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You should probably get the DirectX11 version when it is available, not because it's newer, not because DirectX10 is obsolete (it's not yet), but because the new DirectX11 book has a chapter on animation.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

I don't know that I can help because everyone learns so differently, but I'll say a couple of things. (I'm going to warn you right now that I'm kind of tired and I didn't proofread very much.)

I think my best advice about (1) is to play with small examples. If you're asked to prove something about all real vector spaces then look at R, R\^2, R\^3, and see if you can understand what's going on in those situations. If you're asked to prove something about all differential functions, pick one and play with the statement in that case. Once you get the small examples, move onto bigger examples. What is the craziest, wildest differential function of which you can think? What vector space really pushes the boundaries of the hypotheses that you're given? It's not a fool-proof plan, but I find that working with examples is the single most helpful thing I can do when I'm working on a hard proof.

Also, about number (1), this isn't very concrete advice, but I like to tell my students to try to understand a statement before attempting a proof. Proofs are a linear line of logic between some assumptions and a conclusion. Our brains, however, aren't always compatible with the rigor and linearity of a proof; it can be hard to see everything that's going on from step seven in a proof. If you can really, *really* convince yourself of something and understand it then your intuition will guide you much better during the proof. It can be easy to fall into a trap of saying "this is on my homework so it must be true" and then diving into a proof but it's better to think critically about the statement first. Along the lines of the previous paragraph, an attempt to construct a counterexample (even if you know something is definitely true) can be really helpful.

As to part (2), all I can really say it that it's very frustrating and challenging. I'm about to graduate with my PhD and I still struggle with learning new math from a textbook or paper. You're not alone in that. It really helps to find the right book but it's also very hard. Some books are excellent because they are packed full of content and make a good reference (sort of like an encyclopedia) and other books are excellent because they have great exercises and make a good companion to a course. Rudin's analysis book is notorious for this; the exercises are fantastic which means it gets used in the classroom a lot, but the exposition is pretty brief (and, in my opinion, quite poor at times) which makes it difficult to use when self-teaching. Unfortunately, the best way to sell the most books is to write a book that is meant to be used in a class. It can be really hard to find books that are good for learning on your own. (The best example I know is Gallian's Abstract Algebra book.)

My only real advice for finding good references is to ask your teachers. I'll also say that it's usually easier to learn math in small chunks. It can be really daunting to decide "I'm going to learn all of Real Analysis" but deciding "I want to learn how the real numbers are constructed" is much more doable and it likely won't feel quite as discouraging when you get stuck. Pick a specific topic that interests you (ideally somewhat close to what you already know), and try to understand that one topic. Find videos, find websites, ask a teacher, and look into references. A complaint I often get about this technique is "I don't really know much about Topology so I don't know what interests me" but a good place to start can be "I'd like to understand what the area of Topology is really trying to study."

So, anyway, I've rambled long enough now. Good luck and try to stick with it!

I just started the PhD program this semester at North Carolina State. The program in general isn't ranked well but I'm interested in Environmental and Resource Econ and NCSU is top 10 (arguably top 5) in that field. I thought I'd give you a brief overview of the math that I had to prepare (undergrad rather than a math camp).

1. 3 semester's of calculus and diff eq - Really important for anything you're going to do in terms of optimization.
   
2. Linear Algebra - Important for econometrics stuff. Most applied stuff is easy enough in Stata but most programs will make you derive everything.
   
3. Real Analysis (lower) - I had an intro level class that went over set theory stuff as well as techniques needed to prove a statement. I would highly recommend an intro to proving course. If you're looking to study on your own I would suggest this book.
   
4. Real Analysis (upper) - My other RA courses involved deriving the real numbers, proving calculus, continuity proofs, etc. It's good in terms of practicing methods of proof but the material itself isn't great. That said, an A in RA is a great signal for grad schools. Anything lower than a B+ starts to get uncomfortable.
   
