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I like referring to set theory texts as "the bible", more specifically Hrbacek and Jech just because I was raised on it. I also used to have "The 10 commandments" written on my wall, followed by the 10 axioms of set theory. (Ok, so there's like 7 `core' axioms, but I tossed in some extras like the axiom of foundation to make the word play work).

Linear Algebra Done right by Sheldon Axler.

Sterling K. Berberian's books are super clear.

After using Spivak as the primary text in Freshman Calculus, I still go back every few years and re-read parts of this book for pleasure.. It presents core Calculus concepts and also introduces basic ideas about real analysis. Is this book well known among Mathematicians? Does anyone know about the Author? He seems to keep a low profile.

Calculus by michael spivak :)

Concrete Mathematics by Ronald L. Graham, Donald E. Knuth, Oren Patashnik. I dropped the course that used this textbook in the second week, but decided to keep the book.

I think there are parts of it that are actually quite well done, but in a way that makes the whole project worse, because there is reason to like some of it getting students to swallow the whole thing.

I think the worst part about the book (going to the myopia you mention) is the miseducation of a whole generation of students about determinants. Determinants (and the exterior product if we did not choose base vectors) is literally one of the most if not the central object of all of (multi)-linear algebra and connects the field to so many other topics, and by claiming it's unnecessary and unintuitive students who are educated this way are worse than served badly. Sadly the book is a massive success so the damage is quite substantial.

Seems like a lot of fields have this. Kechris's book is widely regarded to be the Bible of descriptive set theory (and I know a few people in the field who refer to the man himself as the area's God).

What does that make Eisenbud's Commutative Algebra?

It sounds to me like you might be looking to know more about the culture of mathematics. Ironically, for this, math textbooks may not be the best place for you to look! The cultural aspects of mathematics are (in my opinion) just as important as the technical aspects, and much of the former can be lost in expository texts. This includes such things as motivation for a topic, or perhaps the original approach taken by the person who first popularized a concept or topic.

My recommendation is to pick up a math history book: This might help glean some insight into why people worked on the topics they did in the first place. One of my personal favorite is the book about Srinivasa Ramanujan, titled The Man Who Knew Infinity.

If I have to recommend an actual math textbook, I will say to work through Baby Rudin. It's basically Calc 1–3 with tons of proofs and rigor, plus some extra stuff you've probably not seen yet. If you can understand that book and do even some of the exercises, it will give you a strong foothold in getting a "big picture" view of mathematics.

This might be a bit unusual, but I would also recommend Introduction to Mathematical Thinking by Devlin. It starts with history of math and how it evolved from arithmetic to abstract logic. It is a bit basic, but maybe you will give it a try.

While this isn't a book, the youtube account 3Blue1Brown has a calculus playlist with great visuals that can help you understand why stuff works. I'd definitely check it out.

For linear algebra, i heavily recommend you read a book meant for mathematicians, such as serge lang's take on it. (Even 3b1b's series was in a way bounded by the coordinate plane)

I've accidentally picked up a linear algebra book for engineers, and it was not useful at gaining any deeper understanding, only calculative tricks. They were definitely helpful, but it'll only make you hate linear algebra more

Calculus: Early Transcendentals is the only cal book I can recommend because it’s the only one I’ve read.

Prof Leonard on YouTube explains concepts very clearly if you’d like to try that. Really gets to what is going on behind the calculations

I’ll second u/amdpox’s recommendation of Understanding Analysis. And you can get it for free right now because it’s one of the free textbooks that the publisher is offering during the pandemic!

Is there any book you could recommend for this? I’m in almost the same boat as OP, just finishing Calc II instead of III (and while I don’t necessarily feel lost in any way, I would really like to gain a deeper understanding of the content).

Also gilbert strangs book is literally the best linear algebra book there is. I read it for fun. Seriously good.

Charles Pinter's A Book of Abstract Algebra.

The math I knew was limited to what a CS sophomore would have taken (calculus + a discrete math course). When I picked up Pinter's book, it was at the right level for me and written in an inviting way. The topics were completely new to me, so nearly every page kept me glued to the book. This book single-handedly introduced me to and made me love and pursue the world of (abstract) algebra.

Not exactly a textbook, but the Mathematician’s Lament by Paul Lockhart. I can read it over and over again.

Spivak's calculus. In depth analysis of all the basic concepts in an amazingly simple way.

All of the calculus books by James Stewart... That man had a talent for simplifying mathematics AND making them beautiful and interesting. I highly recommend them!

None, but LADR comes close.

[Edit] Linear Algebra Done Right by Axler.

I wish. Axler published solutions for the 2nd edition but none yet for the 3rd.

I know that I should have confidence in my own proofs but as my first attempt at proof based math it would help to have proofs known to be correct and complete to compare to.

