuitive explanation of complex analysis – more so than I’ve seen anywhere else. I guess complex analysis is maybe an inherently unintuitive topic sometimes because you’re dealing with the imaginary numbers and you’re dealing with all these ideas and results and theorems that come out of imaginary numbers, and they can seem really sort of strange and like they just came out of nowhere. So this book, I’d say, is worth reading if you want to actually understand imaginary numbers, and not just be satisfied with saying ‘oh, they are weird and crazy, they’re imaginary’, but actually I guess understanding that area of maths. Now in my last video about the books for learning physics some of you guys left really awesome comments detailing like further book recommendations you have, and even some recommendations for math. So I’m going to read out some of the most recommended books that I saw in the comment section of that video. These are books I haven’t personally read myself, but I know I trust you guys that if you’re recommending them, they’re probably good. So a shout out to ‘Principles of Mathematical Analysis’ by Walter Rudin; ‘Analysis One’ by Terence Tao, the famous mathematician there; ‘Algebraic Topology’ by Allen Hatcher; ‘Mathematical Methods and the Physical Sciences’ by Mary Boas, and I think that’s actually possibly the only book I recommended on here and even in my last video written by a woman, which is kind of sad but at least I’ve got one to include; ‘Abstract Algebra’ by Dummit and Foote. I think that’s actually quite a classic book for abstract algebra; ‘Discrete Math and its Applications’ by Kenneth Rosen; and ‘How to Think Like a Mathematician’ by Houston. And I saw that that last one was recommended if you’re wanting to learn more about formulating proofs and the idea of proofs. And that said, those are some of my recommendations, But I’m sure you guys have even more recommendations than I do, so I’d love you to just sort of talk amongst yourselves in the comments and share some of your further experiences with these books. Also good places to get like really specific book recommendations or reviews are on the math subreddit or even on like math Stack Exchange. People are often talking there about their experiences with certain textbooks, and like I said, there are so many textbooks out there, and they’re often so large and take so long to read that for one person to understand all that’s available is pretty hard Thanks for watching this video, I feel like talking about math textbooks must be inherently one of the most boring things someone can do so, I mean, I’m really grateful that you guys are interested in this content and I’m always open to more ideas. So let me know what you think in the comments and hope you have a good day 🙂