Serious mathematics books that can replace a good teacher?

Been mulling viability of hiring professors and TAs to give 10 day intensive math and physics lecture seminars backed by tutorials in beach settings. Sort of like taking the lectures of theoreticalminimum.com but on the road, where you can join a session with a small group of ~10 students in places like Barbados, BVI, Costa Rica, mostly outdoors. If the economics can work for for yoga, we can probably make them work for an amateur/programmer interest in math. No certifications, maybe just walk through some coursera and khan material as prep.

Want to teach geeky tourists calculus? If we made math tourism a thing, teachers in host countries could skill up pretty fast as well.

You might work yourself through the exercises in "baby Rudin" to learn analysis, and ask questions on Stackexchange, but it'll probably often be very frustrating.

I think there are so many great books on linear algebra, that it is difficult to single one out. I think LA is also picked up much easier by most people, especially with a programming background.

It isn't really about the book imho, it's about being disciplined and do some practice exercises every day, like you're training a muscle.

Edit: and I recommend you never consult solutions, except to verify yours. "Illusion of competence" is a big thing you might accidentally step into in mathematics. My experience is that looking up solutions gives you a short strong eureka feeling, but you learn almost nothing from it. You have to go and arrive there yourself.

Related discussions:

* Susan Rigetti’s “So You Want to Study Mathematics…”: https://news.ycombinator.com/item?id=30591177

* Terry Tao’s “Masterclass on mathematical thinking”: https://news.ycombinator.com/item?id=30107687

* Alan U. Kennington’s “How to learn mathematics: The asterisk method”: https://news.ycombinator.com/item?id=28953781

The best book for me was "Vector Calculus, Linear Algebra, and Differential Forms" by Hubbard and Hubbard.

It is the only book (I know of) that brings you from absolute basics to an integrated development of the subjects from its title. And that integrated developments actually leads to a good didactic method.

The search function will show you many comments on HN recommending it.

Supposing you have at least high school mathematics under the belt:

1. Book of Proof by Hammack [1]

2. Tom Apostol Calculus Vol 1 (stop at Linear Algebra)

3. Linear Algebra by Insel, Spence & Friedberg

4. Understanding Analysis by Stephen Abbott

5. Tom Apostol Calculus Vol 2

These are books with excellent exposition, and solution manuals are available when needed. You are absolutely required to do all the exercises. To learn mathematics is to do mathematics.

Two other excellent books are Spivak - Calculus, the problems in this book are very good, but much more difficult than Apostol. For more advanced analysis there's Walter Rudin's Principles of Mathematical Analysis (baby Rudin), which on its own is too difficult as a first exposure to analysis, but there's a cool professor that recorded an entire semester of baby Rudin lecture videos [2]. This is as close as it gets without being enrolled in a university.

[1] https://www.people.vcu.edu/~rhammack/BookOfProof/

[2] https://www.youtube.com/watch?v=ab41LEw9oiI&list=PLun8-Z_lTk... (the first video is a bit blurry, it gets better)

Let me ask and answer some questions here that doesn't answer your question directly.

What makes a good book and why there are tons of them on the same subjects?

The best book for you is the one that speaks to your technical preparation and perspective. A few hits the sweet spot for a broad audience - perhaps because they are good at drawing analogies with common experiences - but even some obscure books can be good if it aligns with what your background.

How can a mentor help and can you do without one?

A mentor can help lay out the roadmap to build from simple topics to more difficult ones. Maybe more critically, provide rapid feedback on your understanding. They can also explain things in more than one way. Some textbooks do lay out the roadmap reasonably well, provided that it starts from concepts that you are already familiar with (again, you need to find the right book for you). Problem sets in textbooks are meant to provide feedback on your understanding, but it often fails to provide smaller hints if you can't solve the problem outright. You could get a set of solutions for the problems and that could partially help. Grabbing multiple textbook on the same subject can also help understand the most commonly covered (and by implication most essential) elements on the subject, and also give you multiple explanations of the same concept (though not always).

Takeaway message?

You could potentially try to pick up multiple books on the same subject and try to learn this way. Follow the one that speaks to your background most closely, but the others are also likely to help.

