This is a compilation of what I gathered from reading on the internet about self-learning higher maths, I haven't come close to reading all this books or watching all this lectures, still I hope it helps you.

General Stuff:
The books here deal with large parts of mathematics and are good to guide you through it all, but I recommend supplementing them with other books.

1. Mathematics: A very Short Introduction : A very good book, but also very short book about mathematics by Timothy Gowers, a Field medalist and overall awesome guy, gives you a feelling for what math is all about.
   
   
2. Concepts of Modern Mathematics: A really interesting book by Ian Stewart, it has more topics than the last book, it is also bigger though less formal than Gower's book. A gem.
   
   
3. What is Mathematics?: A classic that has aged well, it's more textbook like compared to the others, which is good because the best way to learn mathematics is by doing it. Read it.
   
   
4. An Infinitely Large Napkin: This is the most modern book in this list, it delves into a huge number of areas in mathematics and I don't think it should be read as a standalone, rather it should guide you through your studies.
   
   
5. The Princeton Companion to Mathematics: A humongous book detailing many areas of mathematics, its history and some interesting essays. Another book that should be read through your life.
   
   
6. Mathematical Discussions: Gowers taking a look at many interesting points along some mathematical fields.
   
   
7. Technion Linear Algebra Course - The first 14 lectures: Gets you wet in a few branches of maths.
   
   Linear Algebra: An extremelly versatile branch of Mathematics that can be applied to almost anything, also the first "real math" class in most universities.
   
   
8. Linear Algebra Done Right: A pretty nice book to learn from, not as computational heavy as other Linear Algebra texts.
   
   
9. Linear Algebra: A book with a rather different approach compared to LADR, if you have time it would be interesting to use both. Also it delves into more topics than LADR.
   
   
10. Calculus Vol II : Apostols' beautiful book, deals with a lot of lin algebra and complements the other 2 books by having many exercises. Also it doubles as a advanced calculus book.
    
    
11. Khan Academy: Has a nice beginning LinAlg course.
    
    
12. Technion Linear Algebra Course: A really good linear algebra course, teaches it in a marvelous mathy way, instead of the engineering-driven things you find online.
    
    
13. 3Blue1Brown's Essence of Linear Algebra: Extra material, useful to get more intuition, beautifully done.
    
    Calculus: The first mathematics course in most Colleges, deals with how functions change and has many applications, besides it's a doorway to Analysis.
    
    
14. Calculus: Tom Apostol's Calculus is a rigor-heavy book with an unorthodox order of topics and many exercises, so it is a baptism by fire. Really worth it if you have the time and energy to finish. It covers single variable and some multi-variable.
    
    
15. Calculus: Spivak's Calculus is also rigor-heavy by Calculus books standards, also worth it.
    
    
16. Calculus Vol II : Apostols' beautiful book, deals with many topics, finishing up the multivariable part, teaching a bunch of linalg and adding probability to the mix in the end.
    
    
17. MIT OCW: Many good lectures, including one course on single variable and another in multivariable calculus.
    
    Real Analysis: More formalized calculus and math in general, one of the building blocks of modern mathematics.
    
    
18. Principle of Mathematical Analysis: Rudin's classic, still used by many. Has pretty much everything you will need to dive in.
    
    
19. Analysis I and Analysis II: Two marvelous books by Terence Tao, more problem-solving oriented.
    
    
20. Harvey Mudd's Analysis lectures: Some of the few lectures on Real Analysis you can find online.
    
    Abstract Algebra: One of the most important, and in my opinion fun, subjects in mathematics. Deals with algebraic structures, which are roughly sets with operations and properties of this operations.
    
    
21. Abstract Algebra: Dummit and Foote's book, recommended by many and used in lots of courses, is pretty much an encyclopedia, containing many facts and theorems about structures.
    
    
22. Harvard's Abstract Algebra Course: A great course on Abstract Algebra that uses D&F as its textbook, really worth your time.
    
    
23. Algebra: Chapter 0: I haven't used this book yet, though from what I gathered it is both a category theory book and an Algebra book, or rather it is a very different way of teaching Algebra. Many say it's worth it, others (half-jokingly I guess?) accuse it of being abstract nonsense. Probably better used after learning from the D&F and Harvard's course.
    
    There are many other beautiful fields in math full of online resources, like Number Theory and Combinatorics, that I would like to put recommendations here, but it is quite late where I live and I learned those in weirder ways (through olympiad classes and problems), so I don't think I can help you with them, still you should do some research on this sub to get good recommendations on this topics and use the General books as guides.

Game Engine:

Game Engine Architecture by Jason Gregory, best you can get.

Game Coding Complete by Mike McShaffry. The book goes over the whole of making a game from start to finish, so it's a great way to learn the interaction the engine has with the gameplay code. Though, I admit I also am not a particular fan of his coding style, but have found ways around it. The boost library adds some complexity that makes the code more terse. The 4th edition made a point of not using it after many met with some difficulty with it in the 3rd edition. The book also uses DXUT to abstract the DirectX functionality necessary to render things on screen. Although that is one approach, I found that getting DXUT set up properly can be somewhat of a pain, and the abstraction hides really interesting details about the whole task of 3D rendering. You have a strong background in graphics, so you will probably be better served by more direct access to the DirectX API calls. This leads into my suggestion for Introduction to 3D Game Programming with DirectX10 (or DirectX11).



C++:

C++ Pocket Reference by Kyle Loudon
I remember reading that it takes years if not decades to become a master at C++. You have a lot of C++ experience, so you might be better served by a small reference book than a large textbook. I like having this around to reference the features that I use less often. Example:

namespace
{
//code here
}

is an unnamed namespace, which is a preferred method for declaring functions or variables with file scope. You don't see this too often in sample textbook code, but it will crop up from time to time in samples from other programmers on the web. It's $10 or so, and I find it faster and handier than standard online documentation.



Math:

You have a solid graphics background, but just in case you need good references for math:
3D Math Primer
Mathematics for 3D Game Programming

Also, really advanced lighting techniques stretch into the field of Multivariate Calculus. Calculus: Early Transcendentals Chapters >= 11 fall in that field.



Rendering:

Introduction to 3D Game Programming with DirectX10 by Frank. D. Luna.
You should probably get the DirectX11 version when it is available, not because it's newer, not because DirectX10 is obsolete (it's not yet), but because the new DirectX11 book has a chapter on animation. The directX 10 book sorely lacks it. But your solid graphics background may make this obsolete for you.

3D Game Engine Architecture (with Wild Magic) by David H. Eberly is a good book with a lot of parallels to Game Engine Architecture, but focuses much more on the 3D rendering portion of the engine, so you get a better depth of knowledge for rendering in the context of a game engine. I haven't had a chance to read much of this one, so I can't be sure of how useful it is just yet. I also haven't had the pleasure of obtaining its sister book 3D Game Engine Design.

Given your strong graphics background, you will probably want to go past the basics and get to the really nifty stuff. Real-Time Rendering, Third Edition by Tomas Akenine-Moller, Eric Haines, Naty Hoffman is a good book of the more advanced techniques, so you might look there for material to push your graphics knowledge boundaries.



Software Engineering:

I don't have a good book to suggest for this topic, so hopefully another redditor will follow up on this.

If you haven't already, be sure to read about software engineering. It teaches you how to design a process for development, the stages involved, effective methodologies for making and tracking progress, and all sorts of information on things that make programming and software development easier. Not all of it will be useful if you are a one man team, because software engineering is a discipline created around teams, but much of it still applies and will help you stay on track, know when you've been derailed, and help you make decisions that get you back on. Also, patterns. Patterns are great.

Note: I would not suggest Software Engineering for Game Developers. It's an ok book, but I've seen better, the structure doesn't seem to flow well (for me at least), and it seems to be missing some important topics, like user stories, Rational Unified Process, or Feature-Driven Development (I think Mojang does this, but I don't know for sure). Maybe those topics aren't very important for game development directly, but I've always found user stories to be useful.

Software Engineering in general will prove to be a useful field when you are developing your engine, and even more so if you have a team. Take a look at This article to get small taste of what Software Engineering is about.


Why so many books?
Game Engines are a collection of different systems and subsystems used in making games. Each system has its own background, perspective, concepts, and can be referred to from multiple angles. I like Game Engine Architecture's structure for showing an engine as a whole. Luna's DirectX10 book has a better Timer class. The DirectX book also has better explanations of the low-level rendering processes than Coding Complete or Engine Architecture. Engine Architecture and Game Coding Complete touch on Software Engineering, but not in great depth, which is important for team development. So I find that Game Coding Complete and Game Engine Architecture are your go to books, but in some cases only provide a surface layer understanding of some system, which isn't enough to implement your own engine on. The other books are listed here because I feel they provide a valuable supplement and more in depth explanations that will be useful when developing your engine.

tldr: What Valken and SpooderW said.

On the topic of XNA, anyone know a good XNA book? I have XNA Unleashed 3.0, but it's somewhat out of date to the new XNA 4.0. The best looking up-to-date one seems to be Learning XNA 4.0: Game Development for the PC, Xbox 360, and Windows Phone 7 . I have the 3.0 version of this book, and it's well done.

*****
Source: Doing an Independent Study in Game Engine Development. I asked this same question months ago, did my research, got most of the books listed here, and omitted ones that didn't have much usefulness. Thought I would share my research, hope you find it useful.

I'd like to give you my two cents as well on how to proceed here. If nothing else, this will be a second opinion. If I could redo my physics education, this is how I'd want it done.

If you are truly wanting to learn these fields in depth I cannot stress how important it is to actually work problems out of these books, not just read them. There is a certain understanding that comes from struggling with problems that you just can't get by reading the material. On that note, I would recommend getting the Schaum's outline to whatever subject you are studying if you can find one. They are great books with hundreds of solved problems and sample problems for you to try with the answers in the back. When you get to the point you can't find Schaums anymore, I would recommend getting as many solutions manuals as possible. The problems will get very tough, and it's nice to verify that you did the problem correctly or are on the right track, or even just look over solutions to problems you decide not to try.

Basics

I second Stewart's Calculus cover to cover (except the final chapter on differential equations) and Halliday, Resnick and Walker's Fundamentals of Physics. Not all sections from HRW are necessary, but be sure you have the fundamentals of mechanics, electromagnetism, optics, and thermal physics down at the level of HRW.

Once you're done with this move on to studying differential equations. Many physics theorems are stated in terms of differential equations so really getting the hang of these is key to moving on. Differential equations are often taught as two separate classes, one covering ordinary differential equations and one covering partial differential equations. In my opinion, a good introductory textbook to ODEs is one by Morris Tenenbaum and Harry Pollard. That said, there is another book by V. I. Arnold that I would recommend you get as well. The Arnold book may be a bit more mathematical than you are looking for, but it was written as an introductory text to ODEs and you will have a deeper understanding of ODEs after reading it than your typical introductory textbook. This deeper understanding will be useful if you delve into the nitty-gritty parts of classical mechanics. For partial differential equations I recommend the book by Haberman. It will give you a good understanding of different methods you can use to solve PDEs, and is very much geared towards problem-solving.

From there, I would get a decent book on Linear Algebra. I used the one by Leon. I can't guarantee that it's the best book out there, but I think it will get the job done.

This should cover most of the mathematical training you need to move onto the intermediate level physics textbooks. There will be some things that are missing, but those are usually covered explicitly in the intermediate texts that use them (i.e. the Delta function). Still, if you're looking for a good mathematical reference, my recommendation is Lua. It may be a good idea to go over some basic complex analysis from this book, though it is not necessary to move on.

Intermediate

At this stage you need to do intermediate level classical mechanics, electromagnetism, quantum mechanics, and thermal physics at the very least. For electromagnetism, Griffiths hands down. In my opinion, the best pedagogical book for intermediate classical mechanics is Fowles and Cassidy. Once you've read these two books you will have a much deeper understanding of the stuff you learned in HRW. When you're going through the mechanics book pay particular attention to generalized coordinates and Lagrangians. Those become pretty central later on. There is also a very old book by Robert Becker that I think is great. It's problems are tough, and it goes into concepts that aren't typically covered much in depth in other intermediate mechanics books such as statics. I don't think you'll find a torrent for this, but it is 5 bucks on Amazon. That said, I don't think Becker is necessary. For quantum, I cannot recommend Zettili highly enough. Get this book. Tons of worked out examples. In my opinion, Zettili is the best quantum book out there at this level. Finally for thermal physics I would use Mandl. This book is merely sufficient, but I don't know of a book that I liked better.

This is the bare minimum. However, if you find a particular subject interesting, delve into it at this point. If you want to learn Solid State physics there's Kittel. Want to do more Optics? How about Hecht. General relativity? Even that should be accessible with Schutz. Play around here before moving on. A lot of very fascinating things should be accessible to you, at least to a degree, at this point.

Advanced

Before moving on to physics, it is once again time to take up the mathematics. Pick up Arfken and Weber. It covers a great many topics. However, at times it is not the best pedagogical book so you may need some supplemental material on whatever it is you are studying. I would at least read the sections on coordinate transformations, vector analysis, tensors, complex analysis, Green's functions, and the various special functions. Some of this may be a bit of a review, but there are some things Arfken and Weber go into that I didn't see during my undergraduate education even with the topics that I was reviewing. Hell, it may be a good idea to go through the differential equations material in there as well. Again, you may need some supplemental material while doing this. For special functions, a great little book to go along with this is Lebedev.

Beyond this, I think every physicist at the bare minimum needs to take graduate level quantum mechanics, classical mechanics, electromagnetism, and statistical mechanics. For quantum, I recommend Cohen-Tannoudji. This is a great book. It's easy to understand, has many supplemental sections to help further your understanding, is pretty comprehensive, and has more worked examples than a vast majority of graduate text-books. That said, the problems in this book are LONG. Not horrendously hard, mind you, but they do take a long time.

Unfortunately, Cohen-Tannoudji is the only great graduate-level text I can think of. The textbooks in other subjects just don't measure up in my opinion. When you take Classical mechanics I would get Goldstein as a reference but a better book in my opinion is Jose/Saletan as it takes a geometrical approach to the subject from the very beginning. At some point I also think it's worth going through Arnold's treatise on Classical. It's very mathematical and very difficult, but I think once you make it through you will have as deep an understanding as you could hope for in the subject.

You are in a very special position right now where many interesing fields of mathematics are suddenly accessible to you. There are many directions you could head. If your experience is limited to calculus, some of these may look very strange indeed, and perhaps that is enticing. That was certainly the case for me.

Here are a few subject areas in which you may be interested. I'll link you to Dover books on the topics, which are always cheap and generally good.

• The Nature and Power of Mathematics, Donald M. Davis. This book seems to be a survey of some history of mathematics and various modern topics. Check out the table of contents to get an idea. You'll notice a few of the subjects in the list below. It seems like this would be a good buy if you want to taste a few different subjects to see what pleases your palate.
  
  
• Introduction to Graph Theory, Richard J. Trudeau. Check out the Wikipedia entry on graph theory and the one defining graphs to get an idea what the field is about and some history. The reviews on Amazon for this book lead me to believe it would be a perfect match for an interested high school student.
  
  
• Game Theory: A Nontechnical Introduction, Morton D. Davis. Game theory is a very interesting field with broad applications--check out the wiki. This book seems to be written at a level where you would find it very accessible. The actual field uses some heavy math but this seems to give a good introduction.
  
  
• An Introduction to Information Theory, John R. Pierce. This is a light-on-the-maths introduction to a relatively young field of mathematics/computer science which concerns itself with the problems of storing and communicating data. Check out the wiki for some background.
  
  
• Lady Luck: The Theory of Probability, Warren Weaver. This book seems to be a good introduction to probability and covers a lot of important ideas, especially in the later chapters. Seems to be a good match to a high school level.
  
  
• Elementary Number Theory, Underwood Dudley. Number theory is a rich field concerned with properties of numbers. Check out its Wikipedia entry. I own this book and am reading through it like a novel--I love it! The exposition is so clear and thorough you'd think you were sitting in a lecture with a great professor, and the exercises are incredible. The author asks questions in such a way that, after answering them, you can't help but generalize your answers to larger problems. This book really teaches you to think mathematically.
  
  
• A Book of Abstract Algebra, Charles C. Pinter. Abstract algebra formalizes and generalizes the basic rules you know about algebra: commutativity, associativity, inverses of numbers, the distributive law, etc. It turns out that considering these concepts from an abstract standpoint leads to complex structures with very interesting properties. The field is HUGE and seems to bleed into every other field of mathematics in one way or another, revealing its power. I also own this book and it is similarly awesome. The exposition sets you up to expect the definitions before they are given, so the material really does proceed naturally.
  
  
• Introduction to Analysis, Maxwell Rosenlicht. Analysis is essentially the foundations and expansion of calculus. It is an amazing subject which no math student should ignore. Its study generally requires a great deal of time and effort; some students would benefit more from a guided class than from self-study.
  
  
• Principles of Statistics, M. G. Bulmer. In a few words, statistics is the marriage between probability and analysis (calculus). The wiki article explains the context and interpretation of the subject but doesn't seem to give much information on what the math involved is like. This book seems like it would be best read after you are familiar with probability, say from Weaver's book linked above.
  
  
• I have to second sellphone's recommendation of Naive Set Theory by Paul Halmos. It's one of my favorite math books and gives an amazing introduction to the field. It's short and to the point--almost a haiku on the subject.
  
  
• Continued Fractions, A. Ya. Khinchin. Take a look at the wiki for continued fractions. The book is definitely terse at times but it is rewarding; Khinchin is a master of the subject. One review states that, "although the book is rich with insight and information, Khinchin stays one nautical mile ahead of the reader at all times." Another review recommends Carl D. Olds' book on the subject as a better introduction.
  
  Basically, don't limit yourself to the track you see before you. Explore and enjoy.

My main hobby is reading textbooks, so I decided to go beyond the scope of the question posed. I took a look at what I have on my shelves in order to recommend particularly good or standard books that I think could characterize large portions of an undergraduate degree and perhaps the beginnings of a graduate degree in the main fields that interest me, plus some personal favorites.