5. Topology - Some schools like to see it but no one is expecting it.
   
6. Optimization Theory - A course is unnecessary but its a good idea to look over primal/dual theory.
   
7. Probability Theory - You should, in my opinion, know the cute probability stuff front and back. Make sure to be familiar with compound events and whatnot. A probability class will probably get into random variables towards the end and those turn out to be very important.
   
8. Statistics Theory - More stuff on random variables, transformations, and statistical inference. Very important but unless you want to do econometric theory I think you can get away without knowing testing methods.
   
   One big thing that I didn't work on was programming skills. If you are intending to do applied work rather than theory, you'll want to be a solid programmer. Matlab and/or Maple are valuable and Stata, SAS, and ArcMap don't hurt.
   
   That said, I've met a lot of people in decent PhD programs who do not have much more than Calc, diff EQ, and linear algebra. I don't know if they passed comps or not but they got in. There are a number of good programs ranked 50+ that will teach you the math needed for applied work. However, if you want to go to a top 20 program you should definitely look into a math undergrad.
   
   Good luck to anyone thinking about applying.

I would say that it would depend on the problem. If you cannot solve the first ten, I would be worried, as they can all be solved by simple brute force methods. I have a degree in Astrophysics, and some of the 300 and 400 problems are giving me pause, so if you are stuck there you are in good company.

There are elegant solutions to each problem, if you want to delve into them, but the first handful, the first ten especially, can be simply solved.

Once you get beyond the first ten or so, the mathematical difficulty ratchets up. There are exceptions to that rule of course, but by and large, it holds.

If you are interested in Number Theory, the best place to start is a number theory course at a local university. Mathematics, especially number theory, is difficult to learn by yourself, and a good instructor can expound, not only on the math, but also on the history of this fascinating subject.

Gauss, quite arguably the finest mathematician to ever live loved number theory; of it, he once said:

> Mathematics is the queen of sciences and number theory is the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank.

Although my personal favorite quote of his on the subject is:

> The enchanting charms of this sublime science reveal themselves in all their beauty only to those who have the courage to go deeply into it.

If you are interested in purchasing some books about number theory, here are a handful of recommendations:


Number Theory (Dover Books on Mathematics) by George E. Andrews


Number Theory: A Lively Introduction with Proofs, Applications, and Stories by James Pommersheim, Tim Marks, Erica Flapan


An Introduction to the Theory of Numbers by G. H. Hardy, Edward M. Wright, Andrew Wiles, Roger Heath-Brown, Joseph Silverman


Elementary Number Theory (Springer Undergraduate Mathematics Series) by Gareth A. Jones , Josephine M. Jones

and it's companion


A Classical Introduction to Modern Number Theory (Graduate Texts in Mathematics) (v. 84) by Kenneth Ireland, Michael Rosen

and a fun historical book:


Number Theory and Its History (Dover Books on Mathematics) Paperback by Oystein Ore

I would also recommend some books on

Markov Chains

Algebra

Prime number theory

The history of mathematics

and of course, Wikipedia has a good portal to number theory.

> 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

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.

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!

As I see it there are four kinds of books that fall into the sub $30 zone:

• Dover books which are generally pretty good and cover a wide range of topics
  
  
• Free textbooks and course notes - two examples I can think of are Hatcher's Algebraic Topology (somewhat advanced material but doable if you know basic point-set topology and group theory) and Wilf's generatingfunctionology
  
  
• Really short books—I don't a good example of this, maybe Stanley's book on catalan numbers?
  
  
• Used books—one that might interest you is Automatic Sequences by Allouche and Shallit
  
  You can get a lot of great books if you are willing to spend a bit more however. For example, Hardy and Wright is an excellent book (and if you think about it: is a 600 page book for $60 really more expensive than a 300 page one for 30?). Richard Stanley's books on combinatorics: Enumerative Combinatorics Vol. I and Algebraic Combinatorics are also excellent choices. For algebra, Commutative Algebra by Eisenbud is great. If computer science interests you you can find commutative algebra books with an emphasis on Gröbner bases or on algorithmic number theory.
  