I haven't had many good math teachers, the one that I did was some poor frazzled adjunct who had to run over to CMU after he was done teaching my class at Pitt. (I'm an alumni.)

The textbook isn't the barrier, it's the end of chapter activities.

If I decided to learn, say, geometry, something that wasn't really taught much in my middle school, I'd probably do Khan academy or try to find a set of finished problem sets.

(When you remove the academic dishonesty angle, the real issue with math courses is it's difficult to craft a math question, so then you need to keep the answer and steps leading to it secret lest folks memorize them for an A.

That issue disappears completely if you adopt a more realistic model like "can you figure out which algorithm to use on Wikipedia, search out if it's already used in a common language like Python, then import your data in a simple format like a .csv and use the trustworthy code you found.

- Greg.

I really like Arthur Benjamin's work on mental mathematics. I'm not savant-level, doing division in the thousands or huge floating points in my head yet but I sure am a lot sharper than I was coming out of high school from studying his work, and I guarantee you will just have fun with expanding your capability to think about numbers. [1]

I got a copy of this book from the 1920s which is really cool because it teaches you math lessons you have to actually go out and physically do stuff with like pegs and strings in a field, from the perspective of the history of mathematics where people were limited to such devices in order to do stuff like trigonometry. Very very different approach, probably not for everyone, but for me I just think it's pretty cool. It definitely was written in the 1920s though so you better get used to that particular writing style if you plan on digesting it like a course. It's designed that way, though, and it's got great reviews. Just keep in mind maybe some of the history is subject to have changed over the years. [2]

Ultimately I've self-taught myself a lot more than I ever learned in school for sure but a wide variety of sources is probably more what you're after in terms of getting a grip on what's interesting enough to pursue further for your own means and ends. I think exploring what fascinates you the most and then just going and finding things from that point is a pretty good start as long as you've got elementary understandings up to a point where the fascination actually happens.

[1] https://www.goodreads.com/book/show/83585.Secrets_of_Mental_...

[2] https://www.goodreads.com/book/show/66355.Mathematics_for_th...

The question reminds me of Susan Rigetti's recommendations for math self-study. If your goal is to self-study the equivalent of a university undergraduate mathematics degree, this is one approach: https://www.susanrigetti.com/math

Dummit & Foote is fantastic for algebra. I also really liked Carrothers for analysis. None of the other books I used really stood out, other than Lang's algebra book, but that one was for the opposite reason - once, he said "the following is obvious" and I thought, "oh yes, it is!" and it was one of my proudest moments as a math student. Would not recommend for self-teaching.

More important than a good teacher, imo, is collaborators you're learning with who are around your same level. Sometimes they should figure out exercises faster than you, sometimes you should figure things out faster than them, often together. If that's not happening, it's a lot harder (and more frustrating) to learn. This becomes more important the more advanced you get. I never found lectures useful really but if I had no one to collaborate with I was almost certainly going to drop the class.

For abstract algebra, I'd go with Pinter's "A Book of Abstract Algebra" (ABoAA) [1].

The other recommendations given so far for abstract algebra are fine, but Pinter's organization makes it I think work better for self-study, and it is much more friendly on the wallet because it is a Dover edition. It's currently $14.89 on Amazon ($6.49 eBook).

ABoAA tends to divide the material into short chapters with lots of exercises. A typical chapter is around 5 pages of text and 5 pages of exercises. The ratio of text to exercises varies a bit but mostly will be in the 40-60% range for the text. Chapters are mostly around 10 pages +/- 3.

The exercises for each chapter are split into several sections each section covering a different aspect of the chapter's material. Sometimes there is a section of exercises applying the material to some interesting area. For example, the chapter on groups of permutations has 6 pages of text, then 5 pages of exercises divided into N sections.

The exercise sections for that chapter are computing elements in S6 (5 problems), examples of groups of permutations (4 problems), groups of permutations in R (4 problems), a cyclic group of permutations (4 problems), a subgroup of SR (4 problems), symmetries of geometric figures (4 problems), symmetries of polynomials (4 problems), properties of permutations of a set A (4 problems), and algebra of kinship structures which consists of 9 problems covering how anthropologists have applied groups of permutations to describe kinship systems in primitive societies.