Neuroscience: Theoretical Neuroscience is a good book for the field of that name, though it does require background knowledge in neuroscience (for which, as others mentioned, Kandel's text is excellent, not to mention that it alone can cover the majority of an undergraduate degree in neuroscience if corequisite classes such as biology and chemistry are momentarily ignored) and in differential equations. Neurobiology of Learning and Memory and Cognitive Neuroscience and Neuropsychology were used in my classes on cognition and learning/memory and I enjoyed both; though they tend to choose breadth over depth, all references are research papers and thus one can easily choose to go more in depth in any relevant topics by consulting these books' bibliographies.

General chemistry, organic chemistry/synthesis: I liked Linus Pauling's General Chemistry more than whatever my school gave us for general chemistry. I liked this undergraduate organic chemistry book, though I should say that I have little exposure to other organic chemistry books, and I found Protective Groups in Organic Synthesis to be very informative and useful. Unfortunately, I didn't have time to take instrumental/analytical/inorganic/physical chemistry and so have no idea what to recommend there.

Biochemistry: Lehninger is the standard text, though it's rather expensive. I have limited exposure here.

Mathematics: When I was younger (i.e. before having learned calculus), I found the four-volume The World of Mathematics great for introducing me to a lot of new concepts and branches of mathematics and for inspiring interest; I would strongly recommend this collection to anyone interested in mathematics and especially to people considering choosing to major in math as an undergrad. I found the trio of Spivak's Calculus (which Amazon says is now unfortunately out of print), Stewart's Calculus (standard text), and Kline's Calculus: An Intuitive and Physical Approach to be a good combination of rigor, practical application, and physical intuition, respectively, for calculus. My school used Marsden and Hoffman's Elementary Classical Analysis for introductory analysis (which is the field that develops and proves the calculus taught in high school), but I liked Rudin's Principles of Mathematical Analysis (nicknamed "Baby Rudin") better. I haven't worked my way though Munkres' Topology yet, but it's great so far and is often recommended as a standard beginning toplogy text. I haven't found books on differential equations or on linear algebra that I've really liked. I randomly came across Quine's Set Theory and its Logic, which I thought was an excellent introduction to set theory. Russell and Whitehead's Principia Mathematica is a very famous text, but I haven't gotten hold of a copy yet. Lang's Algebra is an excellent abstract algebra textbook, though it's rather sophisticated and I've gotten through only a small portion of it as I don't plan on getting a PhD in that subject.

Computer Science: For artificial intelligence and related areas, Russell and Norvig's Artificial Intelligence: A Modern Approach's text is a standard and good text, and I also liked Introduction to Information Retrieval (which is available online by chapter and entirely). For processor design, I found Computer Organization and Design to be a good introduction. I don't have any recommendations for specific programming languages as I find self-teaching to be most important there, nor do I know of any data structures books that I found to be memorable (not that I've really looked, given the wealth of information online). Knuth's The Art of Computer Programming is considered to be a gold standard text for algorithms, but I haven't secured a copy yet.

Physics: For basic undergraduate physics (mechanics, e&m, and a smattering of other subjects), I liked Fundamentals of Physics. I liked Rindler's Essential Relativity and Messiah's Quantum Mechanics much better than whatever books my school used. I appreciated the exposition and style of Rindler's text. I understand that some of the later chapters of Messiah's text are now obsolete, but the rest of the book is good enough for you to not need to reference many other books. I have little exposure to books on other areas of physics and am sure that there are many others in this subreddit that can give excellent recommendations.

Other: I liked Early Theories of the Universe to be good light historical reading. I also think that everyone should read Kuhn's The Structure of Scientific Revolutions.

Hello! I'm interested in trying to cultivate a better understanding/interest/mastery of mathematics for myself. For some context:

 




To be frank, Math has always been my least favorite subject. I do love learning, and my primary interests are Animation, Literature, History, Philosophy, Politics, Ecology & Biology. (I'm a Digital Media Major with an Evolutionary Biology minor) Throughout highschool I started off in the "honors" section with Algebra I, Geometry, and Algebra II. (Although, it was a small school, most of the really "excelling" students either doubled up with Geometry early on or qualified to skip Algebra I, meaning that most of the students I was around - as per Honors English, Bio, etc - were taking Math courses a grade ahead of me, taking Algebra II while I took Geometry, Pre-Calc while I took Algebra II, and AP/BC Calc/Calc I while I took Pre-Calc)

By my senior year though, I took a level down, and took Pre-Calculus in the "advanced" level. Not the lowest, that would be "College Prep," (man, Honors, Advanced, and College Prep - those are some really condescending names lol - of course in Junior & Senior year the APs open up, so all the kids who were in Honors went on to APs, and Honors became a bit lower in standard from that point on) but since I had never been doing great in Math I decided to take it a bit easier as I focused on other things.

So my point is, throughout High School I never really grappled with Math outside of necessity for completing courses, I never did all that well (I mean, grade-wise I was fine, Cs, Bs and occasional As) and pretty much forgot much of it after I needed to.

Currently I'm a sophmore in University. For my first year I kinda skirted around taking Math, since I had never done that well & hadn't enjoyed it much, so I wound up taking Statistics second semester of freshman year. I did okay, I got a C+ which is one of my worse grades, but considering my skills in the subject was acceptable. My professor was well-meaning and helpful outside of classes, but she had a very thick accent & I was very distracted for much of that semester.

Now this semester I'm taking Applied Finite Mathematics, and am doing alright. Much of the content so far has been a retread, but that's fine for me since I forgot most of the stuff & the presentation is far better this time, it's sinking in quite a bit easier. So far we've been going over the basics of Set Theory, Probability, Permutations, and some other stuff - kinda slowly tbh.

 




Well that was quite a bit of a preamble, tl;dr I was never all that good at or interested in math. However, I want to foster a healthier engagement with mathematics and so far have found entrance points of interest in discussions on the history and philosophy of mathematics. I think I could come to a better understanding and maybe even appreciation for math if I studied it on my own in some fashion.

So I've been looking into it, and I see that Dover publishes quite a range of affordable, slightly old math textbooks. Now, considering my background, (I am probably quite rusty but somewhat secure in Elementary Algebra, and to be honest I would not trust anything I could vaguely remember from 2 years ago in "Advanced" Pre-Calculus) what would be a good book to try and read/practice with/work through to make math 1) more approachable to me, 2) get a better and more rewarding understanding by attacking the stuff on my own, and/or 3) broaden my knowledge and ability in various math subjects?

Here are some interesting ones I've found via cursory search, I've so far just been looking at Dover's selections but feel free to recommend other stuff, just keep in mind I'd have to keep a rather small budget, especially since this is really on the side (considering my course of study, I really won't have to take any more math courses):
Prelude to Mathematics
A Book of Set Theory - More relevant to my current course & have heard good things about it
Linear Algebra
Number Theory
A Book of Abstract Algebra
Basic Algebra I
Calculus: An Intuitive and Physical Approach
Probability Theory: A Concise Course
A Course on Group Theory
Elementary Functional Analysis

There would have been a time that I would have suggested getting a curriculum
text book and going through that, but if you're doing this for independent work
I wouldn't really suggest that as the odds are you're not going to be using a
very good source.

Going on the typical

Arithmetic > Algebra > Calculus

****

Arithmetic

Arithmetic refresher. Lots of stuff in here - not easy.


I think you'd be set after this really. It's a pretty terse text in general.

*****

Algebra

Algebra by Chrystal Part I

Algebra by Chrystal Part II

You can get both of these algebra texts online easily and freely from the search

chrystal algebra part I filetype:pdf

chrystal algebra part II filetype:pdf

I think that you could get the first (arithmetic) text as well, personally I
prefer having actual books for working. They're also valuable for future
reference. This filetype:pdf search should be remembered and used liberally
for finding things such as worksheets etc (eg trigonometry worksheet<br /> filetype:pdf for a search...).

Algebra by Gelfland

No where near as comprehensive as chrystals algebra, but interesting and well
written questions (search for 'correspondence series' by Gelfand).

Calculus

Calculus made easy - Thompson

This text is really good imo, there's little rigor in it but for getting a
handle on things and bashing through a few practical problems it's pretty
decent. It's all single variable. If you've done the algebra and stuff before
this then this book would be easy.

Pauls Online Notes (Calculus)

These are just a solid set of Calculus notes, there're lots of examples to work
through which is good. These go through calc I, II, III... So a bit further than
you've asked (I'm not sure why you state up to calc II but ok).

Spivak - Calculus

If you've gone through Chrystals algebra then you'll be used to a formal
approach. This text is only single variable calculus (so that might be calc I
and II in most places I think, ? ) but it's extremely well written and often
touted as one of the best Calculus books written. It's very pure, where as
something like Stewart has a more applied emphasis.

**

Geometry

I've got given any geometry sources, I'm not too sure of the best source for
this or (to be honest) if you really need it for the above. If someone has
good geometry then they're certainly better off, many proofs are given
gemetrically as well and having an intuition for these things is only going to
be good. But I think you can get through without a formal course on it.... I'm
not confident suggesting things on it though, so I'll leave it to others. Just
thought I'd mention it.

****

8 to 12 hours is really not that much, but it should be enough to learn something interesting! I would start with category theory if you can. I liked Emily Riehl's categories in context for an intro, but it will go a little slow for how little time you have to learn the basics. Maybe the first chapter of Algebra: Chapter 0 by Aleffi? [EDIT: you might want to find a "reasonably priced" pdf version of this book if you do decide to use it -- it's pretty expensive] If you can get through that, and understand a little about how types fit into the picture, you should be able to present the basic idea behind curry-howard-lambek. IIRC you do not need functors or natural transformations ("higher level" categorical concepts), as important as they usually are, to get through this topic; Aleffi doesn't go over them in his very first intro to categories which is why I'm recommending him. /u/VFB1210 has some very good recommendations above as well.

I am trying to think of a better introduction to type theory than HoTT -- if you can learn about types without getting infinity categories and homotopy equivalence mixed up in them, I would. Type theory is actually pretty cool and sleek.

Here's a selection of intro-to-type theory resources I found:

Programming in Martin-Löf's Type Theory is
pretty long, but you can probably put together a mini-course as follows: read chapters 1 & 2 quickly, skim 3, and then read 19 and 20.

The lecture notes from Paul Levy's mini-course on the typed lambda calculus form a pretty compact resource, but I'm not sure this will be super useful to you right now -- keep it in mind but don't start off with it. Since it is in lecture-note style it is also pretty hard to keep up with if you don't already kind of know what he's talking about.


Constable's Naïve Computational Type Theory seems to be different from the usual intro to types -- it's done in the style of the old Naive Set Theory text, which means you're supposed to be sort of guided intuitively into knowing how types work. It looks like the intuition all comes from programming, and if you know something functional and hopefully strongly typed (OCaml, SML, Haskell, or Lisp come to mind) you will probably get the most out of it. I think that's true about type theory in general, actually.

PFPL by Bob Harper is probably a stretch -- you won't find it useful right at the moment, but if you want to spend 2 semesters really getting to know how type theory encapsulates pretty much any modern programming paradigm (typed languages, "untyped" languages, parallel execution, concurrency, etc.) this book is top-tier. The preview edition doesn't have everything from the whole book but is a pretty big portion of it.

I started from scratch on the formal CS side, with an emphasis on program analysis, and taught myself the following starting from 2007. If you're in the United States, I recommend BookFinder to save money buying these things used.

On the CS side:

• Basic automata/formal languages/Turing machines; Sipser is recommended here.
  
• Basic programming language theory; I used University of Washington CSE P505 online video lectures and materials and can recommend it.
  
• Formal semantics; Semantics with Applications is good.
  
• Compilers. You'll need several resources for this; my personal favorites for an introductory text are Appel's ML book or Programming Language Pragmatics, and Muchnick is mandatory for an advanced understanding. All of the graph theory that you need for this type of work should be covered in books such as these.
  
• Algorithms. I used several books; for a beginner's treatment I recommend Dasgupta, Papadimitriou, and Vazirani; for an intermediate treatment I recommend MIT's 6.046J on Open CourseWare; for an advanced treatment, I liked Algorithmics for Hard Problems.
  
  On the math side, I was advantaged in that I did my undergraduate degree in the subject. Here's what I can recommend, given five years' worth of hindsight studying program analysis:
  
  
• You run into abstract algebra a lot in program analysis as well as in cryptography, so it's best to begin with a solid foundation along those lines. There's a lot of debate as to what the best text is. If you're never touched the subject before, Gallian is very approachable, if not as deep and rigorous as something like Dummit and Foote.
  
• Order theory is everywhere in program analysis. Introduction to Lattices and Order is the standard (read at least the first two chapters; the more you read, the better), but I recently picked up Lattices and Ordered Algebraic Structures and am enjoying it.
  
• Complexity theory. Arora and Barak is recommended.
  
• Formal logic is also everywhere. For this, I recommend the first few chapters in The Calculus of Computation (this is an excellent book; read the whole thing).
  
• Computability, undecidability, etc. Not entirely separate from previous entries, but read something that treats e.g. Goedel's theorems, for instance The Undecidable.
  
• Decision procedures. Read Decision Procedures.
  
• Program analysis, the "accessible" variety. Read the BitBlaze publications starting from the beginning, followed by the BAP publications. Start with these two: TaintCheck and All You Ever Wanted to Know About Dynamic Taint Analysis and Forward Symbolic Execution. (BitBlaze and BAP are available in source code form, too -- in OCaml though, so you'll want to learn that as well.) David Brumley's Ph.D. thesis is an excellent read, as is David Molnar's and Sean Heelan's. This paper is a nice introduction to software model checking. After that, look through the archives of the RE reddit for papers on the "more applied" side of things.
  
• Program analysis, the "serious" variety. Principles of Program Analysis is an excellent book, but you'll find it very difficult even if you understand all of the above. Similarly, Cousot's MIT lecture course is great but largely unapproachable to the beginner. I highly recommend Value-Range Analysis of C Programs, which is a rare and thorough glimpse into the development of an extremely sophisticated static analyzer. Although this book is heavily mathematical, it's substantially less insane than Principles of Program Analysis. I also found Gogul Balakrishnan's Ph.D. thesis, Johannes Kinder's Ph.D. thesis, Mila Dalla Preda's Ph.D. thesis, Antoine Mine's Ph.D. thesis, and Davidson Rodrigo Boccardo's Ph.D. thesis useful.
  
• If you've gotten to this point, you'll probably begin to develop a very selective taste for program analysis literature: in particular, if it does not have a lot of mathematics (actual math, not just simple concepts formalized), you might decide that it is unlikely to contain a lasting and valuable contribution. At this point, read papers from CAV, SAS, and VMCAI. Some of my favorite researchers are the Z3 team, Mila Dalla Preda, Joerg Brauer, Andy King, Axel Simon, Roberto Giacobazzi, and Patrick Cousot. Although I've tried to lay out a reasonable course of study hereinbefore regarding the mathematics you need to understand this kind of material, around this point in the course you'll find that the creature we're dealing with here is an octopus whose tentacles spread in every direction. In particular, you can expect to encounter topology, category theory, tropical geometry, numerical mathematics, and many other disciplines. Program analysis is multi-disciplinary and has a hard time keeping itself shoehorned in one or two corners of mathematics.
  
• After several years of wading through program analysis, you start to understand that there must be some connection between theorem-prover based methods and abstract interpretation, since after all, they both can be applied statically and can potentially produce similar information. But what is the connection? Recent publications by Vijay D'Silva et al (1, 2, 3, 4, 5) and a few others (1 2 3 4) have begun to plough this territory.
  
• I'm not an expert at cryptography, so my advice is basically worthless on the subject. However, I've been enjoying the Stanford online cryptography class, and I liked Understanding Cryptography too. Handbook of Applied Cryptography is often recommended by people who are smarter than I am, and I recently picked up Introduction to Modern Cryptography but haven't yet read it.
  
  Final bit of advice: you'll notice that I heavily stuck to textbooks and Ph.D. theses in the above list. I find that jumping straight into the research literature without a foundational grounding is perhaps the most ill-advised mistake one can make intellectually. To whatever extent that what you're interested in is systematized -- that is, covered in a textbook or thesis already, you should read it before digging into the research literature. Otherwise, you'll be the proverbial blind man with the elephant, groping around in the dark, getting bits and pieces of the picture without understanding how it all forms a cohesive whole. I made that mistake and it cost me a lot of time; don't do the same.

Machine learning is largely based on the following chain of mathematical topics

Calculus (through Vector, could perhaps leave out a subsequent integration techniques course)

Linear Algebra (You are going to be using this all, a lot)

Abstract Algebra (This isn't always directly applicable but it is good to know for computer science and the terms of groups, rings, algebras etc will show up quite a bit)

General Topology (Any time we are going to deal with construction of a probability space on some non trivial manifold, we will need this. While most situations are based on just Borel sets in R^n or C^n things like computer vision, genomics, etc are going to care about Random Elements rather than Random Variables and those are constructed in topological spaces rather than metric ones. This is also helpful for understanding definitions in well known algorithms like Manifold Training)

Real Analysis (This is where you learn proper constructive formulations and a bit of measure theory as well as bounding theorems etc)

Complex Analysis (This is where you will get a proper treatment of Hilbert Spaces, Holomorphic functions etc, honestly unless you care about QM / QFT, P-chem stuff in general like molecular dynamics, you are likely not going to need a full course in this for most ML work, but I typically just tell people to read the full Rudin: Real and Complex Analysis. You'll get the full treatment fairly briefly that way)

Probability Theory (Now that you have your Measure theory out of the way from Real Analysis, you can take up a proper course on Measure Theoretic Probability Theory. Random Variables should be defined here as measurable functions etc, if they aren't then your book isn't rigorous enough imho.)

Ah, Statistics. Statistics sits atop all of that foundational mathematics, it is divided into two main philosophical camps. The Frequentists, and the Bayesians. Any self respecting statistician learns both.

After that, there are lots, and lots, and lots, of subfields and disciplines when it comes to statistical learning.

A sample of what is on my reference shelf includes:

Real and Complex Analysis by Rudin

Functional Analysis by Rudin

A Book of Abstract Algebra by Pinter

General Topology by Willard

Machine Learning: A Probabilistic Perspective by Murphy

Bayesian Data Analysis Gelman

Probabilistic Graphical Models by Koller

Convex Optimization by Boyd

Combinatorial Optimization by Papadimitriou

An Introduction to Statistical Learning by James, Hastie, et al.