  So that's a lot of suggestions, but two of them are free so you can't go wrong with those.

During my sophomore year I took an "intro to proofs" course (known formally at the institution as Foundations of Advanced Mathematics) and I found it to be extremely beneficial in my development as a mathematician. We used Chartrand's "Mathematical Proofs" textbook (here's the link for those who are interested).

The text covered set theory, logic, the various proof methods, and then dug into stuff like elementary number theory, equivalence relations, functions, cardinality (culminating in Cantor's two main results), abstract algebra, and analysis. Obviously the book only scratched the surface on a lot of these topics, but I felt it accomplished its goal.

Part of my satisfaction with the course is likely due to the fact that we had a brilliant professor who taught the course in the spirit of what u/Rtalbert235 spoke of. He was able to clearly articulate the distinction between computation and theory. The way I like to say it is that he taught us the difference between pounding a bunch of nails into a 2X4 (computation) and building a house (proving theorems).

I don't mean to universally praise "intro to proofs" courses, however. I can definitely see how they can be horrible wastes of time if not done properly, and I can also appreciate the idea of "throwing" students into proof-based courses (analysis, algebra, and so on). For me though, I think it's worth the effort to try and optimize these sorts of classes, which will ultimately serve a LOT of math students who need to understand proofs, but don't necessarily have a desire to pursue the subject beyond the undergraduate level.

tl;dr - Given the right combination of textbook and professor, an "intro to proofs" course can be just what the doctor ordered for developing mathematicians.

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.

This turned into kind of a treatise, but you are in the same position I was once, so here goes...

First of all, this is about the best introduction to proofs you can get. It's $17. You should buy this now and read it. Do the problems, too - they're fun and not particularly hard.

As for other advice, if I were you, I'd just graduate so you have a bachelor's and then go back for pure math. That way if you don't end up liking it, at least you'll have something.

You could also just switch majors now if you're sure you want to do it, but take it from me, you're not going to do it in 2 years. The important thing is, even if you could, you wouldn't want to. If you're getting into pure math to go to graduate school, you need to keep in mind that your intense 2 years of studying is exactly what the rest of us do for 4 years. The minimum requirements for a math degree are exactly that - the bare minimum. In fact, I myself switched during the 4th year of an art degree, planning to graduate after 2 years, and am now at the tail end of my 3rd year and no longer have any intention to graduate "early." I'm just doing what I would have done if I had started in math normally, because I realized I want to be my best for graduate schools.

Point is, don't cheat yourself out of this by trying to get some fuckin BA in math. If you decide to do it, do it for real.

(Note: This is assuming you're looking for grad school. If your plan is to stop at bachelor's and then work, consider stats or applied math or double majoring math with something else, cause you ain't doin' shit with only a bachelor's in pure math. That's just a fact.)

This being said, the decision to become a mathematician is the best one I ever made. I was in your position and I am so much happier - even now, when all my old friends have graduated and I'm in "major switch purgatory" - than I would have been if I would have kept trying to be something I'm not. So, I'm not trying to be discouraging. It really is worth a thought.

Here is how you make the decision... next semester, find out if your university has a proofs class. It will probably be for sophomore mathematics majors and use a book similar to the one I linked. Take this class alongside whatever humanities requirements you'd be taking anyway. If it has prerequisites other than 1st year calc (it shouldn't), talk to the math advisor and get them waved. The class probably won't be very hard, but it will give you an idea of what the process of "doing a math problem" evolves into when you get to higher level math. After this, find an introductory abstract algebra class (not a linear algebra class - one that includes group theory), and an introductory analysis class. This way you'll get a taste of two very different flavors of upper level math, and you'll be able to see how doing proofs actually works out. If you find yourself wanting more, then switch (or graduate and go back). If you don't, then don't be a math major. All in all taking three classes is a pretty inexpensive way to find out whether you want to do something, and since you can fit them into your fourth year, it won't fuck up the option of graduating with cinema studies if you decide math isn't your thing.