This combination of small chapters with lots of exercises organized in small groups of related exercises makes it a lot easier to fit this book into a self-study plan if you are like the typical self-study student who has other things (like work) taking up much of their time and so can't get in many long study sessions.

[1] https://www.amazon.com/Book-Abstract-Algebra-Second-Mathemat...

- "A Course of Higher Mathematics", V. I. Smirnov.

- "Differential and Integral Calculus", N. Piskunov.

- "Problems in Mathematical Analysis", B. Demidovich.

These are great books. Demidovich's book is a collection of more than 3000 exercises. Smirnov's book is a course that takes you from what a value means, to advanced topics (5 volumes, in 7 books). Piskunov's book is in-between (course and exercises).

These are the books we've used and went to during the first two years of Engineering (all engineers civil/mechanical/electrical/etc. and phys/maths/chem students go through the maths/phys/chem heavy common core except CS/SE students).

You don't say what you're looking for, so I will assume this is a general inquiry.

The Art of Problem Solving books introduce subjects with a problem-solving, "inquiry-based" approach. As part of the text, you can read the question, attempt it on your own, then read how to do it if you need that guidance. In my experience, this method is great for building up knowledge what's true and how to approach a new unknown problem... which to me is proof of true understanding.

A drawback is that they are organized along more or less traditional US high school lines. Don't be fooled though, "Intermediate Algebra" is lots deeper than expanding (x+y)^2. Read the exercises if you're checking out these books, not just the "content". Intermediate Counting and Probability might have content outside of your academic experience, if you're looking for one to try.

If you post more about your mathematical background and goals, you can get better advice.

Great introductory books along with videos:

  - Mathematics and Its History (John Stillwell)
  - Journey through Genius: The Great Theorems of Mathematics (William Dunham)
  - Proofs Without Words: Exercises in Visual Thinking (Roger B. Nelsen)
  - The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems, and Personalities (William Dunham)
  - Odds & Ends: Introducing Probability & Decision with a Visual Emphasis (Jonathan Weisberg)
  - Introduction to Probability (Joseph K. Blitzstein, Jessica Hwang)
  - The Secret Life of Equations: The 50 Greatest Equations and How They Work (Richard Cochrane)
  - Euler's Gem: The Polyhedron Formula and the Birth of Topology (David S. Richeson)
  - 3Blue1Brown Essence of Linear Algebra: https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab 
  - 3Blue1Brown Differential Equations: https://www.youtube.com/playlist?list=PLZHQObOWTQDNPOjrT6KVlfJuKtYTftqH6 
  - Dynamics: The Geometry of Behavior (Ralph Abraham, Christopher Shaw)
  - Geometry, Relativity and the Fourth Dimension (Rudolf Rucker)
  - 3Blue1Brown Essence of Calculus: https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr 
  - Calculus for the Practical Man (J.E. Thompson)
  - Div, Grad, Curl, and All That: An Informal Text on Vector Calculus (Harry M. Schey)
  - Who Is Fourier? a Mathematical Adventure
  - Physics, Topology, Logic and Computation: A Rosetta Stone (John C. Baez, Mike Stay)
  - An Illustrated Guide to Relativity (Tatsu Takeuchi)
  - The Mathematics Of Quantum Mechanics (Martin Laforest)
  - A Student's Guide to Maxwell's Equations (Daniel Fleisch) or if you're also interested in a visual guide, I wrote one as well: https://github.com/photonlines/Intuitive-Guide-to-Maxwells-Equations 
  - Introduction to Elementary Particles (David J. Griffiths)
  - An Invitation to Applied Category Theory: Seven Sketches in Compositionality (Brendan Fong, David I Spivak)
  - Solving Mathematical Problems: A Personal Perspective (Terence Tao)

I'd highly recommend checking out https://www.mathacademy.us

It's current targeting students, but works well for adults too. There is no better online math training around, and it goes up to graduate level math.

The founder talks about it regularly on his podcast and it's pretty amazing to hear the progress and how comprehensive it is: https://techzinglive.com

Whichever books you choose, one thing I'd recommend is trying to find some good college that used that book and has their syllabus and schedule online. You can use that to get an idea of how to pace yourself through the material.