The Elements of Statistical Learning by Hastie, et al.

Statistical Decision Theory by Liese, et al.

Statistical Decision Theory and Bayesian Analysis by Berger

I will avoid listing off the entirety of my shelf, much of it is applications and algorithms for fast computation rather than theory anyway. Most of those books, though, are fairly well known and should provide a good background and reference for a good deal of the mathematics you should come across. Having a solid understanding of the measure theoretic underpinnings of probability and statistics will do you a great deal--as will a solid facility with linear algebra and matrix / tensor calculus. Oh, right, a book on that isn't a bad idea either... This one is short and extends from your vector classes

Tensor Calculus by Synge

Anyway, hope that helps.

Yet another lonely data scientist,

Tim.

I was going to chime in before in this post with normal subgroups but I didn't feel I knew much about particle physics to comment. But if you're really interested you should pick up a book on basic undergraduate math group theory, it's super simple to pick up and can even be used as a book that you casually go through while riding the bus or something for someone of your caliber.

For a basic understanding, go through chapter 2.

Or if you would like a great, concise and motivating textbook, this dover book on abstract algebra is amazing and a bargain for $10. I can't tell you how much I love this little book.

But here is a thread with someone asking about more advanced textbooks for graduate group theory physics if you would love to get a deeper understanding of it and my comment (the top one) has a list of several free online group theory books that you can check out here.

It's really such a beautiful subject that I recommend anyone learn the basics at least. :)

> Mathematical Logic

It's not exactly Math Logic, just a bunch of techniques mathematicians use. Math Logic is an actual area of study. Similarly, actual Set Theory and Proof Theory are different from the small set of techniques that most mathematicians use.

Also, looks like you have chosen mostly old, but very popular books. While studying out of these books, keep looking for other books. Just because the book was once popular at a school, doesn't mean it is appropriate for your situation. Every year there are new (and quite frankly) pedagogically better books published. Look through them.

Here's how you find newer books. Go to Amazon. In the search field, choose "Books" and enter whatever term that interests you. Say, "mathematical proofs". Amazon will come up with a bunch of books. First, sort by relevance. That will give you an idea of what's currently popular. Check every single one of them. You'll find hidden jewels no one talks about. Then sort by publication date. That way you'll find newer books - some that haven't even been published yet. If you change the search term even slightly Amazon will come up with completely different batch of books. Also, search for books on Springer, Cambridge Press, MIT Press, MAA and the like. They usually house really cool new titles. Here are a couple of upcoming titles that might be of interest to you: An Illustrative Introduction to Modern Analysis by Katzourakis/Varvarouka, Understanding Topology by Shaun Ault. I bet these books will be far more pedagogically sound as compared to the dry-ass, boring compendium of facts like the books by Rudin.

If you want to learn how to do routine proofs, there are about one million titles out there. Also, note books titled Discrete Math are the best for learning how to do proofs. You get to learn techniques that are not covered in, say, How to Prove It by Velleman. My favorites are the books by Susanna Epp, Edward Scheinerman and Ralph Grimaldi. Also, note a lot of intro to proofs books cover much more than the bare minimum of How to Prove It by Velleman. For example, Math Proofs by Chartrand et al has sections about doing Analysis, Group Theory, Topology, Number Theory proofs. A lot of proof books do not cover proofs from Analysis, so lately a glut of new books that cover that area hit the market. For example, Intro to Proof Through Real Analysis by Madden/Aubrey, Analysis Lifesaver by Grinberg(Some of the reviewers are complaining that this book doesn't have enough material which is ridiculous because this book tackles some ugly topological stuff like compactness in the most general way head-on as opposed to most into Real Analysis books that simply shy away from it), Writing Proofs in Analysis by Kane, How to Think About Analysis by Alcock etc.

Here is a list of extremely gentle titles: Discovering Group Theory by Barnard/Neil, A Friendly Introduction to Group Theory by Nash, Abstract Algebra: A Student-Friendly Approach by the Dos Reis, Elementary Number Theory by Koshy, Undergraduate Topology: A Working Textbook by McClusckey/McMaster, Linear Algebra: Step by Step by Singh (This one is every bit as good as Axler, just a bit less pretentious, contains more examples and much more accessible), Analysis: With an Introduction to Proof by Lay, Vector Calculus, Linear Algebra, and Differential Forms by Hubbard & Hubbard, etc

This only scratches the surface of what's out there. For example, there are books dedicated to doing proofs in Computer Science(for example, Fundamental Proof Methods in Computer Science by Arkoudas/Musser, Practical Analysis of Algorithms by Vrajitorou/Knight, Probability and Computing by Mizenmacher/Upfal), Category Theory etc. The point is to keep looking. There's always something better just around the corner. You don't have to confine yourself to books someone(some people) declared the "it" book at some point in time.

Last, but not least, if you are poor, peruse Libgen.

/u/another_user_name posted this list a while back. Actual aerospace textbooks are towards the bottom but you'll need a working knowledge of the prereqs first.

Non-core/Pre-reqs:

Mathematics:

Calculus.

1-4) Calculus, Stewart -- This is a very common book and I felt it was ok, but there's mixed opinions about it. Try to get a cheap, used copy.

1-4) Calculus, A New Horizon, Anton -- This is highly valued by many people, but I haven't read it.

1-4) Essential Calculus With Applications, Silverman -- Dover book.

More discussion in this reddit thread.

Linear Algebra

3) Linear Algebra and Its Applications,Lay -- I had this one in school. I think it was decent.

3) Linear Algebra, Shilov -- Dover book.

Differential Equations

4) An Introduction to Ordinary Differential Equations, Coddington -- Dover book, highly reviewed on Amazon.

G) Partial Differential Equations, Evans

G) Partial Differential Equations For Scientists and Engineers, Farlow

More discussion here.

Numerical Analysis

5) Numerical Analysis, Burden and Faires

Chemistry:

1. General Chemistry, Pauling is a good, low cost choice. I'm not sure what we used in school.
   
   

   Physics:
   

   
   2-4) Physics, Cutnel -- This was highly recommended, but I've not read it.
   
   

   Programming:
   

   
   

   Introductory Programming
   

   
   Programming is becoming unavoidable as an engineering skill. I think Python is a strong introductory language that's got a lot of uses in industry.
   
   

2. Learning Python, Lutz
   
   
3. Learn Python the Hard Way, Shaw -- Gaining popularity, also free online.
   
   

   Core Curriculum:
   

   
   

   Introduction:
   

   
   

4. Introduction to Flight, Anderson
   
   

   Aerodynamics:
   

   
   

5. Introduction to Fluid Mechanics, Fox, Pritchard McDonald
   
   
6. Fundamentals of Aerodynamics, Anderson
   
   
7. Theory of Wing Sections, Abbot and von Doenhoff -- Dover book, but very good for what it is.
   
   
8. Aerodynamics for Engineers, Bertin and Cummings -- Didn't use this as the text (used Anderson instead) but it's got more on stuff like Vortex Lattice Methods.
   
   
9. Modern Compressible Flow: With Historical Perspective, Anderson
   
   
10. Computational Fluid Dynamics, Anderson
    
    

    Thermodynamics, Heat transfer and Propulsion:
    

    
    

11. Introduction to Thermodynamics and Heat Transfer, Cengel
    
    
12. Mechanics and Thermodynamics of Propulsion, Hill and Peterson
    
    

    Flight Mechanics, Stability and Control
    

    
    5+) Flight Stability and Automatic Control, Nelson
    
    5+)[Performance, Stability, Dynamics, and Control of Airplanes, Second Edition](http://www.amazon.com/Performance-Stability-Dynamics-Airplanes-Education/dp/1563475839/ref=sr_1_1?ie=UTF8&qid=1315534435&sr=8-1, Pamadi) -- I gather this is better than Nelson
    
    

13. Airplane Aerodynamics and Performance, Roskam and Lan
    
    

    Engineering Mechanics and Structures:
    

    
    3-4) Engineering Mechanics: Statics and Dynamics, Hibbeler
    
    

14. Mechanics of Materials, Hibbeler
    
    
15. Mechanical Vibrations, Rao
    
    
16. Practical Stress Analysis for Design Engineers: Design & Analysis of Aerospace Vehicle Structures, Flabel
    
    6-8) Analysis and Design of Flight Vehicle Structures, Bruhn -- A good reference, never really used it as a text.
    
    
17. An Introduction to the Finite Element Method, Reddy
    
    G) Introduction to the Mechanics of a Continuous Medium, Malvern
    
    G) Fracture Mechanics, Anderson
    
    G) Mechanics of Composite Materials, Jones
    
    

    Electrical Engineering
    

    
    

18. Electrical Engineering Principles and Applications, Hambley
    
    

    Design and Optimization
    

    
    

19. Fundamentals of Aircraft and Airship Design, Nicolai and Carinchner
    
    
20. Aircraft Design: A Conceptual Approach, Raymer
    
    
21. Engineering Optimization: Theory and Practice, Rao
    
    

    Space Systems
    

    
    

22. Fundamentals of Astrodynamics and Applications, Vallado
    
    
23. Introduction to Space Dynamics, Thomson -- Dover book
    
    
24. Orbital Mechanics, Prussing and Conway
    
    
25. Fundamentals of Astrodynamics, Bate, Mueller and White
    
    
26. Space Mission Analysis and Design, Wertz and Larson

I would guess that career prospects are a little worse than CS for undergrad degrees, but since my main concern is where a phd in math will take me, you should get a second opinion on that.

Something to keep in mind is that "higher" math (the kind most students start to see around junior level) is in many ways very different from the stuff before. I hated calculus and doing calculations in general, and was pursuing a math minor because I thought it might help with job prospects, but when I got to the more abstract stuff, I loved it. It's easily possible that you'll enjoy both, I'm just pointing out that enjoying one doesn't necessarily imply enjoying the other. It's also worth noting that making the transition is not easy for most of us, and that if you struggle a lot when you first have to focus a lot of time on proving things, it shouldn't be taken as a signal to give up if you enjoy the material.

This wouldn't be necessary, but if you like, here are some books on abstract math topics that are aimed towards beginners you could look into to get a basic idea of what more abstract math is like:

• theoretical computer science (essentially a math text)
  
  
• set theory
  
  
• linear algebra
  
  
• algebra
  
  
• predicate calculus
  
  Different mathematicians gravitate towards different subjects, so it's not easy to predict which you would enjoy more. I'm recommending these five because they were personally helpful to me a few years ago and I've read them in full, not because I don't think anyone can suggest better. And of course, you could just jump right into coursework like how most of us start. Best of luck!
  
  (edit: can't count and thought five was four)

I don't say this to be discouraging: Most people don't really have any idea what doing Physics at a high level looks like. I decided in High School that I wanted to be a physicist, and as luck would have it I'm a graduate student and I still enjoy it, but truth be told, the exposure you have in High School doesn't really prepare you for the reality. All that to say: There's no reason to decide at thirteen years old that you need a PhD in Physics! Maybe once you learn math beyond trig you'll decide it isn't for you, or maybe you'll love math and want to switch to a math degree.

All right, now that that's out of the way... You said you're learning trig, that's good, you need it. You also need some basic algebra skills. Then try to teach yourself basic calculus (limits, derivatives, integrals). Then you want to learn Linear Algebra and at least Ordinary Differential Equations.

You can also do some basic physics reading before you've learned the essentials. I really like George Gamow's books for this - he was a very well know and important physicist who also happened to write very accessible books that are very much for lay people but that also don't shy away completely from the math. I really enjoyed this one in particular.

For mathematics, I love Dover books - they're cheap AND good. Shilov, I've found, is clear and readable. This might not be introductory level, but it's inexpensive and let's you see what you're getting yourself into.

Last bit of advice for Physics is what one of my old high school teachers used to say - draw, label, and you can't go wrong. It's still mostly true.

Pick up mathematics. Now if you have never done math past the high school and are an "average person" you probably cringed.

Math (an "actual kind") is nothing like the kind of shit you've seen back in grade school. To break into this incredible world all you need is to know math at the level of, say, 6th grade.

Intro to Math:

1. Book of Proof by Richard Hammack. This free book will show/teach you how mathematicians think. There are other such books out there. For example,
   
   

• How to Prove It: A Structured Approach by Daniel Velleman
  
  
• Mathematical Proofs: A Transition to Advanced Mathematics by Gary Chartrand et al
  
  
• Discrete Mathematics with Applications by Susanna Epp
  
  These books only serve as samplers because they don't even begin to scratch the surface of math. After you familiarized yourself with the basics of writing proofs you can get started with intro to the largest subsets of math like:
  
  Intro to Abstract Algebra:
  
  
• Linear Algebra: Step by Step by Singh
  
  
• Abstract Algebra: A Student-Friendly Approach by the Dos Reis
  
  
• Numbers and Symmetry: An Introduction to Algebra by Johnston/Richman
  
  
• Discovering Group Theory: A Transition to Advanced Mathematics by Barnard/Neil
  
  
• A Friendly Introduction to Group Theory by Nash
  
  
• Linear Algebra Done Right by Sheldon Axler
  
  There are tons more books on abstract/modern algebra. Just search them on Amazon. Some of the famous, but less accessible ones are
  
  
• Algebra: Chapter 0 by Aluffi
  
  
• Algebra by Maclane/Birkhoff
  
  
• Algebra by Lang
  
  
• Advanced Linear Algebra by Steven Roman
  
  Intro to Real Analysis:
  
  
• The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs by Grinberg
  
  
• Understanding Analysis by Abbot
  
  
• Writing Proofs in Analysis by Kane
  
  
• The Lebesgue Integral for Undergraduates by Johnston
  
  Again, there are tons of more famous and less accessible books on this subject. There are books by Rudin, Royden, Kolmogorov etc.
  
  Ideally, after this you would follow it up with a nice course on rigorous multivariable calculus. Easiest and most approachable and totally doable one at this point is
  
  
• Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach by the Hubbards
  
  At this point it's clear there are tons of more famous and less accessible books on this subject :) I won't list them because if you are at this point of math development you can definitely find them yourself :)
  
  From here you can graduate to studying category theory, differential geometry, algebraic geometry, more advanced texts on combinatorics, graph theory, number theory, complex analysis, probability, topology, algorithms, functional analysis etc
  
  Most listed books and more can be found on libgen if you can't afford to buy them. If you are stuck on homework, you'll find help on [MathStackexchange] (https://math.stackexchange.com/questions).
  
  Good luck.

I was in the same position as you in high school (and am finishing my math major this semester). Calculus is not "math" in the sense you're referring to it, which is pure mathematics, without application, just theory and logic. Calculus, as it is taught in high school, is taught as a tool, not as a theory. It is boring, tedious, and has no aesthetic appeal because it is largely taught as rote memorization.

Don't let this bad experience kill your enthusiasm. I'm not sure what specifically to recommend to you to perk your enthusiasm, but what I did in high school was just click around Wikipedia entries. A lot of them are written in layman enough terms to give you a glimpse and you inspire your interest. For example, I remember being intrigued by the Fibonacci series and how, regardless of the starting terms, the ratio between the (n-1)th and nth terms approaches the golden ratio; maybe look at the proof of that to get an idea of what math is beyond high school calculus. I remember the Riemann hypothesis was something that intrigued me, as well as Fermat's Last Theorem, which was finally proved in the 90s by Andrew Wiles (~350 years after Fermat suggested the theorem). (Note: you won't be able to understand the math behind either, but, again, you can get a glimpse of what math is and find a direction you'd like to work in).

Another thing that I wish someone had told me when I was in your position is that there is a lot of legwork to do before you start reaching the level of mathematics that is truly aesthetically appealing. Mathematics, being purely based on logic, requires very stringent fundamental definitions and techniques to be developed first, and early. Take a look at axiomatic set theory as an example of this. Axiomatic set theory may bore you, or it may become one of your interests. The concept and definition of a set is the foundation for mathematics, but even something that seems as simple as this (at first glance) is difficult to do. Take a look at Russell's paradox. Incidentally, that is another subject that captured my interest before college. (Another is Godel's incompleteness theorem, again, beyond your or my understanding at the moment, but so interesting!)

In brief, accept that math is taught terribly in high school, grunt through the semester, and try to read farther ahead, on your own time, to kindle further interest.

As an undergrad, I don't believe I yet have the hindsight to recommend good books for an aspiring math major (there are plenty of more knowledgeable and experienced Redditors who could do that for you), but here is a list of topics that are required for my undergrad math degree, with links to the books that my school uses:

• elementary real analysis
  
• linear algebra
  
• differential equations
  
• abstract algebra
  
  And a couple electives:
  
  
• topology
  
• graph theory
  
  And a couple books I invested in that are more advanced than the undergrad level, which I am working through and enjoy:
  
  
• abstract algebra
  
• topology
  
  Lastly, if you don't want to spend hundreds of dollars on books that you might not end up using in college, take a look at Dover publications (just search "Dover" on Amazon). They tend to publish good books in paperback for very cheap ($5-$20, sometimes up to $40 but not often) that I read on my own time while trying to bear high school calculus. They are still on my shelf and still get use.

This is just my perspective, but . . .

I think there are two separate concerns here: 1) the "process" of mathematics, or mathematical thinking; and 2) specific mathematical systems which are fundamental and help frame much of the world of mathematics.

​

Abstract algebra is one of those specific mathematical systems, and is very important to understand in order to really understand things like analysis (e.g. the real numbers are a field), linear algebra (e.g. vector spaces), topology (e.g. the fundamental group), etc.

​

I'd recommend these books, which are for the most part short and easy to read, on mathematical thinking:

​

How to Solve It, Polya ( https://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069111966X ) covers basic strategies for problem solving in mathematics

Mathematics and Plausible Reasoning Vol 1 & 2, Polya ( https://www.amazon.com/Mathematics-Plausible-Reasoning-Induction-Analogy/dp/0691025096 ) does a great job of teaching you how to find/frame good mathematical conjectures that you can then attempt to prove or disprove.

Mathematical Proof, Chartrand ( https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094 ) does a good job of teaching how to prove mathematical conjectures.

​

As for really understanding the foundations of modern mathematics, I would start with Concepts of Modern Mathematics by Ian Steward ( https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Books/dp/0486284247 ) . It will help conceptually relate the major branches of modern mathematics and build the motivation and intuition of the ideas behind these branches.