Springer is giving free access to download this book right now...

https://link.springer.com/book/10.1007/978-3-319-11080-6

That's something I only discovered recently. I've downloaded all of them.

I used Lie Groups: Beyond an Introduction as a Ph.D. student, and Basic Algebra and Advanced Algebra have been go-to texts since I was given them.

I haven't read his analysis books, but I'd suspect that they're similarly excellent.

Michael Spivak's Calculus. I have still not found a mathematics book with such generosity to the reader. The way each chapter flows into the next often gives me the feeling of reading a novel rather than a textbook.

I was gifted Knapp's Basic Algebra and Advanced Algebra by a professor who retired. I was also gifted Boas' A Primer on Real Functions by another professor who retired.

I also have my mother's copy of Thomas & Finney's Calculus and Analytic Geometry (5th Edition) from when she was in university.

Papadimitriou's "Computational Complexity". The first book I read from cover to cover during my undergrad (and man, do I love the cover!) And one of the few books that I did fell a sense of discovering the proofs along reading them, nothing felt magic, nothing felt pretentious.

Definitely my favorite textbook author so far. Problems in ISM were a little bit easy, but then I read his Riemannian manifolds book and the problems in that book kicked my ass (though they had a great range of difficulties).

Lee's intro to smooth manifolds never stops being useful for me. Also bleecker's gauge theory and variational principles

Had a professor that let me take a course in stochastic calculus meant for PhD students last semester while I was still just an undergraduate student. After one of the lectures he invited me to his office for lunch and to talk. I said that I was interested in pursuing mathematics academically and he gave me advice on what to do and his own experiences. At the end of the discussion he gifted me “Stochastic Differential Equations and Diffusion Processes” by Ikeda and Watanabe and a book on SPDE’s and tells me that If I can read and understand them that I would be prepared to pursue a career in mathematics. Since then I’ve read both books, and bought 5 new ones, and stochastic calculus has become my favourite field of mathematics

https://www.gutenberg.org/ebooks/38769

A Course of Pure Mathematics by GH Hardy

E: This is not an easy book so maybe give it a look just to get an overview.

Calculus by Spivak and Linear Algebra Done Right by Axler are what my pure math courses covered in first year.

Google "dec41 notes" and click on the notes for the Number and Sets course (and anything else that interests you)

Here are some helpful math textbook resources

I'm sure your program will be dense enough not to leave time for reading additional books.

If you want to prepare as much as possible before starting, you could try to build your intuition based on what you will study.

For example, if you are going to study linear algebra in your first year, then a good visual introduction may help you graps a lot of the ideas before being exposed to the formalisms. The introduction to this book seems appropriate. People also swear by the "essence of linear algebra" videos.

It will be much easier to give a recommendation if we know what you have studied and what you are going to study.

The guy who wrote those notes is one of my close friends. He typed them when he was an undergrad at Cambridge.

So I did a ton of self-study/relearning of Math after I realized that I really enjoyed pure math but hadn't really paid attention to the basics. As an adult, I found it hard to find material that didn't feel silly and was also engaging.

Here is a rough set of things that I recommend.

• Gelfand's Algebra. Just as /u/OompaLoompaAssGlands says, the book has a lot of interesting exercises that are helpful, engaging. It is also designed for self-study so there is no assumption that you are following this with a teacher.

• AOPS: Pre-Algebra/Algebra/Intermediate Algebra. Now, I have found these books to be phenomenal in terms of both discovery based learning and also in terms of teaching you how to problem solve. The exercises are hard but have been truly phenomenal in terms of training me to problem solve. I recommend them especially if you wish to learn material while learning how to problem solve.

• H. Wu's books Professor Wu writes books designed for teachers that are intended to provide a rigorous take on elementary Math. Now, I only recommend these if you are far more interested in the why. It made sense to me because like I said, I had a sort of a weird path into Math starting by taking a bunch of abstract Math classes where I studied proof techniques, so from that perspective Wu's work was good. The exercises are fairly easy though.

The Art of Problem Solving (both volume 1 and 2). Amazing books with corresponding solution manuals you can pick up.

Gelfand's Algebra was reccomended to me, it starts literally at the most elementary level of basic addition, but asks clever questions that require that you fully comprehend those essential basic concepts before moving on.