I've found that pacing is one of the hardest parts of self-studying college or graduate level STEM subjects. In many STEM books each chapter expects that you have a certain understanding of and are comfortable applying the previous chapters.

If you move on way too soon, you'll quickly run into a wall and probably realize you weren't ready to go forward, and go back. But there is a terrible spot between "way too soon" and "ready to move on" where you might not notice your struggles with the current chapter are because you aren't as proficient at the earlier material as you should be, and you can get mired down.

It is also possible to get stuck the other way, not moving on when you should.

When you do it right, you move on when you understand enough of the current chapter that it won't hold you back in the new chapter. This generally comes before you are fully comfortable in the current material, but are comfortable in the parts necessary to handle the new chapter, and using the current stuff in the new chapter will increase your comfort in the old material. That gets you learning new material and reinforcing the prior material when you use it with the new material.

A good experienced teacher will know about how long it takes students to get to the "I'm not fully comfortable with this but know enough for the next thing" stage, and the course pacing will be designed around that.

I post this a lot when people ask similar questions: get the Open University books (such as MST124, MST125, M208). They're not cheap but they are designed to teach undergraduate level mathematics without a teacher and they work!

Entirety of math is too large. Are you trying to learn to learn math? Or are you trying to learn actual math? Which topic are you interested in? By that, I mean something like say topology or algebra etc. Of course they have overlaps, but you have to start somewhere. Did you have even more specific ideas in mind? Maybe commutative rings or quasi groups are your jam.

Once you have a vague idea of what you want topic wise, the best thing to do is sample as many books as possible. It seems that some people click with one book and other not so much. Take (baby) Rudin for example. Some people love it as their first analysis book. It was too dense for my liking and I needed something that did more spoon feeding then that. But I can see Rudin making for a good reference book if I wanted such a thing.

Tangentially, the premise is that mentors aren't always available. Sure, but might not have access to them all the time but you'll probably want some one to talk to once in a while. Say you pick a highly recommended book X and you don't like it at all. What do you do? If you find someone who has not only read X but other books on the same topic, they can nudge you in the right direction. It can also help with discussing and understanding concepts. HN seems to have a decent number of mathematically aware people. There is also math stack exchange and friends (which I haven't used a lot). There also exists more niche solutions like the ##math channels in some irc servers (which i used extensively and was the best thing to happen personally).

Lastly, a lot of the popular/most recommended style books never worked out that well for me. I ended up looking at lecture notes scattered across the internet by various professors on very narrow topic and also very niche youtube videos by random professors and such.

Less wrong has a collection of "best" books per subject [1]

[1] https://www.lesswrong.com/posts/xg3hXCYQPJkwHyik2/the-best-t...

If you're like me, of average intelligence and not an abstract thinker, I don't believe any maths book can replace a human until you have a pretty solid experience & background established. There's too many ways to misinterpret what's being said, or just not 'get' it. One needs another mind. IME anyway.

Generally, I think lecture notes + books works much better for this than just books alone. The pacing/style/presentation of both is different for a reason and they work well together. In particular, books are great for a second look, practice and revising, but might be too condensed at first.

So instead of searching for a good book, I would recommend to search for good lecture notes/scripts first and then also get the book(s) these list as references.

- Linear Algebra: Gilbert Strang videos + book

- Calculus and basics of Analysis: Spivak

- Group Theory: Visual Group Theory

- Complex Analysis: Visual Complex Analysis

- Writing proofs: Proofs: A Long Form Textbook by Jay Cummings

- Abstract Algebra: Dummit-Foote book. First few chapters if you like.

___

Pop-Sci:

1. The Joy of X by Steven Strogatz

2. Fermat's Enigma by Simon Singh

(All this books except AA are very highly recommended by me)

Most any textbook should do provided that you read it in full, take notes, and do the assignments at least in part. As a mathematics major in college I didn’t actually understand a thing my professors were talking about. It wasn’t until I spent time with the book that I understood. Sometimes it took multiple semesters to get a single concept like Fourier transforms.