Several good books have already been mentioned in this thread, but some good books are hard to get into as a beginner.

I recommend Elementary Number Theory by Underwood Dudley as a good starting point for a beginner, as well as something like Apostol or Ireland-Rosen if you want more details.

I think it makes sense to start with something like Dudley to get an overall framework, and then rely on more detailed books to flesh out the details of whatever topics you're interested in more.

In particular, I think Dudley's book has an approach to Chebyshev's theorem (i.e. there is always a prime between n and 2n) that's great for beginners, even if someone with a bit more experience can streamline that proof a little.

There is Elementary number theory by William Stein, and A Computational Introduction to Number Theory and Algebra. The latter is better if you are also interested in some of the computation They are both available for free online (legally). I think you would prefer Stein's book but skim through both and see which one you like more.

For something more in depth, I looked at some of the books in this list at mathoverflow. Hardy & Wright , and Niven & Zuckerman's books seem best suited to you (from what I looked at, but go through that list yourself). Many of the other books require some background in abstract algebra.

I haven't read either but just looking through their table of contents I would go with Niven and Zuckerman's book. It seems to go into the more useful things more quickly, and it's not as densely packed with information you probably won't be interested in right now.

TLDR: Start here, or here.

For what it's worth, number theory is a fascinating field. I don't think you'll be disappointed going into it. Good luck!

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 :-)

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/

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.

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.

A matrix is most often used to represent something called a linear transformation, which is a function T satisfying T(v + w) = T(v) + T(w) and also T(av) = aT(v) for all vectors v and w and all scalars a. Under this interpretation of a matrix, multiplying two matrices A representing the linear transformation S and B representing the linear transformation T gives a new matrix AB which represents the composition of functions SoT. Hence, to learn about matrices, it's most helpful to actually learn about linear transformations.

A great book for this is Axler. This may not seem like what you want at first glance - he takes a couple chapters to even introduce matrices, and most of the book eschews them for working with the abstract notion of a linear transformation. However, it will give you a really strong idea of the correspondence between matrices and linear transformations, and the theory behind it.

Axler focuses on a concept called a vector space, specifically vector spaces over the reals and complexes. Matrices can also be used in more general contexts, and there are a couple interpretations which you most likely have not been exposed to yet which do not relate directly to linear transformations. For this, I would recommend my personal favorite math book ever, "Algebra" by Mac Lane and Birkhoff. As the title suggests, this book does much more algebra than just linear algebra. Chapter 7 is specifically on matrices and is probably the best chapter in the book, however you'll have to go through the earlier chapters to understand it. This book is much more challenging than Axler is and the first several chapters don't fit your specific goal, so I would recommend starting with Axler, and keeping this in mind if you ever want to try a more advanced book in the future.

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.

OK, then let's try this again, this time using more calculus and less topology-specific results. I'm going to be using LaTeX markup here; see the sidebar to /r/math for a free browser plugin that'll translate my code into readable mathematics.

The following is from Michael Spivak's Calculus on Manifolds, and it's pretty close to the result you want, but with more restrictions in terms of differentiability and such:

• Problem 2-37.
  
  (a) Let [; f \colon \mathbf{R}^2} \to \mathbf{R} ;] be a continuously differentiable function. Show that [; f ;] is not 1-1. Hint: If, for example, [; D_1 f(x,y) \neq 0 ;] for all [; (x,y) ;] in some open set [; A, ;] consider [; g \colon A \to \mathbf{R}^2 ;] defined by [; g(x,y) = \left( f(x,y), y \right). ;]
  
  (b) Generalize this result to the case of a continuously differentiable function [; f \colon \mathbf{R}^n \to \mathbf{R}^m ;] with [; m<n. ;]
  
  The basic idea for (a) is that if there were such an continuously differentiable injection [; f \colon \mathbf{R}^2 \to \mathbf{R}, ;] then (1) we can find some subset [; A \subseteq \mathbf{R}^2 ;] such that (depending on your convention for notation)
  
  [; D_1 f(x,y) = \partial_1 f(x,y) = \partial_x f(x,y) = \frac{\partial f}{\partial x} (x,y) \neq 0 ;]
  
  for all [; (x,y) \in A, ;] and (2) the function [; g \colon A \to \mathbf{R}^2 ;] must have a local continuously differentiable inverse. (This is by the Inverse Function Theorem.)
  