​

Abstract algebra and analysis are very fundamental to mathematics. There are books on each that I found gave a good conceptual introduction as well as still provided rigor (sometimes at the expense of full coverage of the topics). They are:

​

A Book of Abstract Algebra, Pinter ( https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178 )

​

Understanding Analysis, Abbott ( https://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/1493927116 ).

​

If you read through these books in the order listed here, it might provide you with that level of understanding of mathematics you talked about.

For math you're going to need to know calculus, differential equations (partial and ordinary), and linear algebra.

For calculus, you're going to start with learning about differentiating and limits and whatnot. Then you're going to learn about integrating and series. Series is going to seem a little useless at first, but make sure you don't just skim it, because it becomes very important for physics. Once you learn integration, and integration techniques, you're going to want to go learn multi-variable calculus and vector calculus. Personally, this was the hardest thing for me to learn and I still have problems with it.

While you're learning calculus you can do some lower level physics. I personally liked Halliday, Resnik, and Walker, but I've also heard Giancoli is good. These will give you the basic, idealized world physics understandings, and not too much calculus is involved. You will go through mechanics, electromagnetism, thermodynamics, and "modern physics". You're going to go through these subjects again, but don't skip this part of the process, as you will need the grounding for later.

So, now you have the first two years of a physics degree done, it's time for the big boy stuff (that is the thing that separates the physicists from the engineers). You could get a differential equations and linear algebra books, and I highly suggest you do, but you could skip that and learn it from a physics reference book. Boaz will teach you the linear and the diffe q's you will need to know, along with almost every other post-calculus class math concept you will need for physics. I've also heard that Arfken, Weber, and Harris is a good reference book, but I have personally never used it, and I dont' know if it teaches linear and diffe q's. These are pretty much must-haves though, as they go through things like fourier series and calculus of variations (and a lot of other techniques), which are extremely important to know for what is about to come to you in the next paragraph.

Now that you have a solid mathematical basis, you can get deeper into what you learned in Halliday, Resnik, and Walker, or Giancoli, or whatever you used to get you basis down. You're going to do mechanics, E&M, Thermodynamis/Statistical Analysis, and quantum mechanics again! (yippee). These books will go way deeper into theses subjects, and need a lot more rigorous math. They take that you already know the lower-division stuff for granted, so they don't really teach those all that much. They're tough, very tough. Obvioulsy there are other texts you can go to, but these are the one I am most familiar with.

A few notes. These are just the core classes, anybody going through a physics program will also do labs, research, programming, astro, chemistry, biology, engineering, advanced math, and/or a variety of different things to supplement their degree. There a very few physicists that I know who took the exact same route/class.

These books all have practice problems. Do them. You don't learn physics by reading, you learn by doing. You don't have to do every problem, but you should do a fair amount. This means the theory questions and the math heavy questions. Your theory means nothing without the math to back it up.

Lastly, physics is very demanding. In my experience, most physics students have to pretty much dedicate almost all their time to the craft. This is with instructors, ta's, and tutors helping us along the way. When I say all their time, I mean up until at least midnight (often later) studying/doing work. I commend you on wanting to self-teach yourself, but if you want to learn physics, get into a classroom at your local junior college and start there (I think you'll need a half year of calculus though before you can start doing physics). Some of the concepts are hard (very hard) to understand properly, and the internet stops being very useful very quickly. Having an expert to guide you helps a lot.

Good luck on your journey!

There is a lot in the way of resources for new triathletes these days. For your first tri, grab a free training plan online that matches where you are now. Read Beginner Triathlete in your free time; it's a fantastic resource, and I still refer back to its articles all the time. Train your butt off. You don't need to buy a sweet road bike up front, though you sound like you're pretty sure that you want to get into this stuff.

Feel free to skimp on some of the gear for your first race. No one wants to find out that they dislike triathlon after dumping $3k on tri gear. You can race on an old bike with platform pedals. Unless it's really cold, you don't need a wetsuit. The first race is where you truly find out if this is the sport for you. EDIT: Someone mentioned a bike fit. If you're riding an old bike, Competitive Cyclist's Bike Fit Calculator will get you pretty darn close--good enough to get through your first race. Use the road calculator mode if you don't have aerobars off the bat.

After you finish your first race, sit down and think about what you liked, what you did well with, what needs improvement. Get Joe Friel's Triathlete's Training Bible, read it cover to cover. Read it again. Figure out your long-term training plan for the rest of that season. If you start your base training in the winter/early spring and pick an early first race, you can get a full season of sprints and/or Olympics in.

Look for a triathlon club in your area or find a coach or drag a friend into the insanity of triathlon; the camaraderie is priceless in keeping your spirits up during long seasons packed full of hard training and races.

As far as spending money on triathlon "stuff" goes: Remember during your first couple seasons that gadgets and gizmos and aero gear are great, but what really makes the difference is eating well and training hard.

After that, the gear that makes your races more comfortable is the best place to spend your money (tri shorts if you don't them, cycling kit and proper running shorts for training). Then, points of contact with the bike and pool "toys" will improve your efficiency and form (new bike w/ fit if req'd, clipless pedals, shoes, aerobars, pull buoy, kickboard, fins, paddles... a bike computer probably fits in here, as well). Beyond that, you're at a wetsuit and then the "extras" like aero helmet, race wheels, power meters, GPS, HRM, tri bike, speedsuits, etc., etc. That's the approximate map for spending in my book, anyhow. There's practically no limit to the amount of stuff you can buy for triathlon, and as you train more, you'll know what needs to come next.

Hey! I am a math major at Harvey Mudd College (who went to high school in the Pacific NW!). I'll answer from what I've seen.

1. There seems to be tons. At least I keep being told there are tons! My school has a lot of recruiters come by who are interested in math people!
   
   
2. I can definitely recommend HMC, but I would also consider MIT, Caltech, Carnegie Melon, etc. I've heard UW is good, too!
   
   
3. Most all of linear algebra is important later on. I will say that many texts treat linear algebra the same as "matrix algebra", which it is not. Linear algebra is much more general, and deals with things called vector spaces. Matrix algebra is a specific case of linear algebra. If you want a good linear algebra text (though it might be a bit difficult), check out http://www.amazon.com/Linear-Algebra-Right-Sheldon-Axler/dp/0387982582
   
   End: Also, if you wanna learn something cool, I'd check out Discrete math. It's usually required for both a math or CS major, and it's some of the coolest undergraduate math out there. Oh, and, unlike some other math, it's not terrible to self-teach. :)
   
   Good luck! Math is awesome!

I think the advice given in the rest of the thread is pretty good, though some of it a little naive. The suggestion that differential equations or applied math somehow should not be of interest is silly. A lot of it builds the motivation for some of the abstract stuff which is pretty cool, and a lot of it has very pure problems associated with it. In addition I think after (or rather alongside) your initial calculus education is a good time to look at some other things before moving onto more difficult topics like abstract algebra, topology, analysis etc.

The first course I took in undergrad was a course that introduced logic, writing proofs, as well as basic number theory. The latter was surprisingly useful as it built modular arithmetic which gave us a lot of groups and rings to play with in subsequent algebra courses. Unfortunately the textbook was god awful. I've heard good things about the following two sources and together they seem to cover the content:

How to prove it

Number theory

After this I would take a look at linear algebra. This a field with a large amount of uses in both pure and applied math. It is useful as it will get you used to doing algebraic proofs, it takes a look at some common themes in algebra, matrices (one of the objects studied) are also used thoroughly in physics and applied mathematics and the knowledge is useful for numerical approximations of ordinary and partial differential equations. The book I used Linear Algebra by Friedberg, Insel and Spence, but I've heard there are better.

At this point I think it would be good to move onto Abstract Algebra, Analysis and Topology. I think Farmerje gave a good list.

There's many more topics that you could possibly cover, ODEs and PDEs are very applicable and have a rich theory associated with them, Complex Analysis is a beautiful subject, but I think there's plenty to keep you busy for the time being.

When I first took abstract algebra a couple years ago, we worked out of Fraleigh's A First Course in Abstract Algebra. My classmates and I thoroughly enjoyed it. Well written, well paced, and all around an enlightening introductory read about my most favorite field of math :)

I think it's perfectly tractable for any interested student with a good command of algebra.

Edit: Oh I misread the question. If he's already gone through these elementary parts of abstract algebra, that's about the entire undergraduate coursework I had. The one quarter of graduate algebra I ended up taking went over the orbit-stabilizer theorem, free groups, then dove right into module theory and homologies. We worked out of Artin and Rotman.

Actually now that I think about it, maybe module theory would be a good stepping off point from these parts. I know it gave me a cool new view and appreciation of linear algebra

That depends on your own level, your goals and your ambition. For example, OP wants to learn machine learning. Assuming OP's highschool math is solid, it might be possible for OP to simply download pytorch and immediately start programming neural networks without worrying too much about the hardcore math in the background.

On the other hand, if OP is more serious about improving as a mathematician, and assuming nothing but highschool math, I would start with linear algebra and differential and integral calculus. The famous professor Gil Strang has an excellent book on linear algebra, which is strangely available online. For differential and integral calculus, probably the standard reference is Stewart's book. At this point, OP would have all the basic things needed to start with machine learning. I'm not aware of the literature for machine learning so I can't recommend any specific books.

If OP wanted to get sidetracked learning more things before plunging into machine learning then the obvious choice would be Scientific Computing (my friends wrote an excellent book on the subject). Scientific Computing is the science of calculating things using computers and supercomputers. In addition, the area of Mathematical Optimization is good to know because Stochastic Gradient Descent is omnipresent in machine learning, but I don't know enough about optimization to recommend a book. There is Boyd and Vandenberghe but that is only for convex optimization. Some more areas that are related and useful are Probability and Statistics.

There are essentially "two types" of math: that for mathematicians and everyone else. When you see the sequence Calculus(1, 2, 3) -> Linear Algebra -> DiffEq (in that order) thrown around, you can be sure they are talking about non-rigorous, non-proof based kind that's good for nothing, imo of course. Calculus in this sequence is Analysis with all its important bits chopped off, so that everyone not into math can get that outta way quick and concentrate on where their passion lies. The same goes for Linear Algebra. LA in the sequence above is absolutely butchered so that non-math majors can pass and move on. Besides, you don't take LA or Calculus or other math subjects just once as a math major and move on: you take a rigorous/proof-based intro as an undergrad, then more advanced kind as a grad student etc.

To illustrate my point:

Linear Algebra:

1. Here's Linear Algebra described in the sequence above: I'll just leave it blank because I hate pointing fingers.
   
   
2. Here's a more serious intro to Linear Algebra:
   
   Linear Algebra Through Geometry by Banchoff and Wermer
   
   3. Here's more rigorous/abstract Linear Algebra for undergrads:
   
   Linear Algebra Done Right by Axler
   
   4. Here's more advanced grad level Linear Algebra:
   
   Advanced Linear Algebra by Steven Roman
   
   -----------------------------------------------------------
   
   Calculus:
   
   
3. Here's non-serious Calculus described in the sequence above: I won't name names, but I assume a lot of people are familiar with these expensive door-stops from their freshman year.
   
   
4. Here's an intro to proper, rigorous Calculus:
   
   Calulus by Spivak
   
   3. Full-blown undergrad level Analysis(proof-based):
   
   Analysis by Rudin
   
   4. More advanced Calculus for advance undergrads and grad students:
   
   Advanced Calculus by Sternberg and Loomis
   
   The same holds true for just about any subject in math. Btw, I am not saying you should study these books. The point and truth is you can start learning math right now, right this moment instead of reading lame and useless books designed to extract money out of students. Besides, there are so many more math subjects that are so much more interesting than the tired old Calculus: combinatorics, number theory, probability etc. Each of those have intros you can get started with right this moment.
   
   Here's how you start studying real math NOW:
   
   Learning to Reason: An Introduction to Logic, Sets, and Relations by Rodgers. Essentially, this book is about the language that you need to be able to understand mathematicians, read and write proofs. It's not terribly comprehensive, but the amount of info it packs beats the usual first two years of math undergrad 1000x over. Books like this should be taught in high school. For alternatives, look into
   
   Discrete Math by Susanna Epp
   
   How To prove It by Velleman
   
   Intro To Category Theory by Lawvere and Schnauel
   
   There are TONS great, quality books out there, you just need to get yourself a liitle familiar with what real math looks like, so that you can explore further on your own instead of reading garbage and never getting even one step closer to mathematics.
   
   If you want to consolidate your knowledge you get from books like those of Rodgers and Velleman and take it many, many steps further:
   
   Basic Language of Math by Schaffer. It's a much more advanced book than those listed above, but contains all the basic tools of math you'll need.
   
   I'd like to say soooooooooo much more, but I am sue you're bored by now, so I'll stop here.
   
   Good Luck, buddyroo.

I agree with all the suggestions to start with How to Prove It by Velleman. It's a great start for going deeper into mathematics, for which rigor is a sine qua non.

As you seem to enjoy calculus, might I also suggest doing some introductory real analysis? For the level you seem to be at, I recommend Understanding Analysis by Abbott. It helped me bridge the gap between my calculus courses and my first analysis course, together with Velleman. (Abbott here has the advantage of being more advanced and concise than Spivak, but more gentle and detailed than baby Rudin -- two eminent texts.)

Alternatively, you can start exploring some other fascinating areas of mathematics. The suggestion to study Topology by Munkres is sound. You can also get a friendly introduction to abstract algebra by way of A Book of Abstract Algebra by Pinter.

If you're more interested in going into a field of science or engineering than math, another popular approach for advanced high schoolers to start multivariable calculus (as you are), linear algebra, and ordinary differential equations.

"A Book of Abstract Algebra" by Charles C. Pinter is nice, from what I've seen of it--which is about the first third. I'm going through it in an attempt to relearn the abstract algebra I've forgotten.

I was using Herstein (which was what I learned from the first time), and was doing fine, but saw the Pinter book at Barnes & Noble. I've found it is often helpful when relearning a subject to use a different book from the original, just to get a different approach, so gave it a try (it's a Dover, so was only ten bucks).

What is nice about the Pinter book is that it goes at a pretty relaxed pace, with a good variety of examples. A lot of the exercises apply abstract algebra to interesting things, like error correcting codes, and some of these things are developed over the exercises in several chapters.

You don't have to be a prodigy to be able to understand some real mathematics in middle school or early high school. By 9th grade, after a summer of reading calculus books from the local public library, I was able to follow things like Niven's proof that pi is irrational, for instance, and I was nowhere near a prodigy.

Don't skip proofs and wrestle through them. That's the only way; to struggle. Learning mathematics is generally a bit of a fight.

It's also true that computation theory is essentially all proofs. (Specifically, constructive proofs by contradiction).

You could try a book like this: https://www.amazon.com/Book-Proof-Richard-Hammack/dp/0989472108/ref=sr_1_1?ie=UTF8&qid=1537570440&sr=8-1&keywords=book+of+proof

But I think these books won't really make you proficient, just more familiar with the basics. To become proficient, you should write proofs in a proper rigorous setting for proper material.

Sheldon Axler's "Linear Algebra Done Right" is really what taught me to properly do a proof. Also, I'm sure you don't really understand Linear Algebra, as will become very apparent if you read his book. I believe it's also targeted towards students who have seen linear algebra in an applied setting, but never rigorous and are new to proof-writing. That is, it's meant just for people like you.

The book will surely benefit you in time. Both in better understanding linear algebra and computer science classics like isomorphisms and in becoming proficient at reading/understanding a mathematical texts and writing proofs to show it.

I strongly recommend the second addition over the third addition. You can also find a solutions PDF for it online. Try Library Genesis. You don't need to read the entire book, just the first half and you should be well-prepared.

A graph theory project! I just started today (it was assigned on Friday and this is when I selected my topic). I’m on spring break but next month I have to present a 15-20 minute lecture on graph automorphisms. I don’t necessarily have to, but I want to try and tie it in with some group theory since there is a mix of undergrads who the majority of them have seen some algebra before and probably bored PhD students/algebraists in my class, but I’m not sure where to start. Like, what would the binary operation be, composition of functions? What about the identity and inverse elements, what would those look like? In general, what would the elements of this group look like? What would the group isomorphism be? That means it’s a homomorphism with a bijective function. What would the homomorphism and bijective function look like? These are the questions I’m trying to get answers to.

Last semester I took a first course in Abstract Algebra and I’m currently taking a follow up course in Linear Algebra (I have the same professor for both algebra classes and my graph theory class). I’m curious if I can somehow also bring up some matrix representation theory stuff as that’s what we’re going over in my linear algebra class right now.

This is the textbook I’m using for my graph theory class: Graph Theory (Graduate Texts in Mathematics) https://www.amazon.com/dp/1846289696?ref=yo_pop_ma_swf

Here are the other graph theory books I got from my library and am using as references: Graph Theory (Graduate Texts in Mathematics) https://www.amazon.com/dp/3662536218?ref=yo_pop_ma_swf

Modern Graph Theory (Graduate Texts in Mathematics) https://www.amazon.com/dp/0387984887?ref=yo_pop_ma_swf

And for funsies, here is my linear algebra text: Linear Algebra, 4th Edition https://www.amazon.com/dp/0130084514?ref=yo_pop_ma_swf

But that’s what I’m working on! :)

And I certainly wouldn’t mind some pointers or ideas or things to investigate for this project! Like I said, I just started today (about 45 minutes ago) and am just trying to get some basic questions answered. From my preliminary investigating in my textbook, it seems a good example to work with in regards to a graph automorphism would be the Peterson Graph.

Group theory is important for theoretical physics and crystallography, but I think it takes a back seat to the topics I listed. I've survived to grad school without learning anything beyond the basics, though I would love to study it eventually. Unfortunately, I couldn't fit in an abstract algebra course during my undergrad, so I don't have a textbook to personally recommend, although Dummit and Foote is popular with others.

Also, pretty much every branch of math (except maybe number theory?) is useful in physics (category theory, combinatorics, topology, measure and probability theory etc) so it's hard to make a comprehensive list.

You should spend some lovely evenings with my friend, Stewart. If you find my friend Stewart too hard on you, take some exercises from my little friend Thomas! And if you want even more fun, my friend Piskunov has some lovely exercises for you!