I recently saw Sheldon Axler's Linear Algebra Done Right recommended here, and have been working through it. In my nonexpert opinion, it's really good, and has a great series of videos reviewing each section in a clear and understandable fashion.

https://linear.axler.net/

Depends on what mathematics you want to learn and whether it's specific to a particular domain.

Many years ago I used Engineering Mathematics[1] when I was doing electrical engineering and my GCSE maths knowledge wasn't cutting it.

Fabulous book.

[1] https://www.amazon.co.uk/Engineering-Mathematics-K-Stroud/dp...

To quote from a reviewer

> Firstly, it's not just for university engineering/science students. The first half of the book would be great for A level. I wish I'd known about it when I did my mine! I wouldn't have needed a teacher.

The book that opened my eyes to what Math really was is a biography on Paul Erdös, The man who loved only numbers. Another really invaluable resource for me was How to solve it by Polya.

Books that are given nicknames by grad students. :) For a foundational understanding of Calculus, and just becoming a better mathematician, a good start would be ->

Baby Rudin (Principles of Mathematical Analysis, Walter Rudin)

Papa Rudin (Real and Complex Analysis, Walter Rudin)

Grandpa Rudin (Functional Analysis, Walter Rudin)

Haven’t read this one, but has been recommended highly to those trying to pick up rigorous math starting from non-quant training:

https://longformmath.com/analysis-home

>This book is the first of a series of textbooks which I am calling “long-form textbooks.” Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation and explanation. The proofs are not terse, and aim for understanding over economy. Furthermore, dozens of proofs are preceded by "scratch work" or a proof sketch to give students a big-picture view and an explanation of how they would come up with it on their own. Examples often drive the narrative and challenge the intuition of the reader. The text also aims to make the ideas visible, and contains over 200 illustrations.

>The writing is relaxed and includes interesting historical notes, periodic attempts at humor, and occasional diversions into other interesting areas of mathematics.

Have read this one, a book on advanced high-school math with a ‘problem-solving’ bent from a passionate teacher.

https://www.amazon.in/Educative-Jee-mathematics-PB-Joshi/dp/...

> Mathematics is best learned under the guidance of a mentor.

I disagree. The most important thing is curiosity.

For math, you need to think, question, and play with the concepts. It is much more active than just reading. Develop an intuition by seeing what happens when you maximize or minimize a parameter. Think about the consequences if something were different. Try to derive relations for fun.

This also applies to music and code.

If you figure out how things work on your own, you will retain it far better than going through the motions mechanically.

Spivak's Calculus! (It's really a book about real analysis.) It's extremely well written and starts by rebuilding your understanding of your fundamental mathematical building blocks, only using things you can prove. It also teaches you how to prove them.

I'm not sure what it's but all books suggested for all self learning seem to be not like books I am used to having as textbooks. I expect textbook to have lot of diagrams, colors (as dumb it sounds, pictures/colors help me remember things), tips, questions and answers in the back. But all recommended books seem to boring novel style books and everyone swearing by it. How can you learn math by reading novels?

We need to replace teachers with mentors. People you can actually get in touch with on a regular basis that can drive you in a specific direction while letting you study alone most of the time

If someone has a nonstandard analysis text that they endorse as really good, I would love to hear about that. It would be a shame to tell a kid about deltas and epsilons.

One reason for a good tutor is having some direction. I feel this is like asking for a "serious computer book". Do you want a book on C++ or Dummies guide to Windows or regular expressions or Turing machines or chip design or functional programming or hash functions or ...?

Mathematics is very broad. You might try something like "Mathematics and its history" by John Stillwell to get some perspective on the subject.

No. Books can't understand how you think and learn and help guide you. There is no substitute for a good teacher. Is there a best book for that student? Probably, but who is that student? We don't know!

Beyond just learning mathematics, seeing the art and beauty in it is also best taught by someone who knows the subject and the student. Without the beauty, it's just what could be in a textbook, assuming you found the right book.

If you're asking if you can find a good book to teach someone, that depends on your style and theirs . . .

We will almost certainly have good student-focused AI teachers during our lifetimes for things like math and languages. Will AI be able to show us the art? I can't say for sure, but I bet so . . .

I find brilliant.org to be pretty good. It may not go as deeply into each topic as a good book, but it's a great start. I recommend going through all the math courses there, starting from the level you're at and then go deeper with some books.