  The problem, however, arises when you consider the actual form of a local inverse for [; g, ;] since [; g^{-1} ;] will be independent of the second coordinate. Accordingly, [; g ;] cannot be injective, whence [; f ;] cannot be injective.
  
  I imagine the generalization to part (b) is similar. The important thing here is that given a function
  
  [; f \colon \mathbf{R}^m \times \mathbf{R}^n \to \mathbf{R}^m, \text{ where } m<n, ;]
  
  one can construct the associated function
  
  [; \begin{align*}<br /> g \colon \mathbf{R}^m \times \mathbf{R}^n &amp;amp;\to \mathbf{R}^m \times \mathbf{R}^n\\<br /> (\mathbf{x}, \mathbf{y}) &amp;amp;\mapsto \left( f(\mathbf{x},\mathbf{y}), \mathbf{y} \right).<br /> \end{align*} ;]
  
  In the above example, we're considering the case [; m=n=1, ;] and we're considering the equivalence [; \mathbf{R}^1 \times \mathbf{R}^1 \simeq \mathbf{R}^2. ;]
  
  The advantage is that [; g ;] now maps between two spaces of the same dimension, so one can often apply the Inverse Function Theorem. (In fact, this is a common way to deduce the Implicit Function Theorem from the Inverse Function Theorem, so you see this technique often enough that it's worth your time to remember it.)
  
  These exercises require stronger assumptions—i.e., continuous differentiability rather than mere continuity—but perhaps this'll at least be a bit more accessible because it doesn't invoke quite so much topology. Hope this helps, and 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.

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!

There's really no easy way to do it without getting yourself "in the shit", in my opinion. Take a course on multivariate calculus/analysis, or else teach yourself. Work through the proofs in the exercises.

For a somewhat grounded and practical introduction I recommend Multivariable Mathematics: Linear Algebra, Calculus and Manifolds by Theo Shifrin. It's a great reference as well. If you want to dig in to the theoretical beauty, James Munkres' Analysis on Manifolds is a bit of an easier read than the classic Spivak text. Munkres also wrote a book on topology which is full of elegant stuff; topology is one of my favourite subjects in mathematics,

By the way, I also came to mathematics through the study of things like neural networks and probabilistic models. I finally took an advanced calculus course in my last two semesters of undergrad and realized what I'd been missing; I doubt I'd have been intellectually mature enough to tackle it much earlier, though.

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!

Honestly - through rigor. I would suggest studying logic, some philosophy (this is about the structure of arguments, and deduction in a general sense) and then something applied, like policy analysis or program evaluation. <- those last two are just related to my field so I know about them, plenty of others around.

Some suggested books that could be interesting for you:

Intro to Logic by Tarski

The Practice of Philosophy by Rosenberg

Thank you for Arguing by Heinrichs

Policy Analysis is instructive in that you have to define a problem, define its characteristics, identify the situation it exists in, plot possible solutions (alternatives), and create criteria for selecting the alternative you like most.

Program Evaluation is really just tons of fun and will teach a bunch about how to appraise things. Eval can get pretty muddy into social research but honestly you can skip a lot of that and just learn the principles.

The key to this is that you're either very smart and can learn this stuff through your own brains and force of will, or, more likely, you'll need people to help beat it into you WELCOME TO GRADSCHOOL.

I'm a physicist/mathematician, but I think it could be useful for you. Exterior algebra (differential forms) in particular is worth learning because it makes the theory of multivariable calculus much more elegant and simple. With exterior algebra you can see that the fundamental theorem of calculus, Green's theorem, and the divergence theorem are special cases of a generalized Stoke's theorem. Spivak's Calculus on Manifolds book (which is actually not a manifolds book despite the name) teaches calculus at an undergrad level using exterior algebra and differential forms if you're interested in learning this stuff.