And ask your questions here :-)

I learned a lot from getting a copy of Rudin (however, this book is very challenging and probably not the best to self study from. I was able to get to about continuity before taking my analysis course and it was challenging, but worth while). You can probably find it online somewhere for free.

A teacher lent Introduction to Analysis to me and suggested I use it instead of the book by Rudin. It was a well written book and had exercises which were much more approachable (although it included very difficult ones as well). The layout of this book (and I'd bet many others) is quite similar to that of Rudin. It was nice to be able to read them together.

For linear algebra, I can't speak to the quality of many books, but there are plenty which can fairly easily be found online. You will likely be recommended Linear Algebra Done Right however I found it a bit challenging as a first introduction to linear algebra and never got quite far.

My university course used Larson, Falvo Linear Algebra and it was enjoyable and helps you learn the computations very well and gives a decent understanding of proofs.

You need some grounding in foundational topics like Propositional Logic, Proofs, Sets and Functions for higher math. If you've seen some of that in your Discrete Math class, you can jump straight into Abstract Algebra, Rigorous Linear Algebra (if you know some LA) and even Real Analysis. If thats not the case, the most expository and clearly written book on the above topics I have ever seen is Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers.

Some user friendly books on Real Analysis:

1. Understanding Analysis by Steve Abbot
   
   
2. Yet Another Introduction to Analysis by Victor Bryant
   
   
3. Elementary Analysis: The Theory of Calculus by Kenneth Ross
   
   
4. Real Mathematical Analysis by Charles Pugh
   
   
5. A Primer of Real Functions by Ralph Boas
   
   
6. A Radical Approach to Real Analysis by David Bressoud
   
   
7. The Way of Analysis by Robert Strichartz
   
   
8. Foundations of Analysis by Edmund Landau
   
   
9. A Problem Book in Real Analysis by Asuman Aksoy and Mohamed Khamzi
   
   
10. Calculus by Spivak
    
    
11. Real Analysis: A Constructive Approach by Mark Bridger
    
    
12. Differential and Integral Calculus by Richard Courant, Edward McShane, Sam Sloan and Marvin Greenberg
    
    
13. You can find tons more if you search the internet. There are more superstars of advanced Calculus like Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra by Tom Apostol, Advanced Calculus by Shlomo Sternberg and Lynn Loomis... there are also more down to earth titles like Limits, Limits Everywhere:The Tools of Mathematical Analysis by david Appelbaum, Analysis: A Gateway to Understanding Mathematics by Sean Dineen...I just dont have time to list them all.
    
    Some user friendly books on Linear/Abstract Algebra:
    
    
14. A Book of Abstract Algebra by Charles Pinter
    
    
15. Matrix Analysis and Applied Linear Algebra Book and Solutions Manual by Carl Meyer
    
    
16. Groups and Their Graphs by Israel Grossman and Wilhelm Magnus
    
    
17. Linear Algebra Done Wrong by Sergei Treil-FREE
    
    
18. Elements of Algebra: Geometry, Numbers, Equations by John Stilwell
    
    Topology(even high school students can manage the first two titles):
    
    
19. Intuitive Topology by V.V. Prasolov
    
    
20. First Concepts of Topology by William G. Chinn, N. E. Steenrod and George H. Buehler
    
    
21. Topology Without Tears by Sydney Morris- FREE
    
    
22. Elementary Topology by O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev and and V. M. Kharlamov
    
    Some transitional books:
    
    
23. Tools of the Trade by Paul Sally
    
    
24. A Concise Introduction to Pure Mathematics by Martin Liebeck
    
    
25. How to Think Like a Mathematician: A Companion to Undergraduate Mathematics by Kevin Houston
    
    
26. Introductory Mathematics: Algebra and Analysis by Geoffrey Smith
    
    
27. Elements of Logic via Numbers and Sets by D.L Johnson
    
    Plus many more- just scour your local library and the internet.
    
    Good Luck, Dude/Dudette.

Math is essential the art pf careful reasoning and abstraction.
Do yes, definitely.
But it may be difficult at first, like training anything that’s not been worked.

Note: there are many varieties of math. I definitely recommend trying different ones.

A couple good books:

An Illustrated Theory of Numbers

Foolproof (first chapter is math history, but you can skip it to get to math)

A Book of Abstract Algebra

Also, formal logic is really fun, imk. And excellent st teaching solid thinking. I don’t know a good intro book, but I’m sure others do.

Since your current knowledge is limited to calculus only, your goal seems kind of out of reach, at least in my opinion (but it depends on your progress/motivation). Writing good proofs is not something that you learn in a day by reading notes, it's something that comes with lots of experience reading and writing mathematics.

That being said, if you put a lot of focus on your studies it is certainly possible to learn the basics of algebra pretty fast. Linear algebra is an excellent tool, but it isn't required for learning abstract algebra. You can take both linear algebra and group theory classes at once and see where you want to go from there. It is a beautiful field of study for sure!

I'd strongly recommend Herstein's Topics in Algebra for a very solid introduction to most everything algebra-related. It covers Group Theory, Ring Theory, Vector Spaces and Modules, Fields, Linear Transformations, and some selected special topics.

For calc MATH 1300/1014 and 1310/1014 you need , buy it new from the bookstore cause you will need the online code for assignments also it’s useful for calc 3 if you wanna take that. Man Wong is a good prof I had him for both 1300 and 13010

For EECS 1019 you need it’s not that useful and PDF can be found online for free and no online assignments so no need to buy it new. I had Zhihua Chang he’s a new prof but really nice but his lectures are boring. Trev tutor on YouTube is really helping with the course.

For Math 1025/1021 you need I found the book helpful but unlike calc some profs tend not to use this book so I’d hold out of buying it but most profs use lyryz which is an online assignment program so you will need to buy that. I had Paul Skoufranis, amazing prof but had tests. The book is also useful for linear 2 but again depend if the prof uses it

For EECS 1022 you need
It’s a good book and the guy you wrote it teaches the class.

PM if you have any other questions

I would advise you not to start with category theory, but abstract algebra. Mac Lane and Birkhoff's book Algebra is excellent and well worth the money in hardback. It covers things like monoids, groups, rings, modules and vector spaces, all of which are -- not coincidentally -- typical examples of structures that form categories. Saunders Mac Lane invented category theory along with Samuel Eilenberg, and Birkhoff basically founded universal algebra, so you cannot find a more authoritative text.

Edit: The other thing that will really help you is a basic understanding of preorders and posets. I don't have a book that deals exclusively with this topic, but any introduction to lattice theory, logical semantics or denotational semantics of programming languages will treat it. I would recommend Paul Taylor's Practical Foundations of Mathematics, though the price on Amazon is very steep. You can look through it here: http://paultaylor.eu/~pt/prafm/

There's a lot of orders you could study mathematics in, and it's hard to say you should definitely pick one over the others.

One thing I can say pretty assuredly, though, is you should get a good background in algebra before you do much else. It's really the backbone of everything else. You can pick a bunch of different subjects after that, but study algebra first.

There are good online resources. Khan Academy is pretty good, as is Alcumus and Purple Math. Khan Academy has tests, and Alcumus is basically a big test.

Personally, though, I've learned way more from good books like this one than I tend to learn from websites.

Well, Hardy & Wright is the classic book for elementary stuff. It has almost everything there is to know. There is also a nice book by Melvyn Nathanson called Elementary Methods in Number Theory which I really like and would probably be my first recommendation. Beyond that, you need to decide which flavour you like. Algebraic and analytic are the big branches.

For algebraic number theory you'll need a solid grounding in commutative algebra and Galois theory - say at the level of Dummit and Foote. Lang's book is pretty classic, but maybe a tough first read. I might try Number Fields by Marcus.

For analytic number theory, I think Davenport is the best option, although Montgomery and Vaughan is also popular.

Finally, Serre (who is often deemed the best math author ever) has the classic Course in Arithmetic which contains a bit of everything.

First, please make sure everyone understands they are capable of teaching the entire subject without a textbook. "What am I to teach?" is answered by the Common Core standards. I think it's best to free teachers from the tyranny of textbooks and the entire educational system from the tyranny of textbook publishers. If teachers never address this, it'll likely never change.

Here are a few I think are capable to being used but are not part of a larger series to adopt beyond one course:
Most any book by Serge Lang, books written by mathematicians and without a host of co-writers and editors are more interesting, cover the same topics, more in depth, less bells, whistles, fluff, and unneeded pictures and other distracting things, and most of all, tell a coherent story and argument:

Geometry and solutions

Basic Mathematics is a precalculus book, but might work with some supplementary work for other classes.

A First Course in Calculus

For advanced students, and possibly just a good teacher with all students, the Art of Problem Solving series are very good books:
Middle & high school:
and elementary linked from their main page. I have seen the latter myself.

Some more very good books that should be used more, by Gelfand:

The Method of Coordinates

Functions and Graphs

Algebra

Trigonometry

Lines and Curves: A Practical Geometry Handbook

People will give me flack for this but I think Stewart is a great text for an intro to calc, and moreover, one that a person with little math experience can feasibly use for self study. Obviously buying it new is expensive but I've heard rumors of PDF's flying around on torrent sites and stuff, and there's always a few used copies of it in like a 1 mile radius of wherever you are. Working through the first 8 chapters and maybe chapter 11 (infinite sequences and series) will give you a pretty thorough understanding of all of a first year calculus course, and the sections on multivariable calculus aren't bad either. Once you actually know some basics you'll want to find a more advanced text, but I find myself turning back to this text constantly when I need to remember how to do something basic that I've forgotten from first year.

Do the problems. You'll get stuck on lots of them. /r/learnmath is great for that—if you post a problem from this book up there you'll have a detailed answer in about 45 seconds. http://math.stackexchange.com is also great for that.

As for statistics, there's only so far you can go in traditional statistics without knowing any calculus. You can learn the extreme basics like descriptive statistics and basic probability, but at some point, probability theory requires that you know how to take a derivative or an integral, so you'll need to have those skills under your belt. So I'd start on Stewart's book and just try to work through it.

For real analysis I really enjoyed Understanding Analysis for how clear the material was presented for a first course. For abstract algebra I found A book of abstract algebra to be very concise and easy to read for a first course. Those two textbooks were a lifesaver for me since I had a hard time with those two courses using the notes and textbook for the class. We were taught out of rudin and dummit and foote as mainly a reference book and had to rely on notes primarily but those two texts were incredibly helpful to understand the material.

​

If any undergrads are struggling with those two courses I would highly recommend you check out those two textbooks. They are by far the easiest introduction to those two fields I have found. I also like that you can find solutions to all the exercises so it makes them very valuable for self study also. Both books also have a reasonable amount of excises so that you can in theory do nearly every problem in the book which is also nice compared to standard texts with way too many exercises to realistically go through.

Cool.

Linear Algebra Don't waste your time with anything other than Lay, pretty much. Sounds like you're 100% new to LinAlg (it's not about polynomial equations) so it may be a bit tough to get off the ground working by yourself, but not impossible. It'd be worth finding a MOOC on the subject, there should be plenty. Otherwise, it's a pretty standard freshman maths course and a lot of people struggle with it (not because it's hard, just because it's different to HS maths), so there's a ton of resources on the internet.

Calculus Kinda just gotta slog away with where you're at tbh. I had Stewart as a freshman, didn't think it was overly great though. Still, that's the kind of level you need, so search for "alternatives to Stewart calculus" and anything that comes up should be appropriate. I wouldn't be able to tell you which to pick though.

Stats Basically, completing both of the above is pretty much a prerequisite for being able to understand linear regression properly, so don't expect to gain much by diving straight into stats. You could probably find a "business analytics" style textbook that would let you do more stats without understanding what's really going on under the hood, but if you want to stick with it in the long term you'll benefit more from getting stuff right at the beginning.

What do you want to do, though? Is your goal to read math textbooks and later, maybe, math papers or is it for science/engineering? If it's the former, I'd simply ditch all that calc business and get started with "actual" math. There are about a million books designed to get you in the game. For one, try Book of Proof by Richard Hammack. It's free and designed to get your feet wet. Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand/Polimeni/Zhang is my favorite when it comes to books of this kind. You'll also pick up a lot of math from Discrete Math by Susanna Epp. These books assume no math background and will give you the coveted "math maturity".

There is also absolutely no shortage of subject books that will nurse you into maturity. For example, check out [The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs by Grinberg](https://www.amazon.com/Real-Analysis-Lifesaver-Understand-Princeton/dp/0691172935/ref=sr_1_1?ie=UTF8&qid=1486754571&sr=8-1&keywords=real+analysis+lifesaver() and Book of Abstract Algebra by Pinter. There's also Linear Algebra by Singh. It's roughly at the level of more famous LADR by Axler, but doesn't require you have done time with lower level LA book first. The reason I recommend this book is because every theorem/lemma/proposition is illustrated with a concrete example. Sort of uncommon in a proof based math book. Its only drawback is its solution manual. Some of its proofs are sloppy, messy. But there's mathstackexchange for that. In short, every subject of math has dozens and dozens of intro books designed to be as gentle as possible. Heck, these days even grad level subjects are ungrad-ized: The Lebesgue Integral for Undergraduates by Johnson. I am sure there are such books even on subjects like differential geometry and algebraic geometry. Basically, you have choice. Good Luck!

For algebra, I'd recommend Mac Lane/Birkhoff. They may not be as comprehensive as some other texts but to me, they are more motivating, and will probably provide a better introduction to categorical thinking.

For linear algebra, I'm going to suggest something slightly unusual: Kreyszig's Introductory Functional analysis with applications. Functional analysis is essentially linear algebra on infinite dimensional spaces, and it generalizes a lot of the results in finite dimensions. Kreyszig does a good job motivating the reader. I can definitely sit down and read it for hours, much longer than I can for other books, and I definitely don't consider myself an analyst. However, it could be difficult if you are not familiar with basic topology and never seen linear algebra before.

How long ago did you do it? I work with calculus and statistics a lot and I often go back to earlier concepts to make sure my foundations are still strong.

I would recommend looking at this book and just quickly running through the exercises. That will give you a good idea about what you need to focus on. If you feel comfortable with those, something like this might be good to check out since it's made for self-teaching as opposed to being used in conjunction with a class.

Edited to add: math is like any language, in that the more you practice and manipulate numbers the better you'll be at it!

Linear algebra is about is about linear functions and is typically taken in the first or second year of college. College algebra normally refers to a remedial class that covers what most people do in high school. I highly recommend watching this series of videos for getting an intuitive idea of linear algebra no matter what book you go with. The book you should use depends on how comfortable you are with proofs and what your goal is. If you just want to know how to calculate and apply it, I've heard Strang's book with the accompanying MIT opencourseware course is good. This book also looks good if you're mostly interested in programming applications. A more abstract book like Linear Algebra Done Right or Linear Algebra Done Wrong would probably be more useful if you were familiar with mathematical proofs beforehand. How to Prove it is a good choice for learning this.

I haven't seen boolean algebra used to refer to an entire course, but if you want to learn logic and some proof techniques you could look at How to Prove it.

Most calculus books cover both differential and integral calculus. Differential calculus refers to taking derivatives. A derivative essentially tells you how rapidly a function changes at a certain point. Integral calculus covers finding areas under curves(aka definite integrals) and their relationship with derivatives. This series gives some excellent explanations for most of the ideas in calculus.

Analysis is more advanced, and is typically only done by math majors. You can think of it as calculus with complete proofs for everything and more abstraction. I would not recommend trying to learn this without having a strong understanding of calculus first. Spivak's Calculus is a good compromise between full on analysis and a standard calculus class. It's possible to use this as a first exposure to calculus, but it would be difficult.

ProfRobBob Youtube - This sir has great videos. His playlists are in order and very useful for Calculus. Loved his pre calculus playlist.

Patrick JMT - I could not have passed Calculus 2 without this guy. For the most part, his Calculus section is in order on his website.

KhanAcademy - Nice courses with problems available for you. Really easy to use and navigate. I worked through Algebra and only watched his videos on Trigonometry and Calculus.

Hope you get back on track buddy. Don't give up.


I self taught myself Algebra through Precalculus, here are books I used:

1. Practical Algebra - This helped when doing KhanAcademy Algebra course
   
   
2. Precalculus Demystified - Easy to understand w/o having any knowledge of precalculus.
   
   
3. Precalculus by Larson - The demystified book above helped form a foundation that allowed me to understand this book fairly well
   
   
4. Calculus for Dummies by PatrickJMT - This goes great for soliving problems in PatrickJMT's 1000 problem book.
   

Category theory is not easy to get into, and you have to learn quite a bit and use it for stuff in order to retain a decent understanding.

The best book for an introduction I have read is:

Algebra (http://www.amazon.com/Algebra-Chelsea-Publishing-Saunders-Lane/dp/0821816462/ref=sr_1_1?ie=UTF8&qid=1453926037&sr=8-1&keywords=algebra+maclane)

For more advanced stuff, and to secure the understanding better I recommend this book:

Topoi - The Categorical Analysis of Logic (http://www.amazon.com/Topoi-Categorial-Analysis-Logic-Mathematics/dp/0486450260/ref=sr_1_1?ie=UTF8&qid=1453926180&sr=8-1&keywords=topoi)

Both of these books build up from the basics, but a basic understanding of set theory, category theory, and logic is recommended for the second book.

For type theory and lambda calculus I have found the following book to be the best:

Type Theory and Formal Proof - An Introduction (http://www.amazon.com/Type-Theory-Formal-Proof-Introduction/dp/110703650X/ref=sr_1_2?ie=UTF8&qid=1453926270&sr=8-2&keywords=type+theory)

The first half of the book goes over lambda calculus, the fundamentals of type theory and the lambda cube. This is a great introduction because it doesn't go deep into proofs or implementation details.

Yeah either of those are easier. I don't like Fraleigh cause I think it lacks motivation (also the chapters on splitting/separable fields really suck) but I love Herstein. If you're set on cheap, this guy ain't too bad. If I were self studying though I would try to find a cheap older edition of Artin, as he's very example motivated, and it can sometimes be hard to wrap your head around all the abstraction without a class.

EDIT: Also you might want to find a cheap number theory text, since elementary number theory is probably the most accessible way to see groups and rings in action. And for "how do I prove xxx" questions I always recommend starting with this.