Do you have specific mathematical topics you're interested in?

I have tried reading textbooks a few times to teach myself but found it hard to stay motivated, so I found a tutor on Upwork to assign and grade homework problems and answer my questions. Along with math textbooks and YouTube videos, this has been super helpful for fleshing out my knowledge of college-level math that I never learned properly. It's also great because they can go at the pace you want and focus on topics you find challenging, interesting, or useful.

There are a couple of textbooks I haven't seen mentioned elsewhere in this thread which I found very useful as a first year undergraduate - they are concise in style but not to the point of being too difficult to follow (it's personal preference but I found I got on much better with this style of textbook):

Davenport - The Higher Arithmetic

Burkill - A First Course in Mathematical Analysis (and also A Second Course in Mathematical Analysis by Burkill and Burkill)

Hardy and Wright - An introduction to the theory of numbers

Walter Rudin's books are the only ones you need. Start with Principles of Mathematical Analysis and when you're done with it read the more advanced ones.

Back in 1960 I learned Boolean algebra from Suzanne Langer's Symbolic Logic Dover paperback. In '63-'64 I learned calculus from Johnson and Kiokemeister's Calculus (with some tutoring in the summer of '63 from a future MacArthur grantee).

https://archive.org/details/in.ernet.dli.2015.523033

What type of mathematics? What level?

I taught myself all of A-level maths and further maths here in the UK from the standard text books at the time (Bostock and Chandler) before I started in the sixth form and then maybe about half of the first year university material before I went. Still better when you have somebody to teach you but not impossible. I did have access to somebody whom I could ask questions but didn't really use that.

Jeffreys and Jeffreys is an old favourite of mine, and the copy shown at the link is in reasonably good condition.

Jeffreys, Harold, and Bertha Swirles Jeffreys. Methods Of Mathematical Physics. Cambridge At The University Press, 1950. http://archive.org/details/methodsofmathema031187mbp.

Nothing can replace a teacher, much less a good one, but I recently stumbled upon an introduction book on calculus from 1910 (yes last century) that had a really nice way to explain things.

I just read the beginning so I don't know how far it goes but anyway here the link https://calculusmadeeasy.org/

Geometry by Brennan. Even it was writen for self study at the Open University.

Concrete Mathematics by Knuth.

Information Theory, Inference and Learning Algorithms by Mackay.

It's not just Mathematics, the discussion is about teaching vs learning.

Ideally the recipe is: good mentor + good student + good teaching tools

IMO for most disciplines, in the year 2022 CE, the critical point is student, not the mentor, not the tools.

Internet resources can hack the mentor/tools stack, when WE are eager and hungry student WE will find a way.

Since this thread is very general, I'd like to specifically request anything that'd help one dive into Model Theory. Everything I've found labeled introductory is very dense, and I feel like I'm missing some prereqs it assumes. It's not easy to know what those are though.

I'd honestly spend my energy on finding a platform which gets you closest to a mentor. Books are good, but in no way, shape, or form a substitute for a mentor. Mentoring is two-way communication, a books is almost always a one-way.

Forums, chat groups/channels, pen pal, whatever it takes.

It's going to very much depend on which field you are interested in. The subject is huge. If you want to study statistics or number theory, you're going to be looking at very different skills and knowledge, with only basically high school maths in common.

The Schaum's Outline Series. It has tons of solved problems and exercises, which is exactly what is critical for getting a handle on mathematical concepts and methods. There's also the Demystified series, which is good but has fewer exercises.

Not sure if they can replace a good math teacher but some top picks for math books include "Geometry" by Euclid, "Algebra" by Harold Jacobs, "Trigonometry" by Charles P. McKeague, and "Calculus" by Michael Spivak.

George Polya's "How to solve it" taught me how to think better, in engineering, mathematics & problem solving. After going through the whole thing, and solving a few of the exercises, I feel like I get stuck less compared to before.

A teacher has two fundamental roles: first, to explain the concepts in a way that is accessible to a given person, and second, to check if they actually understood.

As for 1), there are so many resources to choose from and everybody has their favorites. But 2) is crucial, so today I tend to choose resources with exercises AND answers/key/solutions so that I can verify if I understood the material correctly myself.