Exterior algebra can be considered as part of geometric algebra, so you could continue on to learn geometric algebra if you enjoy exterior algebra.

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.
   

There's a couple options. You could pick up a basic elementary number theory book, which will have basically no prerequisites, so you'll be totally fine going into it. For instance Silverman has an elementary number theory book that I've heard great things about. I haven't read most of it myself, but I've read other things Silverman has written and they were really good.

There's a couple other books you might consider. Hardy and Wright wrote the classic text on it, which I've heard still holds up. I learned my first number theory from a book by Underwood Dudley which is by far the easiest introduction to number theory I've seen.

Another route you might take is that, since you have some background in calculus, you could learn a little basic analytic number theory. Much of this will still be out of your reach because you haven't taken a formal analysis class yet, but there's a book by Apostol whose first few chapters really only require knowledge of calculus.

If you decide you want to learn more number theory at that point, you're going to want to make sure you learn some basic algebra and analysis, but these are good places to start.

One thing I like to remind people, is that Linear Algebra is really cool and though it tends to come "after" calculus for some reason, it really has no explicit calc prerequisite.

I highly recommend Dr. Gilbert Strang's lectures on it, available on youtube and ocw.mit.edu (which has problems, solutions, etc, also)

I think it's a great topic for right around late HS, early college. And he stresses intuition and imo has the right balance of application and theory.

I'd also say that contrary to most peoples' perceptions, a student's understanding of a math topic will vary greatly depending on the teacher. And some teachers will be better for some students, others for others. That's just my opinion, but I firmly believe it. So if you find yourself struggling with a topic, find another teacher/resource and perhaps it will be more clear. Of course this shouldn't diminish the effort needed on your part, learning math isn't a passive activity, one really has to do problems and work with the material.

And finally, proofs are of course the backbone of mathematics. Here is an intro text I like on that.

Oh okay, one more thing, physics is a great companion to math. I highly recommend "Classical Mechanics" by Taylor, in that regard. It will be challenging right now, but it will provide some great accompaniment to what you'll learn in upcoming years.

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.

There are a few options. Firstly, if you are more familiar using infinity in the context of Calculus, then you might want to look into Real Analysis. These subjects view infinity in the context of limits on the real line and this is probably the treatment you are probably most familiar with. For an introductory book on the subject, check out Baby Rudin (Warning: Proofs! But who doesn't like proofs, that's what math is!)

Secondly, you might want to look at Projective Geometry. This is essentially the type of geometry you get when you add a single point "at infinity". Many things benefit from a projective treatment, the most obvious being Complex Analysis, one of its main objects of study is the Riemann Sphere, which is just the Projective Complex Plane. This treatment is related to the treatment given in Real Analysis, but with a different flavor. I don't have any particular introductory book to recommend, but searching "Introductory Projective Geometry" in Amazon will give you some books, but I have no idea if they're good. Also, look in your university library. Again: Many Proofs!

The previous two treatments of infinity give a geometric treatment of the thing, it's nothing but a point that seems far away when we are looking at things locally, but globally it changes the geometry of an object (it turns the real line into a circle, or a closed line depending on what you're doing, and the complex plane into a sphere, it gets more complicated after that). But you could also look at infinity as a quantitative thing, look at how many things it takes to get an infinite number of things. This is the treatment of it in Set Theory. Here things get really wild, so wild Set Theory is mostly just the study of infinite sets. For example, there is more than one type of infinity. Intuitively we have countable infinity (like the integers) and we have uncountable infinity (like the reals), but there are even more than that. In fact, there are more types of infinities than any of the infinities can count! The collection of all infinities is "too big" to even be a set! For an introduction into this treatment I recommend Suppes and Halmos. Set Theory, when you actually study it, is a very abstract subject, so there will be more proofs here than in the previous ones and it may be over your head if you haven't taken any proof-based courses (I don't know your background, so I'm just assuming you've taken Calc 1-3, Diff Eq and maybe some kind of Matrix Algebra course), so patience will be a major virtue if you wish to tackle Set Theory. Maybe ask some professors for help!