Intro Calculus, in American sense, could as well be renamed "Physics 101" or some such since it's not a very mathematical course. Since Intro Calculus won't teach you how to think you're gonna need a book like How to Solve Word Problems in Calculus by Eugene Don and Benay Don pretty soon.

Aside from that, try these:

Excursions In Calculus by Robert Young.

Calculus:A Liberal Art by William McGowen Priestley.

Calculus for the Ambitious by T. W. KORNER.

Calculus: Concepts and Methods by Ken Binmore and Joan Davies

You can also start with "Calculus proper" = Analysis. The Bible of not-quite-analysis is:

[Calculus by Michael Spivak] (http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?s=books&ie=UTF8&qid=1413311074&sr=1-1&keywords=spivak+calculus).

Also, Analysis is all about inequalities as opposed to Algebra(identities), so you want to be familiar with them:

Introduction to Inequalities by Edwin F. Beckenbach, R. Bellman.

Analytic Inequalities by Nicholas D. Kazarinoff.

As for Linear Algebra, this subject is all over the place. There is about a million books of all levels written every year on this subject, many of which is trash.

My plan would go like this:

1. Learn the geometry of LA and how to prove things in LA:

Linear Algebra Through Geometry by Thomas Banchoff and John Wermer.

Linear Algebra, Third Edition: Algorithms, Applications, and Techniques
by Richard Bronson and Gabriel B. Costa.

2. Getting a bit more sophisticated:

Linear Algebra Done Right by Sheldon Axler.

Linear Algebra: An Introduction to Abstract Mathematics by Robert J. Valenza.

Linear Algebra Done Wrong by Sergei Treil.

3. Turn into the LinAl's 1% :)

Advanced Linear Algebra by Steven Roman.

Good Luck.

For classes like number theory and abstract algebra, I would suggest just picking up a book and attempting to read it. It will be hard, but the main prerequisite for courses like this is some mathematical maturity. That only comes with practice.

Realistically there is probably no preparation that you could have which would prepare you in such a way that a book on advanced mathematics would be super easy.

I like this number theory book
http://www.amazon.com/Number-Theory-Dover-Books-Mathematics/dp/0486682528/ref=sr_1_1?ie=UTF8&qid=1348165257&sr=8-1&keywords=number+theory

I like this abstract algebra book
http://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178/ref=sr_1_2?s=books&ie=UTF8&qid=1348165294&sr=1-2&keywords=abstract+algebra

Congrats! And sorry about the DNF.

My opinion (for whatever its worth i guess), if your right on the edge of cut off times then you have to look at 3 things: age, weight, time spent training.

Unfortunately not much we can do about age, at a certain point no one is finishing a half ironman. I assume that you are not at that age yet.

Weight is probably the hardest thing to adjust. You can't out run a bad diet. So knowing nothing about your weight, are you satisfied with your weight or do you think that there is room for improvement?

Time spent training is the easy stuff! Woooo! More specifically, effective training and an effective training plan is probably your biggest gap. I (and others) suggest a book called The Triathlete's Training Bible by Joel Friel. This gets into how to spend your time to be more effectively training with self guided training plans etc etc. If you give more information about what you did to train for this specific event then maybe we could have more in-depth conversation about what you should be doing.

https://www.amazon.com/Triathletes-Training-Bible-Joe-Friel/dp/1934030198/ref=sr_1_2?ie=UTF8&qid=1491248736&sr=8-2&keywords=triathletes+training+bible

I would say that the best way to start is to pick a single book in Calculus, such as this or this or even this, and work all the way through it.

Then it is up to you; you could go straight towards Real Analysis; I recommend starting with a book that bears Intro in the name.

Or you could pursue a more collegiate curriculum and move onto Differential Equations and Linear Algebra, then Real Analysis.

I assume you are doing this all independently, so you should look at college sequences for math majors and the likes. You can mimic those, and look for online syllabi of the courses to make sure you are covering the appropriate material. This helps because it gives a nice structure to your learning.

Whatever the case, work through a calculus book, then decide what further direction you wish to take.

Working through Griffiths is a good idea, but I strongly suggest working through an abstract linear algebra book before you do anything else. It will make your life much better. Doing some of Griffiths in advance might make your homework a bit easier, but you'll be repeating material when you could be learning new things. And learning real linear algebra will benefit you in pretty much every class.

I recommend this book as your primary text and this one for extra problems and and a second opinion.

Start with arithmetic. Make sure you are comfortable in adding, subtracting, multiplying, and dividing integers, negatives, fractions, and decimals. Old school books are great for this. This shouldn't take too long and for your sake DO NOT USE A CALCULATOR. You have no idea how the introduction of a calculator early on ruins kids, they become dependent on it. Train yourself to not need one.

Once you are comfortable with arithmetic, move on to algebra. I recommend reading both Lang's Basic Mathematics and the series of books by I.M. Gelfand for this. These are great books for their subjects and will introduce rigor into your math. The going will be a bit difficult at first but you will come out better than before. You want to focus on UNDERSTANDING math, not just doing it.

When you are done with algebra, you can move on to geometry and trigonometry. Both of the authors I mentioned cover these. for the most part, you want to understand basic things like area, volume, congruence theorems, and whatnot.

A bit of advice: practice is the secret to being successful. If you do enough practice problems, you will eventual reach a point where you will question how the hell you didn't know this stuff before. Also, feel free to find other books to supplement these, there is nothing wrong with getting multiple explanations so long as it all benefits you.

I am a master's student with interests in algebraic geometry and number theory. And I have a good collection of textbooks on various topics in these two fields. Also, as part of my undergraduate curriculum, I learnt abstract algebra from the books by Dummit-Foote, Hoffman-Kunze, Atiyah-MacDonald and James-Liebeck; analysis from the books by Bartle-Sherbert, Simmons, Conway, Bollobás and Stein-Shakarchi; topology from the books by Munkres and Hatcher; and discrete mathematics from the books by Brualdi and Clark-Holton. I also had basic courses in differential geometry and multivariable calculus but no particular textbook was followed. (Please note that none of the above-mentioned textbooks was read from cover to cover).

As you can see, I didn't learn much geometry during my past 4 years of undergraduate mathematics. In high school, I learnt a good amount of Euclidean geometry but after coming to university geometry appears very mystical to me. I keep hearing terms like hyperbolic/spherical geometry, projective geometry, differential geometry, Riemannian manifold etc. and have read general maths books on them, like the books by Hartshorne, Ueno-Shiga-Morita-Sunada and Thorpe.

I will be grateful if you could suggest a series of books on geometry (like Stein-Shakarchi's Princeton Lectures in Analysis) or a book discussing various flavours of geometry (like Dummit-Foote for algbera). I am aware that Coxeter has written a series of textbooks in geometry, and I have read Geometry Revisited in high school (which I enjoyed). If these are the ideal textbooks, then where to start? Also, what about the geometry books by Hilbert?

Sweet. I think the best curriculum to approach this with, assuming you're in this for the long haul, would be to start with building a good understanding of calculus, cover basic classical mechanics, then cover electricity and magnetism, and finally quantum mechanics. I'm going to leave math and mechanics mostly for someone else, because no textbooks come to mind at the moment. I'll leave you with three books though:

For Math, unless someone else comes up with something better, the bible is Stewart's Calculus

The other two are by the same author:

Griffith's Introduction to Electrodynamics

Griffith's Introduction to Quantum Mechanics

I think these are entirely reasonable to read cover to cover, work through problems in, and come out with somewhere near an undergraduate level understanding. Be careful not to rush things. One of the biggest barriers I've run into trying to learn physics independently is to try and approach subjects I don't have the background for yet: it can be a massive waste of time. If you really want to learn physics in its true mathematical form, read the books chapter by chapter, make sure you understand things before moving on, and do problems from the books. I'd recommend buying a copy of the solutions manuals for these books as well. It can also be helpful to look up the website for various courses from any university and reference their problem sets/solutions.

Good luck!

I think the most important part of being able to see beauty in mathematics is transitioning to texts which are based on proofs rather than application. A side effect of gaining the ability to read and write proofs is that you're forced to deeply understand the theory of the math you're learning, as well as actively using your intuition to solve problems, rather than dry route calculations found in most application based textbooks. Based on what you've written, I feel you may enjoy taking this path.

Along these lines, you could start of with Book of Proof (free) or How to Prove It. From there, I would recommend starting off with a lighter proof based text, like Calculus by Spivak, Linear Algebra Done Right by Axler, or Pinter's book as you mentioned. Doing any intro proofs book plus another book at the level I mentioned here would have you well prepared to read any standard book at the undergraduate level (Analysis, Algebra, Topology, etc).

I'm not sure if you mean abstract algebra or linear algebra, but if it's the former, I liked Herstein's Topics in Algebra. There's also Abstract Algebra by Herstein as well, which I think is a cheaper slimmed down version. I used these books for self study and found Herstein's exposition, particularly at the beginning of chapters, very helpful. He isn't as verbose as your typical 7th edition mass market textbook author though.

For linear algebra, I hear Axler's Linear Algebra Done Right is good. I haven't read it, but I read his paper "Down with Determinants" which is, I think, written in the same style and enjoyed the alternative perspective a lot.

There are some really good books that you can use to give yourself a solid foundation for further self-study in mathematics. I've used them myself. The great thing about this type of book is that you can just do the exercises from one side of the book to the other and then be confident in the knowledge that you understand the material. It's nice! Here are my recommendations:

First off, three books on the basics of algebra, trigonometry, and functions and graphs. They're all by a guy called Israel Gelfand, and they're good: Algebra, Trigonometry, and Functions and Graphs.

Next, one of two books (they occupy the same niche, material-wise) on general proof and problem-solving methods. These get you in the headspace of constructing proofs, which is really good. As someone with a bachelors in math, it's disheartening to see that proofs are misunderstood and often disliked by students. The whole point of learning and understanding proofs (and reproducing them yourself) is so that you gain an understanding of the why of the problem under consideration, not just the how... Anyways, I'm rambling! Here they are: How To Prove It: A Structured Approach and How To Solve It.

And finally a book which is a little bit more terse than the others, but which serves to reinforce the key concepts: Basic Mathematics.

After that you have the basics needed to take on any math textbook you like really - beginning from the foundational subjects and working your way upwards, of course. For example, if you wanted to improve your linear algebra skills (e.g. suppose you wanted to learn a bit of machine learning) you could just study a textbook like Linear Algebra Done Right.

The hard part about this method is that it takes a lot of practice to get used to learning from a book. But that's also the upside of it because whenever you're studying it, you're really studying it. It's a pretty straightforward process (bar the moments of frustration, of course).

If you have any other questions about learning math, shoot me a PM. :)

> btw ce crezi de masterul asta de la unibuc http://fmi.unibuc.ro/ro/pdf/2008/curs_master/informatica/4InteligentaArtificialaEnachescuSite.pdf , e din 2008,nu am gasit o varianta mai buna.Daca voi avea posibilitatea sa fiu acceptat l;a o facultate mai moderna care face cercetare din afara o voi face,dar mai intai trebuie sa capat o diploma din Romania).

Acum, trebuie sa intelegi ca ML si AI sunt 2 lucruri diferite. AI includes ML, si ce ai tu aici e un master general de AI. Nu pot sa iti spun cat de bun e masterul, dar vad ca faci 1 curs de ML doar in anul 2, ceea ce pentru mine ar fi un motiv sa nu il fac. Information retrieval si NLP sunt interesante, dar eu as incerca sa invat ML la nivel teoretic first, si apoi sa abordez probleme specifice domeniilor.

> Eu ma gandeam ca Unibuc e mai potrivit pt ca la Poli voi face multa electronica si programare low-level si nu cred ca le voi folosi

Ar putea fi utile daca te gandesti la un moment dat ca te intereseaza mai degraba sa fii Research engineer si sa nu lucrezi atat de mult pe teorie, cat pe implementare. Toate librariile de scientific programming sunt implementate in C/C++. Dar pe langa asta, in general programarea low-level ar fi interesant sa o inveti pentru ca te ajuta sa intelegi cum functioneaza lucrurile at a more basic level, fara x abstractii construite pentru a fi totul beginner-friendly. Daca nu vrei sa continui cu asta dupa 1-2 cursuri e ok, tot cred ca iti va folosi mai incolo. Sa inveti python si c++ in paralel e un challenge interesant :).

> Va veni vacanta de vara si voi avea mult timp liber si vreau sa ma apuc de machine learning de-acum.Ce crezi de planul asta de invatare?

Iti va lua mai mult decat 1 vara sa termini ce ai listat aici. Sfatul meu ar fi sa imbini programare aplicata cu matematica. Cursurile sunt ok, dar eu pentru matematica as incepe cu single variable calculus -> multiple variable calculus inainte de altceva (daca ai cunostintele necesare sa abordezi cursul). Uite o carte pe care ti-o recomand: https://www.amazon.com/Calculus-Early-Transcendentals-James-Stewart/dp/1285741552

Are in jur de 8 sectiuni care reprezinta pre-requisites (lucruri pe care ar trebui sa le stii inainte sa abordezi cartea), algebra, geometrie de baza, etc. Fiecare invata diferit, eu prefer cartile.

Legat de programare, incearca sa faci probleme de aici: https://projecteuler.net/, te va ajuta mai incolo :). Si daca te plictisesti incearca construiesti lucruri care ti-ar fi utile. Vei invata destule din proiecte de genul.

no problem. enjoy the journey! :)

as i mentioned in another reply, it's easily my favorite area of math, and i am frustrated that these books were never recommended by anyone (or didn't exist in some cases) while i was in my undergraduate studies.

i feel i should also at least mention conceptual mathematics: a first introduction to categories. it is very approachable for an undergraduate, and having knowledge of basic category theory and the intuitions it provides would provide exceedingly helpful for the very "functional" experience that is modern differential geometry. a primer of infinitesimal analysis could also provide a unique viewpoint when learning the traditional material, as it really explores the idea of what we mean by infinitesimal or differential and the continuum of the real line. it's less approachable than the conceptual mathematics book (which is extremely approachable and excellent), but you have high potential of gaining some very unique insights.

i feel like all i do is recommend these books on here, but that's really because they are excellent and unique.

One thing I found useful for doing this is Stewart's Calculus (many people will disagree with me here, but it was my old Calc book, so I didn't have to buy a new one, and I thought it was pretty decent). Don't worry about buying the latest version. you can probably find an old one in a used book store, or ebay or something, which will save you some bucks. The thing that kills Calc students is their poor algebra, so make sure you are rock-solid on that. You should be able to solve linear equations, quadratic equations, rational equations, and equations involving square-roots without a problem. You should also be able to graph all of these, and you should have a good understanding of exponents and logs. Don't spend much time reading the book, spend your time practicing, doing problem after problem until you really nail each one. If you can find a study-buddy, this will help a lot, as they will be able to point out where you are going wrong, and you will be able to teach them things (which is one of the best ways to learn).

Anyway, that's just some random advice, but I hope it helps. Good luck!

Usual hierarchy of what comes after what is simply artificial. They like to teach Linear Algebra before Abstract Algebra, but it doesn't mean that it is all there's to Linear Algebra especially because Linear Algebra is a part of Abstract Algebra.

Example,

Linear Algebra for freshmen: some books that talk about manipulating matrices at length.

Linear Algebra for 2nd/3rd year undergrads: Linear Algebra Done Right by Axler

Linear Algebra for grad students(aka overkill): Advanced Linear Algebra by Roman

Basically, math is all interconnected and it doesn't matter where exactly you enter it.

Coming in cold might be a bit of a shocker, so studying up on foundational stuff before plunging into modern math is probably great.

Books you might like:

Discrete Mathematics with Applications by Susanna Epp

Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers

Building Proofs: A Practical Guide by Oliveira/Stewart

Book Of Proof by Hammack

Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand et al

How to Prove It: A Structured Approach by Velleman

The Nuts and Bolts of Proofs by Antonella Cupillary

How To Think About Analysis by Alcock

Principles and Techniques in Combinatorics by Khee-Meng Koh , Chuan Chong Chen

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!) by Carol Ash

Problems and Proofs in Numbers and Algebra by Millman et al

Theorems, Corollaries, Lemmas, and Methods of Proof by Rossi

Mathematical Concepts by Jost - can't wait to start reading this

Proof Patterns by Joshi

...and about a billion other books like that I can't remember right now.

Good Luck.

Yes, they're awesome. Brought up pretty frequently on /r/math, too. I'm pretty sure I have at least 10 Dover books. Two excellent titles that come to mind are Pinter's A Book of Abstract Algebra and Rosenlicht's Introduction to Analysis.

I'd recommend hitting up somewhere like half-price books and grabbing a textbook for like $10-$15. I purchased this book for probably $12 when I needed to brush up. I know it's not online, but it will provide good direction, offer a solid foundation, provide sample problems to test your knowledge, and can easily be supplemented by online materials. As someone else mentioned, Khan Academy is also great, but I would highly recommend using them as a supplement, and using a book as your base.

It depends on your interests. I thought the machine learning course on coursera was great. Antirez sometimes blogs about the internals of Redis on his blog, and he is a great writer. If you like math, this is the best math book I've read. Finally, you can always start contributing code to an open source project -- learn by doing!

the "vanilla" books are IMO quite boring to read - especially when you don't know more than Set/Functions.

but I really enjoy P. Aluffi; Algebra: Chapter 0 that builds up algebra using CT from the go instead of after all the work

----

remark I don't know if this will really help you understanding Haskell (I doubt it a bit) but it's a worthy intellectual endeavor all in itself and you can put on a knowing smile whenever you hear those horrible words after

This is one of the best books of abstract algebra I've seen, very well explained, favoring clear explanations over rigor, highly recommended (take your time to read the reviews, the awesomeness of this book is real :P): http://www.amazon.com/Book-Abstract-Algebra-Edition-Mathematics/dp/0486474178/ref=sr_1_6?ie=UTF8&qid=1345229432&sr=8-6&keywords=introduction+to+abstract+algebra

On a side note, trust me, Dummit or Fraileigh are not what you want.

Herstein's Topics in Algebra is the book I learned both group and field theory from. It's a very easy read with lots of good examples and problems that help you develop and learn about the topics.

Also, the field of quaternions with integer coefficients is pretty cool. You can use it to prove that every natural number can be written as the sum of four squares, almost for free just by examining the field.