A group of things I sometimes share, a list of free resources

Understanding Math

http://www.sumizdat.org/arith_6_8.pdf A fine basic explanation of mathematical abstraction

Why is Math Important Part 1

Why is Math Important Part 2

The Importance of Mathematics 8 Parts from this thought provoking post.

http://www.themathpage.com/

http://algebrafree.com/

https://archive.uea.ac.uk/jtm/contents.htm/

https://mathstrek.blog/contents/

The Craft of Word Problems

http://whyslopes.com/index.php/4020Volume_1_Elements_of_Reason/

http://whyslopes.com/index.php/5020Algebra_Starter_Lessons/

http://www.shelovesmath.com/

http://betterexplained.com/articles/category/math/

http://www.reddit.com/r/explainlikeimfive/comments/1rsqkz/eli5what_is_math_and_why_does_it_work/

http://www.youtube.com/user/mathinreallife

http://www.whenwilliusemath.com/

http://en.wikipedia.org/wiki/Language_of_mathematics

http://en.wikipedia.org/wiki/Outline_of_mathematics

Two sites worth digging around on http://math.stackexchange.com , matheducators.stackexchange.com

And some deeper thoughts on the subject here http://philosophy.stackexchange.com/questions/tagged/philosophy-of-mathematics

Precalculus by blitzer

I found Charles Pinters a book of abstract algebra to be fun, also either Janich or Munkres topology!

In some respects I have a very strong love of Oyestein Ore's "Number Theory and Its History." I read this book in high school (it is highly readable in general with no background requirement) and it is a major influence on what made me become a number theorist.

Concrete Mathematics by Graham, Knuth, and Patashnik is pretty amazing.

Nonlinear Dynamics & Chaos, by Steven Strogatz. It’s informative, interesting, and clearly written with excellent analogies and applications.

Sure. I don't think it is the best introductory number theory textbook out there for any audience. Some of its advantages are precisely because it assumes so little background that a high school student can read it. But it doesn't cover a lot of things we'd expect a general basic number theory course to cover; for example, there's basically no coverage of quadratic extensions, nor does it cover quadratic reciprocity, and the coverage of continued fractions is not great either.

In that regard, it is a bit unlike say Hardy and Wright which I mostly read a few years later while also in high school, but consult regularly as a reference work. I don't think I'd ever use Ore's book as a reference that way.

But if one has no exposure or minimal exposure to number theory, it is absolutely a great place to start.

You need some background (differential equations [and therefore calculus and linear algebra]), but that's the kind of stuff you'll learn as an engineering student. He gives you a full overview/introduction, so you don't need any background in dynamical systems.

It's more pop-sciencey, but Douglas Hofstadter's Gödel, Escher, Bach: An Eternal Golden Braid is what made me pursue a career in mathematics. It's a beautiful book that is just perfect in every single way and covers so many topics incredibly insightingly, not to mention humorously. If I had to rate all the books that I've ever read on a 10-point scale, then the books I didn't like would be 1, books that I did like would be 2, and GEB would be 10.

Flatland ... without any doubt :)

Serge Lang and Murrow's Geometry is in my opinion best, especially for a beginner. The other recommendations seem to skip over that qualifier you posted. Coxeter is absolutely not a beginner's book and though Gelfand's is aimed at school kids, it's also not a beginners book. The Art of Problem Solving is aimed at the more advanced math student, so you might like it, but it is also not really aimed at students that might struggle with math, in fact the whole series is dedicated to math proficiency that will lead to math competitions. Lang's book is approachable to a beginner and meant to be a working highschool math book, and I think because Lang is actually a very good mathematician and teacher of math, his book will be precise and clear to help you understand, plus he goes further in this respect than any normal school math book.

After reading your regular geometry book, you could try Gelfand's Geometry(Superb and for motivated high-schoolers). For a demanding and harcore oldie, try Kiselev's Geometry. Also, Lang & Murrow High-School Geometry is very good modern book. If you're interested in a bird-view of the several interesting branches in geometry, Coxeter's Introduction to Geometry is excellent. Good Luck!

Thank you! :D

The Art of Problem Solving's book Introduction to Geometry is what you are looking for. Here's the link.

Lol that's way to advanced

It sounds like you want a pop book about math, not an actual math book, so I'll recommend A History of Pi and Dr. Euler's Fabulous Formula.

Proofs from the Book (named after the erdos concept) is a great collection of beautiful proofs.

The man who loved only numbers (Erdos biography) is also great if you're looking for a book about math, rather than a math book.

The Housekeeper and the Professor. Fiction, but very beautiful.

"What Is Mathematics?" by Richard Courant. It's getting a bit forgotten now, but in the last century it was probably the most influential popular math book. It's a bit dense in terms of content, but since you've liked GEB, maybe that's the kind of thing that would be up your alley. For example it's discussing mathematical induction at like page 30, just after the basic arithmetics.

As far as pop maths books go, everything written by Simon Singh is pure gold.

A book that was really influential on me when I was younger was "e: The Story of a Number" by Eli Maor. It tells the history of the discovery of a lot of very important concepts like logarithms, integrals, hyperbolic trig functions, and of course e, in a lively and engaging way.