There are also online books with solutions integrated, such as[0].

[0] http://linear.ups.edu/fcla/section-WILA.html

I repeat my comment from https://news.ycombinator.com/item?id=11274264

„A single book is enough to learn mathematics: Riley, Hobson, Bence: Mathematical Methods for Physics and Engineering: A Comprehensive Guide It has a whopping 1300 pages, but it has everything you need.

And if that is not enough for you get Cahill: Physical Mathematics This will give you advanced topics like differential forms, path integrals, renormalization group, chaos and string theory.“

I believe Richard Feynman has mentioned in interviews that he learned calculus from "Calculus Made Easy".

IIRC there might have been other books in a "Made Easy" series about math, but I'm not sure.

I had two acceptable maths teachers in my life. Good teachers seem so rare.

For independent study, get workbooks with exercises and solutions. Best way of learning is by doing. Solving problems is an invaluable way of proving to yourself that you understood the material.

“Unknown Quantity: A Real and Imaginary History of Algebra”

This is a great history of algebra. Math comes alive for me when I know more about how things come to be, the problems people we’re working on, etc.

Not an overall answer, but some parts of an answer:

- Notes on Discrete Mathematics (James Aspnes)

- Mathematical Proofs (Chartrand, Polimeni et al)

- TrevTutor videos on YouTube

- Professor Leonard videos on YouTube

A good alternative to a book would be Khan Academy.

https://www.khanacademy.org

I would recommend looking not only at books, but at online video lecture recordings (from reputable sources). There is so much intuition you get from someone actually discussing and problem-solving in front of you, not to mention (especially in higher pure mathematics at least) a lot of intuition which is spoken or drawn on the board, which never makes it into a book.

Burn Math Class by Jason Wilkes is interesting.

Mathematics is extremely broad. Find translations of books used for mathematics courses at Russian universities.

I am skeptical that a book can substitute for a mentor (or teacher) at the start of someone's dive into mathematics.

Once you know your basics, sure, go study a book. But by this point, you already know whether a given book helps you or not...

I'd really want books that cover topics of pre-requisites of General Relativity with the following attributes:

- Accessible, does not go too deep into the topic

- Exercises with printed answer (could be in separate student solution manual)

- Well written for self study

It is a long time since I learned, so I dont have specific book, but dont underestimate exercises. Whatever math topic you are learning, look for one of those books with only exercises in them and solutions in the end.

I am not sure if they can replace a teacher, but some good mathematics books are "Euclid's Elements", "Introduction to Algebra" by Richard Rusczyk, and "Calculus" by Michael Spivak.

I try to find lectures online of professors who wrote books. Good example is Mr. Gilbert Strang. Good books and lectures. For theory of computation Sipser Michael. Signals and Systems by Opprnheimer. And so on.

If you're willing to pay a monthly subscription, check out Jason Roberts (of https://techzinglive.com) online Math Academy.

Visual Complex Analysis.

After reading all the comments I am sharing my choices

1. The foundations of mathematics by Stewart and Tall

2. Problem Solving Strategies by Arthur Engels

3. Journey into Mathematics by Rotman

4. Concrete Mathematics by Knuth et al.

art of problem solving

J.Steward's "Calculus: Early Transcendentals" is pretty on point for calculus. Mathematics is more than calculus though.

Bluntly, with only occasional exceptions, have to learn from the books. Period. No biggie: Once are in grad school or a prof, have to learn from papers, and the good books are typically MUCH easier to read than the papers!

For first calculus, read one of the well respected books. In my experience and opinion, don't need high school advanced placement calculus -- regard it as a waste of time. Also the last time I looked at Khan Academy, I concluded that it was a waste or worse and by people who didn't have a good background in calculus. Generally the Internet sources are too elementary. Solution: Just use some good books. Protter and Morrey is good (I taught from it), but it is a bit elementary. I learned from Johnson and Kiokmeister; I like it. At the time it was also being used at Harvard. The exercises are especially good. It's old, but calculus hasn't changed much since 1950 or so about when it gave up on teaching infinitesimals.

But might check out the Maxima software for symbolic indefinite integration -- as I recall, its algorithms are so good they do any indefinite integration that can be done in closed form.