There is no "one fastest" method to solving them.

Systems of equations are systematic, and it really depends on the problem. The only real way to learn about this is to take a course in Linear Algebra. That is all about systems of linear equations.

But these show up all of the time, here is what I usually do:

If I just need one of the 2-3 variables, Cramer's Rule is a good way to test solvability and extract a single value.

On normal 2x2 systems, I usually do a quick determinant/matrix inverse. Checks the rank as well as the det, and it is always going to work.

On 3x3 or higher systems, it depends. This is why Linear Algebra is important.

Supposedly Linear Algebra Done Right is a good book on the subject, so if you're interested there is one way. The book I used was A custom edition of this one. I thought it was very good as well.

Yes!

If you don't know any calculus Stewart Calculus is the typical primer in colleges. Combine this with Khan Academy for easy mode cruise control.

After that, you want to look at The Big Orange Book, which is essentially the bible for undergrad astrophysics and 100% useful beyond that. This book could, alone, tell you everything you need to know.

As for other topics like differential equations and linear algebra you can shop around. I liked Linear Algebra Done Right for linear personally. No recommendations from me on differential equations though, never found a book that I loved.

if you want determinants, Shilov's is supposed to be "Determinants done right" I wouldn't recommend the other Dover LA book by Stoll

http://www.amazon.com/Linear-Algebra-Dover-Books-Mathematics/product-reviews/048663518X/

-----------

Anyway: Free!

http://www.math.ucdavis.edu/~anne/linear_algebra/

http://www.math.ucdavis.edu/~linear/linear.pdf

http://www.cs.cornell.edu/courses/cs485/2006sp/LinAlg_Complete.pdf (Dawkins notes that were recently pulled off lamar.edu site, gentle intro like Anton's)

http://joshua.smcvt.edu/linearalgebra/

http://www.ee.ucla.edu/~vandenbe/103/reader.pdf

http://www.math.brown.edu/%7Etreil/papers/LADW/LADW.pdf

https://math.byu.edu/~klkuttle/Linearalgebra.pdf

---------

Or, google "positive definite matrix" or "hermitian" or "hessian" or some term like that and it will show you lecture notes from dozens of universities after the inevitable wikipedia and Wolfram hits

If you're doing both applied and pure abstract algebra rather than elementary algebra, then you'll probably need to learn to write proofs for the pure side. I recommend Numbers, Groups, and Codes by J. F. Humphreys for an introduction to the basics and to some applied abstract algebra. If you need more work on proofs, the free Book of Proofs can help, and Fraleigh's A First Course in Abstract Algebra is a good book for pure abstract algebra. If you want something more advanced, I recommend the massive Abstract Algebra by Dummit and Foote.

Here's three very good books:

1. De Morgan, On the Study and Difficulty of Mathematics. This is a free book available on the internet. Read the parts you find interesting.
   
   
2. Gelfand, Algebra.
   
   
3. Chrystal, Algebra: An Elementary Text-Book. This is available online for free. A lot of the greatest mathematicians and physicists of the last century lauded this (erdos, feynman...)

There's no single book that's right for everyone: a suitable book will depend upon (1) your current background, (2) the material you want to study, (3) the level at which you want to study it (e.g., undergraduate- versus graduate-level), and (4) the "flavor" of book you prefer, so to speak. (E.g., do you want lots of worked-out examples? Plenty of exercises? Something which will be useful as a reference book later on?)

That said, here's a preliminary list of titles, many of which inevitably get recommended for requests like yours:

1. Undergraduate Algebra by Serge Lang
   
   
2. Topics in Algebra, 2nd edition, by I. N. Herstein
   
   
3. Algebra, 2nd edition, by Michael Artin
   
   
4. Algebra: Chapter 0 by Paolo Aluffi
   
   
5. Abstract Algebra, 3rd edition, by David S. Dummit and Richard M. Foote
   
   
6. Basic Algebra I and its sequel Basic Algebra II, both by Nathan Jacobson
   
   
7. Algebra by Thomas Hungerford
   
   
8. Algebra by Serge Lang
   
   Good luck finding something useful!

I don't claim to know Category Theory, but I came across it when doing exercises in the beginning part of Chapter 0 by Aluffi. It was very terse, but still understandable. The video seems to be much more relaxed in comparison. It is even more relaxed than Awodey's book which is a much better intro to CT than Aluffi's Chapter 0. In short, it reminds me of Conceptual Mathematics: A First Introduction to Categories by Lawvere/Schnauel a little.

Here are some great books that I believe you may find helpful :)

• Elements of Algebra by Leonhard Euler
  
• Foundations of Mathematics by Ian Stewart
  
• Geometry Revisited by H. S. M. Coxeter
  
• Excursions in Calculus by Robert M. Young
  
• New Horizons in Geometry by Tom M. Apostol and Mamikon A. Mnatsakanian
  
• The Calculus Gallery: Masterpieces From Newton to Lebesgue by William Dunham
  
• A Primer of Real Functions by Ralph P. Boas
  
• Understanding Analysis by Stephen Abbot
  
• Chaos: Making a New Science by James Gleick
  
• Logic: An Introduction to Elementary Logic by Wilfred Hodges
  
  and last but definitely not least:
  
  
• A Mathematical Gift I, II, III by Kenji Ueno et al.
  
  Later on:
  
  
• Conceptual Mathematics: A First Introduction to Categories by F. William Lawvere
  
• Twelve Landmarks of Twentieth-Century Analysis by D. Choimet & H. Queffélec
  
• Theory of Complex Functions by Reinhold Remmert
  
• Algebraic Combinatorics. Walks, Trees, Tableaux & More by Richard P. Stanley

If you're currently at the pre-calc level, you could probably get away with learning from khan academy for a little while. After that (and building some familiarity with proof writing), you'd be ready for some of the pure math classes like abstract algebra and real analysis. For those courses, you'll probably want to check out some Open Courseware. You'd want to treat it like a real class; watch the lectures online and read from the textbooks, while working on problem sets.

While you're working your way through the khan academy stuff, you might want to check out Stewart's calculus book; it's pretty solid for making your way through the calculus sequence.
I'd ask around for a good linear algebra book, since I haven't encountered a decent one that's at that level.

A couple good ones to get started are:

• A Classical Introduction to Modern Number Theory - Ireland & Rosen
  
  
• Primes of the form x^(2)+ny^(2) - David Cox
  
  These will introduce a lot of the main ideas in a way that you can understand with only the basics of abstract algebra. Having background in Galois Theory helps (you can use Dummit & Foote for a solid intro). Cox's book is probably one of the best reads in the subject. Then there are a few that introduce the basic formalism rigorously:
  
  
• First two/three chapters of Algebraic Number Theory - Neukirch
  
  
• Milne's Online Notes (PDF)
  
  
• A Course in Arithmetic - Serre (PDF)
  
  These cover the basics and can be found from many sources via amazon and google. After this, you can look at some more specialized stuff, though the two main directions are Analytic Number Theory and Langlands Program:
  
  
• Analytic Number Theory: Introduction to Analytic Number Theory - Apostal
  
  
  
• Local Fields: Local Fields - Serre
  
  
• Cyclotomic Fields: Introduction to Cyclotomic Fields - Washington
  
  
• Class Field Theory: Algebraic Number Theory - Cassels & Frolich, the final chapters of Neukirch, Artin L-Functions: A Historical Approach - Snyder (PDF)
  
  
• Elliptic Curves: Rational Points on Elliptic Curves - Silverman (undergrad level), The Arithmetic of Elliptic Curves - Silverman
  
  
• Modular Forms: A First Course in Modular Forms - Diamond & Shurman, Introduction to Elliptic Curves and Modular Forms - Koblitz, Modular Functions and Modular Forms - Milne (PDF), Automorphic Forms and Representations - Bump
  
  
• Langlands Program: (The theory that ties absolutely everything together) An Introduction to Langlands Program - Various, An Elementary Introduction to the Langlands Program - Gelbart (PDF)
  
  
• Prerequisites for the Langlands Program - Knapp (PDF) - This has a basic overview of most everything encountered in algebraic number theory along with many more references. Check it out!
  
  This may be looking far into the future, for someone just getting into Algebraic Number Theory, but I figure it's good to have a reference to point people to. I have my own biases and preferences, if anyone else wants to add something, go for it!

I strongly suggest you take your time learning calculus, because anything you don't grasp completely will come back to haunt you.

But the good news is that there are lots of great resources you can use. MIT OCW has a full course with lectures, notes, and exams. Here are three free online books. If you're looking to buy a textbook, some good choices are Thomas, Stewart, and Spivak. (You can find dirt-cheap copies of older editions at abebooks.com.)

If you want more guidance, another great place to find it is at /r/learnmath.

Sometimes you can find what textbook your school uses before the semester starts (I'm also the weird kid that emails the professor asking about books if I cant find it >.>). Some of my professors have what material they use for each class on their personal web pages though. For calculus, you'll most likely use this book. My brother used it at his Uni my friend at another and I myself used it at mine. Not sure if you're registered yet though. I had a weird case going into my Uni because I did community college then took summer courses so I was enrolled earlier than students who transfer and probably the freshman. YouTube videos will also be your best friend. People I liked for my math classes are TrevTutor (I don't think he ever finished his Calc 2 series) and PatrickJMT. Hope this helps a bit if you have any other questions or need more clarifications don't hesitate to ask.

You're not really doing higher math right now as much as you're learning tricks to solve problems. Once you start proving stuff that'll be a big jump. Usually people start doing that around Real Analysis like your father said. Higher math classes almost entirely consist of proofs. It's a lot of fun once you get the hang of it, but if you've never done it much before it can be jarring to learn how. The goal is to develop mathematical maturity.

Start learning some geometry proofs or pick up a book called "Calculus" by Spivak if you want to start proving stuff now. The Spivak book will give you a massive head start if you read it before college. Differential equations will feel like a joke after this book. It's called calculus but it's really more like real analysis for beginners with a lot of the harder stuff cut out. If you can get through the first 8 chapters or so, which are the hardest ones, you'll understand a lot of mathematics much more deeply than you do now. I'd also look into a book called Linear Algebra done right. This one might be harder to jump into at first but it's overall easier than the other book.

A lot of people recommend Khan Academy, but you cannot really learn from the Khan Academy; there is just too much material to cover. I recommend either going into an algebra class at your local community college, and/or get some good algebra/maths books. This one gets a lot of praise on Amazon.com:

http://www.amazon.com/Practical-Algebra-Self-Teaching-Guide-Second/dp/0471530123/ref=sr_1_fkmr0_1?ie=UTF8&qid=1288684060&sr=8-1-fkmr0

and, this one is the one nobel laureate Richard Feynman taught himself with:

http://www.amazon.com/Algebra-practical-Mathematics-self-study/dp/B0007DZPT6

Linear algebra is an essential tool in many areas of mathematics. Computations with matrices aren't always that important; far more important are the concepts of vector space and linear transformation. Pretty much any time you work with coordinates, dimension, changes of coordinates, vectors, linear relations, or anything like that, you're going to need some linear algebra.

If you're interested, I recommend taking a look at Axler's Linear Algebra Done Right. Axler has very clear exposition and proofs, and if you've only seen the computational aspect of linear algebra, it'll provide a different, more abstract and conceptual perspective.

It should also be noted that simpler subjects and introductory texts tend to be common knowledge to the point that citation is often not needed. You don't need to cite that water is wet, not even on Wikipedia.

Journals and modern texts about modern subjects tend to be very well cited, because they're building heavily on other sources of information.

When you don't do this, you have to have an enormous amount of backgrounding. For instance, check out this algebra text. It's 944 pages because it doesn't tend to cite much of exposition and instead states it all directly. It includes an enormous amount of information -- it's meant to be used as fundamental education material. It's not just high level conclusions that could fit in 20-50 pages.

So the amount of citation depends a great deal on the purpose of the text and how close it is to common knowledge. However, Anita's criticism is clearly not common knowledge because nobody but her sees it the way she does. Therefore, she should be explaining how she comes to her conclusions, and citing information. She should also be citing the direct quotes she uses, because it's plagiarism otherwise (and we have huge volumes of evidence that she outright plagiarizes a great deal). Plagiarism in academia is something that ends your career.

Can you run on the deck of the ship?

If you are already pretty fit (which I assume you are since you are in the Navy), you shouldn't have too much of an issue finishing an Oly. If you are shooting for a specific time goal you will be a bit more constrained however.

You have quite a bit of time until early summer so I would build up a strong aerobic base and maybe incorporate a bit of weights in for lower body and upper body. I would be careful with maximal weights at this point. Try to go for low weight and a lot of reps. Try to avoid putting on a ton of mass -- keep it lean.

Joe Friel writes some amazing books that you would find very interesting and helpful in structuring your plan. See the Triathlete's Training Bible.

No worries for the timeliness!

For Measure and Integration Theory I recommend Elements of Integration and Measure by Bartle.

For Functional Analysis I recommend Introductory Functional Analysis with Applications by Kreyszig.

And for Topology, I think it depends on what flavor you're looking for. For General Topology, I recommend Munkres. For Algebraic Topology, I suggest Hatcher.

Most of these are free pdf's, but expensive ([;\approx \$200;]) to buy a physical copy. There are some good Dover books that work the same. Some good ones are this, this, and this.

Personally, I would take the time to read them both. A strong linear algebra background will be very helpful in ML. Its especially useful if you want to expand out a little bit more into other areas of signal processing. Make sure you also spend some time getting a good background in probability and statistics.

EDIT: I haven't actually read Axler's book but me and some of my friends are partial to this book.

The Joe Friel Books are great. The Triathletes Training Bible by Joe Friel is fantastic (https://www.amazon.com/Triathletes-Training-Bible-Joe-Friel/dp/1934030198) in addition I found a subscription to training peaks with a training plan to be great for accountability.

I love Aluffi! It's a fun read, and more "modern" than texts like Dummit and Foote (in that it uses basic category theory freely). I like category theory, so I really enjoy Aluffi's approach.

Here's a couple of books I'd recommend.

1. Slow Fat Triathlete - This book is the beginner's book.
   amazon
   
   
2. Triathlete's Training Bible - This is the encyclopedia of triathlon. It can help you build a plan from an Olympic to an Ironman race.
   amazon
   
   You might check out the Minneapolis area for a tri club. I'm certain there is a good one up there. Some clubs have New Triathlete programs that can be really good.

I've never taught the course, but a couple of my colleagues are very fond of Linear Algebra Done Wrong and would willingly teach from it if (1) the title wouldn't immediately turn students off of it and (2) the school would be okay with sacrificing some income from students having to purchase a book.

If you're curious, the book title is a play on the title of another well-known linear algebra book.

These are, in my opinion, some of the best books for learning high school level math:

• I.M Gelfand Algebra {[.pdf] (http://www.cimat.mx/ciencia_para_jovenes/bachillerato/libros/algebra_gelfand.pdf) | Amazon}
  
• I.M. Gelfand The Method of Coordinates {Amazon}
  
• I.M. Gelfand Functions and Graphs {.pdf | Amazon}
  
  These are all 1900's Russian math text books (probably the type that /u/oneorangehat was thinking of) edited by I.M. Galfand, who was something like the head of the Russian School for Correspondence. I basically lived off them during my first years of high school. They are pretty much exactly what you said you wanted; they have no pictures (except for graphs and diagrams), no useless information, and lots of great problems and explanations :) There is also I.M Gelfand Trigonometry {[.pdf] (http://users.auth.gr/~siskakis/GelfandSaul-Trigonometry.pdf) | Amazon} (which may be what you mean when you say precal, I'm not sure), but I do not own this myself and thus cannot say if it is as good as the others :)
  
  
  I should mention that these books start off with problems and ideas that are pretty easy, but quickly become increasingly complicated as you progress. There are also a lot of problems that require very little actual math knowledge, but a lot of ingenuity.
  
  Sorry for bad Englando, It is my native language but I haven't had time to learn it yet.

Ok this is not going to be very original, but I'd start getting a foundation in algebra, linear algebra and analysis. My suggestions for those topics are Fraleigh, Gilbert Strang's Video Lectures (I'd suggest Heuser for learning analysis but that's german and won't help you).

I guess the most important thing to remember is that you don't have to understand everything when you read it for the first time. Try to get a feel for functions and matrices, sets and maps, etc, because you'll need those all the time.

Good Luck!

tl;dr: you need to learn proofs to read most math books, but if nothing else there's a book at the bottom of this post that you can probably dive into with nothing beyond basic calculus skills.

Are you proficient in reading and writing proofs?

If you aren't, this is the single biggest skill that you need to learn (and, strangely, a skill that gets almost no attention in school unless you seek it out as an undergraduate). There are books devoted to developing this skill—How to Prove It is one.

After you've learned about proof (or while you're still learning about it), you can cut your teeth on some basic real analysis. Basic Elements of Real Analysis by Protter is a book that I'm familiar with, but there are tons of others. Ask around.

You don't have to start with analysis; you could start with algebra (Algebra and Geometry by Beardon is a nice little book I stumbled upon) or discrete (sorry, don't know any books to recommend), or something else. Topology probably requires at least a little familiarity with analysis, though.

The other thing to realize is that math books at upper-level undergraduate and beyond are usually terse and leave a lot to the reader (Rudin is famous for this). You should expect to have to sit down with pencil and paper and fill in gaps in explanations and proofs in order to keep up. This is in contrast to high-school/freshman/sophomore-style books like Stewart's Calculus where everything is spelled out on glossy pages with color pictures (and where proofs are mostly absent).

And just because: Visual Complex Analysis is a really great book. Complex numbers, functions and calculus with complex numbers, connections to geometry, non-Euclidean geometry, and more. Lots of explanation, and you don't really need to know how to do proofs.

AH HA, one of the few times I will link a dover book in good heart!

http://www.amazon.com/Book-Abstract-Algebra-Edition-Mathematics/dp/0486474178

Pinter offers a fine introduction to abstract algebra.

Gelfand's Algebra is interesting, encourages mathematical thinking, and has the additional advantage of being much more approachable than the books you've listed.

This is probably a much better place to start for someone who's interested in "starting from the basics."