(Note: I wrote this elsewhere)

Discrete Mathematics. It teaches the basics of the following 5 key concepts in theoretical computer science:

• function
  
• set
  
• recursion
  
• proposition
  
• proof
  
  When you master these concepts, you will see that all hairy, formal definitions boil down to functions, sets, and propositions (with quantifiers). Recursion appears in many interesting structures in computer science, as well as in proofs of theorems (which are themselves propositions).
  
  I don't know of any good online introduction to discrete math, but here are a few book ideas.
  
  
• For a good textbook on discrete mathematics, check Schaum's Outline of Discrete Mathematics. It's also cheap.
  
• For a less textbook-ish overview written in an entertaining manner, Discrete Mathematics: Elementary and Beyond is my personal favourite.
  
• If you want to learn-by-doing, then The Haskell Road to Logic, Maths and Programming teaches you discrete math by having you manipulate discrete math concepts in the programming language Haskell. Here is a good free online Haskell tutorial.

What is the "Highest Level" of mathematics you have taken?
Math is substantially more like a foreign language than popular culture would lead you to believe. It takes practice and what I like to call 'settle time.'
If you feel like you have a strong grasp on the concepts of algebra I highly recommend starting from 'scratch' (first principals) and getting a book like http://www.amazon.com/gp/offer-listing/0321390539/ref=sr_1_2_twi_har_1_olp?ie=UTF8&qid=1450533541&sr=8-2&keywords=mathematical+proofs

It was the first textbook that made me really start to understand what is needed to think like a mathematician. Start at the beginning work though problems, set theory is so much more important than most people realize. It will be cloudy and frustrating but really try to work some problems, put it down for a week let it stew and come back to the problems you had trouble with. Do that over and over.
While you are doing that pick up any elementary Stats/Prob and/or Linear Algebra book and start flipping through from the beginning you will see all the tools you are learning in Mathematical Proofs in those books as well. Try to take what you are learning and see it applied in those books to add some extra hooks to attach things to in your brain.

For Numerical Analysis you are going to want to build a strong base in proofs, linear algebra, set theory, and calculus as you go forward. Don't let this stop you from starting to read up it is a great way to stay excited when you are learning things to know fun ways that they are applied but don't get discouraged. My Numerical Analysis class was a Sr level college course that started the semester with 24 Math and CS majors about half gave up before the mid term/

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

Hey mathit.

I'm 32, and just finished a 3 year full-time adult education school here in Germany to get the Abitur (SAT-level education) which allows me to study. I'm collecting my graduation certificate tomorrow, woooo!

Now, I'm going to study math in october and wanted to know what kind of extra prep you might recommend.

I'm currently reading How to Prove It and The Haskell Road to Logic, Maths and Programming.
Both overlap quite a bit, I think, only that the latter is more focused on executing proofs on a computer.

Now, I've just been looking into books that might ease the switch to uni-level math besides the 2 already mentioned and the most promising I found are these two:
How to Study for a Mathematics Degree and Bridging the Gap to University Mathematics.

Do you agree with my choices? What else do you recommend?

I found online courses to be ineffective, I prefer books.

What's your opinion, mathit?

Cheers and many thanks in advance!

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.

For me, a "good read" in mathematics should be 1) clear, 2) interestingly written, and 3) unique. I dislike recommending books that have, essentially, the same topics in pretty much the same order as 4-5 other books.

I guess I also just disagree with a lot of people about the
"best" way to learn topology. In my opinion, knowing all the point-set stuff isn't really that important when you're just starting out. Having said that, if you want to read one good book on topology, I'd recommend taking a look at Kinsey's excellent text Topology of Surfaces.

If you're interested in a sequence of books...keep reading.