For linear algebra, sure, Nering (a student of E. Artin at Princeton) and then Halmos (a finite dimensional introduction to Hilbert space and written when Halmos was an assistant to von Neumann, office not far from Einstein), etc. For still more, R. Bellman has a good book -- he went all over active and put in everything including the kitchen sink. Then for numerical work, old but a start, Forsythe and Moler and/or the documentation of LINPAK.

For multivariate calculus through the divergence theorem and as needed, e.g., for Maxwell's equations, start with the most elementary treatments of vector analysis and f'get about the lack of proofs. Actually notice, can see via a Google search, that the divergence theorem on a box is just a dirt simple application of the fundamental theorem of freshman calculus. Then the difficulty is proving the thing on other shapes. Then do multivariate calculus again via, say, Apostol, Mathematical Analysis: A Modern Approach to Advanced Calculus. That he uses the Jordan curve theorem is a bit much but just go with the flow. Fleming's Functions of Several Variables does it with both measure theory and exterior algebra. Or W. Rudin's, as I recall, third edition, Principles of Mathematical Analysis. For calculus done via Lebesgue's measure theory, Royden's Real Analysis and/or the first half of Rudin's Real and Complex Analysis.

My favorite author on ordinary differential equations is Coddington, and his elementary An Introduction to Ordinary Differential Equations is nicely done and likely enough for nearly any applications might encounter. Will want some of this if do A/C circuit theory or deterministic optimal control.

For basic abstract algebra Herstein, Topics in Algebra should usually be enough and is well written. With that will be well prepared for, say, algebraic coding theory which actually has some important applications.

Once have a background in measure theory, for probability, of course, Loeve or either of his students Breiman or Neveu. Breiman is really nice to read. Neveu is a good candidate for the most polished and elegantly done math book or writing of any kind in all of civilization. Be sure to get to the good stuff on the Radon-Nikodym theorem, conditional expectation, martingales, and ergodic processes.

For relatively good results, better than hardly any students get just from courses, study the theorems and proofs until can do them yourself and can also explain them to any common man in the street and also explain where they fit in more generally. Also be able to do the exercises. E.g., Rudin's Principles has the inverse and implicit function theorems as exercises! Fleming applies them to Lagrange multipliers. These two theorems are the main prerequisites of differential geometry -- some parts of physics, now popular, touch on this math!

In my Ph.D. program I led the class in 4 of the five qualifying exams, and for the analysis and linear algebra exams, the books above were the secret. After I did that, the department tried to have the students do better by offering a course from Rudin's Principles; the course didn't work; that book is too hard for a course and needs weeks of quiet time where a student can chew on the text slowly line by line. Rudin has some of the most precise presentations on Fourier theory. So when physics uses Fourier theory or engineers use the fast Fourier transform, you will likely know Fourier theory quite a lot better than they do. Right, the uncertainty principle in quantum mechanics is just a basic result in Fourier theory. With Fourier theory and linearity, Schroedinger's equation starts to make sense. A good exercise in early measure theory is differentiation under the integral sign, Leibniz's rule -- be sure to cover that with a good proof.

With the background from the books above will be well prepared for a wide range of more narrow topics in pure and applied math.

Nearly all these books are available used in nearly new condition for low prices. In a move I lost some of my library and rebuilt with used copies.

Nothing can replace a good teacher, as nothing can replace natural interest and motivation. A good teacher makes topics graspable... Shows why this is an awesome thing to know and how this can propel the child and society further... Most stories of people I know that are great in something was a person who lead them to find the purpose and interest in a thing

Probability Theory: The Logic of Science by E. T. Jaynes.

Only a better teacher can replace a good teacher.

In order:

1. Book of Proof

1a. How to Prove It: A Structured Approach

2. Elements of Set Theory - Enderton

2a. Naive Set Theory - Halmos

3. Foundations of Analysis - Landau

4. Basic Mathematics - Lang

5. Principles of Mathematics - Allendoerfer & Oakley

6. A Transition to Advanced Mathematics

6a. Concept of Modern Mathematics - Stewart

7. Calculus - Apostol or Spivak

Don't skip any