Also, Zolv mentioned the book Practical Algebra (A Self Teaching Guide), by Peter Selby and Steve Slavin. I concur. It's cheap, about $11, and has great reviews on Amazon. I found it extremely helpful when I was getting started. Practical Algebra

I think this sample paragraph is something you'd agree with (from page 79 of the second edition):
"We have some good news and some bad news. This chapter and the two that follow [about factoring] introduce some fairly difficult concepts. That's the bad news. The good news is that if you can learn about 75 or 80 percent of this material, you're way ahead of the game....Remember, you're teaching yourself math, and the only thing that's helping you is this book, which is kind of like doing open heart surgery over the phone. So don't get down on yourself if you don't comprehend something the first time - or even the second time. If you get stuck, go on to the next frame..."

A Book of Abstract Algebra by Charles C. Pinter

I really enjoyed reading the book, almost reads like a novel. There is a great first chapter laying out the history of the subject and it just builds from there.

The answer is "virtually all of mathematics." :D

Although lots of math degrees are fairly linear, calculus is really the first big branch point for your learning. Broadly speaking, the three main pillars of contemporary mathematics are:

1. Analysis
   
2. Algebra
   
3. Topology
   
   You might also think of these as the three main "mathematical mindsets" — mathematicians often talk about "thinking like an algebraist" and so on.
   
   Calculus is the first tiny sliver of analysis and Spivak's Calculus is IMO the best introduction to calculus-as-analysis out there. If you thought Spivak's textbook was amazing, well, that's bread-n-butter analysis. I always thought of Spivak as "one-dimensional analysis" rather than calculus.
   
   Spivak also introduces a bit of algebra, BTW. The first few chapters are really about abstract algebra and you might notice they feel very different from the latter chapters, especially after he introduces the least-upper-bound property. Spivak's "properties of numbers" (P1-P9) are actually the 9 axioms which define an algebraic object called a field. So if you thought those first few chapters were a lot of fun, well, that's algebra!
   
   There isn't that much topology in Spivak, although I'm sure he hides some topology exercises throughout the book. Topology is sometimes called the study of "shape" and is where our most general notions of "continuous function" and "open set" live.
   
   Here are my recommendations.
   
   Analysis If you want to keep learning analysis, check out Introductory Real Analysis by Kolmogorov & Fomin, Principles of Mathematical Analysis by Rudin, and/or Advanced Calculus of Several Variables by Edwards.
   
   Algebra If you want to check out abstract algebra, check out Dummit & Foote's Abstract Algebra and/or Pinter's A Book of Abstract Algebra.
   
   Topology There's really only one thing to recommend here and that's Topology by Munkres.
   
   If you're a high-school student who has read through Spivak in your own, you should be fine with any of these books. These are exactly the books you'd get in a more advanced undergraduate mathematics degree.
   
   I might also check out the Chicago undergraduate mathematics bibliography, which contains all my recommendations above and more. I disagree with their elementary/intermediate/advanced categorization in many cases, e.g., Rudin's Principles of Mathematical Analysis is categorized as "elementary" but it's only "elementary" if your idea of doing math is pursuing a PhD. Baby Rudin (as it's called) is to first-year graduate analysis as Spivak is to first-year undergraduate calculus — Rudin says as much right in the introduction.

We used the Dover textbook by Pinter. It's my favorite math textbook ever, the writing was just so clear, and even entertaining and funny. We had a good professor too.

Honestly, if she has passion for math to the extent that she wants to learn calculus over the summer, she'll find the classroom pace annoyingly slow. AP Calculus can be taught in 2 months, but they stretch it into 8 months.

I always recommend Stewart's calculus book,
http://www.amazon.com/Calculus-6th-Edition-Stewarts-Series/dp/0495011606/ref=sr_1_1?ie=UTF8&qid=1372050088&sr=8-1&keywords=stewart+calculus+6th+edition
It's a college-level textbook, but it starts where a high school student should be comfortable. Only the first 7-8 chapters apply to AP Calculus.

Conceptual Mathematics by Lawvere and Schanuel is a good low level introduction to category theory (and a bit of set theory) if you are feeling shaky on those grounds. From there lots of books open up to you.

The best books I know on how to "think" like a functional programmer are all written by Richard Bird. http://www.amazon.com/gp/product/1107452643/ref=pd_lpo_sbs_dp_ss_1?pf_rd_p=1944579842&pf_rd_s=lpo-top-stripe-1&pf_rd_t=201&pf_rd_i=0134843460&pf_rd_m=ATVPDKIKX0DER&pf_rd_r=090NKMWKY6078Z0WPCTW http://www.amazon.com/Pearls-Functional-Algorithm-Design-Richard/dp/0521513383

Not much is available in book form, especially that I can recommend on the FRP front.

Dependent types is a broad area, you're going to find yourself reading a lot of research papers. You might be able to get by with something more practical like Chlipala's Certified Programming with Dependent Types, but if you want a more theoretical treatment then perhaps Zhaohui Luo's Computation and Reasoning might be a better starting point.

Here's my rough list of textbook recommendations. There are a ton of Dover paperbacks that I didn't put on here, since they're not as widely used, but they are really great and really cheap.

Amazon search for Dover Books on mathematics

There's also this great list of undergraduate books in math that has become sort of famous: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm

Pre-Calculus / Problem-Solving

• The Art of Problem Solving - Lehoczky/Ruczyk - in two volumes, both have full solutions available separately
  
  Calculus
  
  
• Calculus - Stewart - ignore the bad reviews, it's very clear and covers all of basic calculus well - it's just the book that most intro students use and as a consequence gets a bad rap
  
  
• Calculus - Spivak - this will take your understanding of calculus to the next level
  
  
• Calculus - Apostol - there are two volumes and they are very comprehensive, albeit difficult
  
  Linear Algebra
  
  
• Linear Algebra - Lay
  
  
• Linear Algebra Done Right - Axler
  
  Differential Equations
  
  
• Elementary Differential Equations - Boyce/DePrima
  
  
• Ordinary Differential Equations - Tenenbaum
  
  Number Theory
  
  
• Number Theory - Pommersheim for a change of pace and proof skills
  
  Proof-Writing
  
  
• Transition to Higher Mathematics - Dumas/McCarthy - helps to bridge the gap between "lower" and higher math
  
  Analysis
  
  
• The Way of Analysis - Strichartz
  
• Principles of Mathematical Analysis - Rudin
  
• Real and Complex Analysis - Rudin
  
  Complex Analysis
  
  
• Complex Analysis - Ahlfors
  
  Functional Analysis
  
  
• Functional Analysis - Rudin
  
  
• Introductory Functional Analysis with Applications - Kreyszig
  
  Partial Differential Equations
  
  
• Partial Differential Equations for Scientists and Engineers - Farlow
  
  Higher-dimensional Calculus and Differential Geometry
  
  
• Calculus on Manifolds - Spivak - for a modern take on multivariable calculus
  
  
• Differential Geometry - Kreyszig
  
  Abstract Algebra
  
  
• A First Course in Abstract Algebra - Fraleigh
  
• Abtract Algebra - Dummit and Foote
  
  Geometry
  
  
• The Elements - Euclid - for culture and for understanding where math has come from, plus it's still an awesome book after all these years; don't need to go through all of it
  
• Euclidean and Non-Euclidean Geometries - Greenberg
  
  Topology
  
  
• Introduction to Topology - Mendelson
  
  
• Topology - Munkres
  
  Set Theory and Logic
  
  
• Introduction to Set Theory - Hrbacek/Jech
  
  
• Elements of Set Theory - Enderton
  
  
• Mathematical Logic - Kleene
  
  
• Set Theory and the Continuum Hypothesis - Cohen
  
  Combinatorics / Discrete Math
  
  
• Combinatorics - Cameron
  
  
• Concrete Math - Graham/Knuth/Patashnik
  
  
• Discrete Math - Biggs
  
  Graph Theory
  
  
• A First Course in Graph Theory - Chartrand/Zhang
  
  P. S., if you Google search any of the topics above, you are likely to find many resources. You can find a lot of lecture notes by searching, say, "real analysis lecture notes filetype:pdf site:.edu"
  

I'm not sure what kind of shape you're in, but I'm guessing that the ironman requires a lot more planning just to finish it. I'd suggest getting a copy of this book which will help you plan out and train for all three sports.

Depending on the area you're in, I'd suggest joining a club that does group worksouts (runs, rides, swims, etc). Very useful for all sorts of things, but especially for organized pool workouts. If you're in the DC area, I'll suggest (Team Z)[http://www.triteamz.com/], but I'm sure there are other teams out there.

I'm soon starting my trek through every problem in the algebra text that Harvard's PhD prelim recommends for study:

Abstract Algebra by Dummit and Foote

I've started the first section of the first chapter, but that was only in a few hours of spare time. I'll be posting solutions by chapter soon and post my stories/insights on Hacker News. Here's section 1.1 (except the last problem, 36):

http://therobert.org/alg/1.1.pdf

Comments are appreciated. Better now than when I start the real journey. :)

I had a nice book called Precalculus Mathematics in a Nutshell but it is not geared to starting from scratch. It's a good book if you remember some of your algebra, geometry, and trigonometry.

I've known some people who had good experiences with Practical Algebra

Try to find entry points that interest you personally, and from there the next steps will be natural. Most books that get into the nitty-gritty assume you're in school for it and not directly motivated, at least up to early university level, so this is harder than it should be. But a few suggestions aimed at the self-motivated: Lockhart Measurement, Gelfand Algebra, 3blue1brown's videos, Calculus Made Easy, Courant & Robbins What Is Mathematics?. (I guess the last one's a bit tougher to get into.)

For physics, Thinking Physics seems great, based on the first quarter or so (as far as I've read).

>When university starts, what can I do to ensure that I can compete and am just as good as the best mathematics students?

Read textbooks for mathematics students.

For example for Linear Algebra I heard that Axler's book is very good (I studied Linear Algebra in another language, so I can't really suggest anything from personal experience). For Calculus I personally suggest Spivak's book.

There are many books that I could suggest, but one of the greatest books I've ever read is The Art and Craft of Problem Solving.

Yes: Dummit and Foote. I used it in my freshman algebra class. It has excellent proofs and exercises. It will teach you the mathematical maturity faster than analysis and will most likely be more useful to you later on.

I don't know much about AI, though I do know that (there's a theme, here) linear algebra gets a starring role. So, if you're currently enjoying linear algebra, continue with that. Axler is frequently recommended, if you want a textbook to go through.

After that it's really up to you what you want to go for next, since you have many paths available. Sipser is a great intro to theoretical CS, but, again, don't spend $200 on it. Try to find it in a library, or use something like this to find a much cheaper international edition.

Edit: Forgot to mention, CLRS is the standard for algorithms, but I'm not sure how useful it is as a primary source for learning. Maybe try to borrow a copy to see if you like it.

How about some nice, inexpensive classics from Dover Publications?

For number theory, Andrews, Number Theory or Leveque, Elementary theory of numbers or the more advanced Leveque, Fundamentals of Number Theory

For linear algebra, Cullen, Matrices and Linear Transformations.

I bet you haven't read Edwards, Riemann's Zeta Function.

Edit: Oops! Now I see that you wanted to avoid linear algebra. Cullen might still be good as a second source. Maybe Pinter, A book of Abstract Algebra would appeal to you for a taste of field theory. However, vector spaces just naturally go with fields, so you may want to wait until after you have studied linear algebra.

Also, this book is a tough piece of work, for sure, but it's very helpful. It probably goes deeper than your class will, and may present ideas/methods in a different way, but if you grapple w/ this one, it'll really help you figure out L.A.

If your priority is training for the Tri, a muscle building program like SL will not be very helpful.

You would be much better off following an endurance program that peaks on your event date. You still have a couple months to establish base and then another couple months added anaerobic and intervals.

Read this entire book- it will help you plan a good peak - http://www.amazon.com/Triathletes-Training-Bible-Joe-Friel/dp/1934030198/ref=dp_ob_title_bk

This Book Is a great read. Explains every part of training and competing at your best.

I had a friend who went through the program. I don't think there was a pre-assessment as Academic Math itself is a prerequisite to other stuff, but don't take my word as law on that. The course resource appears [to be here] (https://www.nscc.ca/learning_programs/programs/PlanDescr.aspx?prg=ACC&pln=ACCONNECT) and doesn't mention pre-assessments. [This PDF] (http://gonssal.ca/documents/AcadMathIVCurr2010.pdf) should cover a fair bit of what the course is about.

As an aside, [this book] (https://www.amazon.ca/Practical-Algebra-Self-Teaching-Peter-Selby/dp/0471530123) is a fantastic way to get yourself up to speed on algebra. I can't recommend it highly enough.

Please, simply disregard everything below if the info is old news to you.

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Algebraic geometry requires the knowledge of commutative algebra which requires the knowledge of some basic abstract algebra (consists of vector spaces, groups, rings, modules and the whole nine yards). There are many books written on abstract algebra like those of Dummit&Foote, Artin, Herstein, Aluffi, Lang, Jacobson, Hungerford, MacLane/Birkhoff etc. There are a million much more elementary intros out there, though. Some of them are:

Discovering Group Theory: A Transition to Advanced Mathematics by Barnard/Neil

A Friendly Introduction to Group Theory by Nash

Abstract Algebra: A Student-Friendly Approach by the Dos Reis

Numbers and Symmetry: An Introduction to Algebra by Johnston/Richman

Rings and Factorization by Sharpe

Linear Algebra: Step by Step by Singh

As far as DE go, you probably want to see them done rigorously first. I think the books you are looking for are titled something along the lines of "Analysis on Manifolds". There are famous books on the subject by Sternberg, Spivak, Munkres etc. If you don't know basic real analysis, these books will be brutal. Some elementary analysis and topology books are:

Understanding Analysis by Abbot

The Real Analysis Lifesaver by Grinberg

A Course in Real Analysis by Mcdonald/Weiss

Analysis by Its History by Hirer/Wanner

Introductory Topology: Exercises and Solutions by Mortad

I haven't yet started practicing for the GRE, but does it include Linear Algebra or Modern/Abstract Algebra? Also is there Calculus on it? I'm taking (or have taken, or will take by the time of the GRE) all of those classes and they're all very interesting. I just bought this book on Abstract Algebra, if you're interested.

If you are enjoying your Calc 3 book, I highly recommend reading Topology, which provides the foundations of analysis and calculus. Two other books I would highly recommend to you would be Abstract Algebra and Introduction to Algorithms, though I suspect you're well aware of the latter.

I am not sure that a pure math textbook is what you want. A lot of the problems that mathematicians think about may not be what you need. Let's take functional analysis for example. Most textbooks focus on bounded/ compact operators, and they only have one chapter at the end dedicated to unbounded operators. Unfortunately, the derivative (momentum) is an unbounded operator, so the part that has the least detail is what you need.

I would recommend a "math for physics students" book. A nice book that tries to paint the intuitive idea of most branches of math relevant to physics (and then some) and show you how to calculate is Goldbart and Stone's book, which they have made freely available online. This book assumes familiarity with linear algebra. If you are weak on this subject, I would highly recommend the book by Friedberg, Insel, and Spence. This is a more traditional math textbook, but it gets you very comfortable with the details of linear algebra (except for tensor products, but you should understand their construction with this background).

If you are serious about learning, Linear Algebra by Friedberg Insel and Spence, or Linear Algebra by Greub are your best bets. I love both books, but the first one is a bit easier to read.

I think category theory is best learned when taught with a given context. The first time I saw category theory was in my first abstract algebra course (rings, modules, etc.), where the notion of a category seemed like a necessary formalism. Given you already know some algebra, I'd suggest glancing through Paolo Aluffi's Algebra: Chapter 0. It is NOT a book on category theory, but rather an abstract algebra book that works with categories from the ground level. Perhaps it could be a good exercise to prove some statements about modules and rings that you already know, but using the language of category theory. For example, I'd get familiar with the idea of Hom(X,-) as a "functor"from the category of R-modules to the category of abelian groups, which maps Y \to Hom(X,Y). We can similarly define Hom(-,X). How do these act on morphisms (R-module homomorphisms)? Which one is covariant and which one is contravariant? If one of these functors preserves short exact sequences (i.e. is exact), what does that tell you about X?

For a very good textbook, I would recommend Calculus Early transcendentals by Stewart. He goes through every concept in single variable calculus (there's also a version with multi variable calculus) and proves almost every concept he teaches. Its one of my favorite textbooks in general.

I recommend reading the Triathlete's Training Bible (http://www.amazon.com/The-Triathletes-Training-Bible-Friel/dp/1934030198) which quite extensively covers the base training period.


If I recall correctly, he speaks about doing lots of leg and core strength training, swimming drills concentrating heavily on technique, hill repeats on the treadmill, etc... Things that would serve as a good base for other training later on.

You might find this book to be a good place to start: Algebra, by Gelfand and Shen.

Another book in a similar vein might be Basic Mathematics by Serge Lang.

I haven't used either of these books myself, but I came across them recently, and it looks like they might be among the few titles that cover high-school math in the way that you describe (they were written by prominent research mathematicians).

You might consider using the materials on Khan Academy (articles, videos, and exercises) to structure your studies, since these may be more closely aligned with current standards in the U.S. Then, as you go along, you can use these books as supplements (e.g. if you feel that a different perspective on a particular topic might be helpful).

Dummit (or just D&F), Artin, [Lang] (https://www.amazon.com/Algebra-Graduate-Texts-Mathematics-Serge/dp/038795385X), [Hungerford] (https://www.amazon.com/Algebra-Graduate-Texts-Mathematics-v/dp/0387905189). The first two are undergraduate texts and the next two are graduate texts, those are the ones I've used and seen recommended, although some people suggest [Pinter] (https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178) and Aluffi. Please don't actually buy these books, you won't be able to feed yourself. There are free versions online and in many university libraries. Some of these books can get quite dry at times though. Feel free to stop by /r/learnmath whenever you have specific questions

The most popular calculus book for college classes in the United States is Stewart, Calculus: Early Transcendentals. A typical Calculus II course starts somewhere in chapter 5 or 6 (picking up wherever Calculus I left off) and ends with chapter 11.

This book has answers to all of the odd-numbered exercises in the back, so it works reasonably well to read the book and then try the exercises. Typically the first 3/4 of the exercises in each section are straightforward, and the remaining 1/4 are more difficult and would only be assigned in an honors class.