Generatingfunctionology, Wilf.

How to solve it, Polya.

EDIT:

I'm going to sneak in "Computer Networking, a Top down Approach" by Kurose and Ross, I'm reading through it now while planning out my home HPC Cluster, and it is one of the most lucid and down to earth textbooks I've seen. Extremely practical, legitimately interesting, and carefully laid out. Bit off topic for /r/math but not out of the realm entirely and the question was textbooks, not math textbooks.

For me it's Book of Proof by Richard Hammack. It's lucid with just the right amount of handholding and a pretty linear increase in difficulty, so you always feel challenged, without feeling overwhelmed.

The book by Tu on smooth manifolds. His other books are apparently similar

Stochastic Differential Equations, by Oksendal

Random Matrix Theory, by Deift and Gioev

A Computational Introduction To Number Theory, by Shoup

An Introduction To Mathematical Cryptography, by Hoffstein, Pipher and Silverman

Quantum Computing, by Yanofsky and Mannucci

Neural Networks, by Haykin

Atiyah & MacDonald. Anything by Halmos. Anything by Serre.

I remember Serre talking about his approach to writing papers. Write the paper. Put it in a drawer. Ruminate. Reread it when you don't remember the content. At that point, you'll think "this idiot is explaining this basic stuff really badly". Rewrite and submit.

Linear Algebra Done right by Sheldon Axler.

Sterling K. Berberian's books are super clear.

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).

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).

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.

My Professor calls Grothendieck's EGA, SGA etc the Bible of AG, and Hartshorne it's cathechism.

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Springer has a long list of free eBooks available right now.

https://www.reddit.com/r/FreeEBOOKS/comments/g34xi5/408_free_ebooks_from_springer/?utm_medium=android_app&utm_source=share

I don't know a lot of math, but I'm self-taught. It is tough and I wish I had a teacher and classmates that could help me clarify things when I'm confused about something, but it's still doable.

I only saw around 40 free maths books on a link from a sub the other day. Have they expanded since or was I missing something the other day?

You can get Schaum's Outlines and Dover Publications for dirt cheap. Tons of practice problems.

Learn online - 3B1B, Khan, etc. Then practice with the books. My 2 cents.

Learn a little LaTeX, whenever you get stuck you can use Math Stack Exchange to post your doubts. A ton of people will help you.

Not sure whereabouts you are in your self-education, but once you get through Linear Algebra and Differential Equations (usually second year math courses) you will find less and less video/interactive resources online and you will/should be mainly focusing on books.

Learning to read textbooks is essential, all advanced mathematics is found almost solely in textbooks, lecture notes and the brains of mathematicians. Because of this, textbooks are the 'fountain of life' for continued progressions in mathematics when you are outside academia.

At that point I highly recommend you become a member of the Mathematics Stack Exchange and begin to read and (if no one else has asked it) ask questions. This will be a interactive way of learning that will aide you in learning from books.

Similar communities can be found on Discord servers I hear, though I don't know of any myself -- perhaps some other commenter can help with that.

Seconding the other commenter to say that your best bet will usually be lecture notes or textbooks. For a broad overview of key ideas in different areas of university mathematics, you can check out An Infinite Napkin by Evan Chen.

Since the paper you linked was about deep learning, I'd like to make a separate point. I used to do research in machine learning and generally you could get an undergrad to the point where they could read papers in a given area relatively quickly (after just a few courses). On the other hand, despite having a degree in math, I basically can't read any math research papers at all. The amount of background knowledge and training you need is just way more because mathematical topics tend to be much more well-explored than most other subjects.

The best introductory "papers" are lecture notes written by professors.

Google "math X lecture notes Y", where X is the subject and Y is the optional school name. I know Y = MIT or Oxford works great.

The Princeton Companion to Mathematics is a whole book full of overviews of various different mathematical subjects, written by experts in these fields, which are maybe a little shorter than what you're looking for (maybe 5-10 pages rather than 15-20). There's variation in how the articles are pitched, but they're generally aimed somewhere between a "general mathematical" audience and an "educated layperson". So it's comparable to wikipedia, except that (1) The articles are nicely curated, aiming to somewhat accurately reflect the scope of mathematics at the time of writing (published in 2008); (2) Each article is written by one person, so that it tells a coherent story about its subject in (3) a bit more detail than you'll find on wikipedia and (4) guaranteed to be written by an expert.

Another thing I like about the Companion is that they have articles organized a few different ways -- e.g. in one part of the book they have articles about different mathematical concepts while in another part of the book they have articles about different mathematical subfields, and in yet another part they have articles about particular theorems and conjectures. It's a nice balance.

Evan Chen’s An Infinitely Large Napkin is nice.

Thanks! Plan on going back through all of clac after this summer as a refresher, this is really helpful!

Thank you and God bless you my good sir!

thanks

Thank you!!!!

You my good sir, are god send!

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

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.

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

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

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.

I highly recommend the AoPS (art of problem solving) program.

I purchased the pre-algebra, introduction to algebra, intermediate algebra, pre-calclus, and calculus e-books and worked through them.

You learn math from the ground up in a rigorous manner, which helps with mathematical intuition and bridging concepts.

The e-books come with solutions and plenty of exercises. There are also videos embedded right in the text providing explanations to concepts with examples. Aside from that, they have a thing called Alcumus on their site as well, where you can continue to drill practice problems of increasing difficulty.

For how good they are, the e-books are quite inexpensive.

I was in your position before and tried many different textbooks, YouTube series, Khan Academy, etc...

AoPS is far an away the best of the lot and has given me much deeper levels of understanding.

As a self-learner, the e-book and Alcumus is all you need, so ignore the online classes stuff on the site, unless you decide you’d prefer guided teaching.

Here is the site: https://artofproblemsolving.com

Best of luck to you!

I have also struggled with math my whole life — but it got better after a professor said to me that math is a language.

It’s not some arcane, complicated thing whereby people manipulate numbers to get an “answer”.

It is a language which people use in order to talk in precise terms about how the universe and the things in it work.

In beginning algebra from where you’re at in Pre-Algebra, up through college-level Trigonometry and Pre-Calculus, you have essentially three tasks:

1. Learn the basic grammar, syntax, and vocabulary of the mathematical language.
2. Learn how to find a missing word in a sentence given context clues.
3. Learn how to take an overly-complicated or poorly-fitting sentence and work out a simplified or more useful sentence which has an identical meaning.

Along the way, you’ll learn to proofread the math of others and to recognize and fix common grammatical and syntax errors. As with any language, you’ll stumble and struggle to say certain things — that’s normal. Nobody becomes fluent in a new language overnight. But it has been immensely helpful for me to think about math this way, instead of the rote-memorizing, procedural method I was taught in school.

Get Textbook "Lang's Basic Mathematics" and go through Khan's Academy in parallel

You're assuming that there's an "end", or a unique order in which to do things. It makes about as much sense as asking for a map to the "end of English literature". That's far from being true. Once you've got a basic grounding as others have mentioned, pick something that sounds interesting and work your way back through prerequisistes until you get to something approachable, then repeat until the interesting thing is approachable.

I highly recommend you review basic concepts of set theory and mathematical philosophy in order to have an intuitive grasp of the subject. Mathematics in school is all rigour and is taught very formulaic making math seem very bland and unbearable to sit through.

Once you found the beauty of mathematics and how it permeates some high platonic mathematical dimension that basically projects our spatial reality. You can see how you can conjure math as some elderly language from the cosmic deities that you can use to unravel questions on a series of logic. It all sounds funny but visualizing and being passionate is what fueled mathematicians to solve the greatest problems of our time.

Also, when you learn a thing in math. Always learn it in grassroots level of understanding. For example, the quadratic equations and how it translates to the cartesian plane. See how moving and changing variables cause te graph to move. Always ask the why and make sure you can answer it in a way that you truly understood why that works.

There's really no absolute starting point in math. You really just have to start doing it. And by that i dont just mean solve integrals all day. You have to enter the mindset as well and while youre doing nothing always think of some mathematical problems you can tinker with.

Khan Academy

Thank me later

There is a video that handles this exact topic.

https://youtu.be/pTnEG_WGd2Q

Personally I'm not a fan of Khan Academy or really any video medium to learn math because the textbooks has practice problems and usually the explanations are more detailed and math terminology definitions are more precise. One the other hand, video learning is more colloquial and simplified, so they both have their uses.

I would start with textbooks as the main source of learning. If the concept is deemed to difficult, only then I would watch the videos to make the learning the new concept less difficult because you primed yourself with simplified information and you can learn the details using the textbook.

OpenStax has it. They let you download their pdfs from Pre-Algebra or basically Arithmetics, up to Calculus III.

Worth to look up if you want to study from ground up, as I'm currently doing.

Start Here

Professor Leonard

see the pinned posts

Mathematics:

Analysis I&II by Tao

Linear Algebra by Hoffmann and Kunze

Complex Analysis by Stein and Shakarchi

Algebra by Artin

Topology by Munkres

Real Analysis by Royden and Fitzpatrick

Algebra by Lang

Differential Topology: First Steps by Wallace

Physics:

Introduction to Classical Mechanics by Morin

Introduction to Electrodynamics by Griffiths

Waves by Crawford

Introduction to Quantum Mechanics by Griffiths

Thermal Physics by Schroeder

Modern Classical Mechanics by Helliwell and Sahakian

Modern Electrodynamics by Zangwill

Quantum Mechanics Lecture Notes by Littlejohn

Statistical Physics of Particles by Kardar

Spacetime and Geometry by Carroll

Quantum Field Theory in a Nutshell by Zee

Geometry, Topology and Physics by Nakahara

String Theory Vols.1&2 by Polchinski

Conformal Field Theory by Di Francesco et al

I have a lot of books, but there are only a handful which, if I had to get rid of most of them, I’d reaaaally like to keep. Those are:

-Linear Algebra Done Right (Axler)

-Abstract algebra (dummit and foote)

-Differential equations (ritger and rose)

-Topology (munkres)

-Theory of numbers (Hardy and wright)

-Analysis on real and complex manifolds (narasimhan)

-Tensor calculus for physics (neuenschwander)

Calculus on Manifold by Micheal Spivak, well-written, abstract and deep understanding of the subject.

An Introduction to Geometric Algebra and Geometric Calculus by Micheal D. Taylor, a very thorough introduction to geometric algebra, which is the modern interpretation of multi-variable calculus.

Guillemin and Pollack as preparation for modern geometry.

i can recommend pinter for abstract algebra (very gentle) and Pugh for analysis (less gentle, but very good; it's like Rudin if it weren't so painful to read)