If you are confident with calculus (I'm assuming through multivariable or vector calculus) and linear algebra, then I'd suggest picking up a copy of Edwards' Advanced Calculus: A Differential Forms Approach. Read that at about the same time as Spivak's Calculus on Manifolds. Next up is Milnor Topology from a Differentiable Viewpoint, Kinsey's book, and then Fulton's Algebraic Topology. At this point, you might have to supplement with some point-set topology nonsense, but there are decent Dover books that you can reference for that. You also might be needing some more algebra, maybe pick up a copy of Axler's already-mentioned-and-excellent Linear Algebra Done Right and, maybe, one of those big, dumb algebra books like Dummit and Foote.

Finally, the books I really want to recommend. Spivak's A Comprehensive Introduction to Differential Geometry, Guillemin and Pollack Differential Topology (which is a fucking steal at 30 bucks...the last printing cost at least $80) and Bott & Tu Differential Forms in Algebraic Topology. I like to think of Bott & Tu as "calculus for grown-ups". You will have to supplement these books with others of the cookie-cutter variety in order to really understand them. Oh, and it's going to take years to read and fully understand them, as well :) My advisor once claimed that she learned something new every time she re-read Bott & Tu...and I'm starting to agree with her. It's a deep book. But when you're done reading these three books, you'll have a real education in topology.

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'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.

> 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.

This one's well-known and highly regarded as a good source.

I'm also going to start learning number theory because it's a pretty fun subject. So far, Hardy's been pretty good (I've only read excerpts of the 1st chapter though).

As for your background, you would only need to know basic facts about numbers (divisibility/primes etc) when starting Hardy so you should be fine I think.

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!

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.

I wouldn't bother with Apostol's Calculus. For analysis, you should really look at the first two volumes of Stein and Shakarchi's Princeton Lectures in Analysis.

Vol I: Fourier Analysis
Vol II: Complex Analysis

Then, you should pick up:

Munkres, Analysis on Manifolds or something similar, you could try Spivak's book but it's a bit terse. (on a personal note, I tried doing Spivak's book when I was a freshman. It was a big mistake).

In truth, most introductory undergrad analysis texts are actually more invested in trying to teach you the rigorous language of modern analysis than in expositing on ideas and theorems of analysis. For example, Rudin's Principles is basically to acquaint you with the language of modern analysis -- it has no substantial mathematical result. This is where the Stein Shakarchi books really shines. The first book really goes into some actual mathematics (fourier analysis even on finite abelian groups and it even builds enough math to prove Dirichlet's famous theorem in Number Theory), assuming only Riemann Integration (the integration theory taught in Spivak).

For Algebra, I'd suggest you look into Artin's Algebra. This is truly a fantastic textbook by one of the great modern algebraic geometers (Artin was Grothendieck's student and he set up the foundations of etale cohomology).

This should hold you up till you become a sophomore. At that point, talk to someone in the math department.

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.

Possible path:

Learn to think like mathematicians because you'll need it. For example, Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand et al is a good book for that. When you got the basics of math argumentation down, it's time for abstract algebra with emphasis on vector spaces(you really need good working knowledge of linear algebra). People like Axler's Linear Algebra Done Right. Maybe, study that. Or maybe work through Maclane's Algebra or Chapter 0 by Aluffi.

After that you want to get familiar with more or less rigorous calculus. One possibility is to study Spivak's Calculus, then pick up Munkres Analysis on Manifolds.

Up next: differential geometry which is your main goal. At this point your mathematical sophistication will have matured to the level of a grad student of math.

Good luck.

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.

Proofs: Hammack's Book of Proof. Free and contains solutions to odd-numbered problems. Covers basic logic, set theory, combinatorics, and proof techniques. I think the third edition is perfect for someone who is familiar with calculus because it covers proofs in calculus (and analysis).

Calculus: Spivak's Calculus. A difficult but rewarding book on calculus that also introduces analysis. Good problems, and a solution manual is available. Another option is Apostol's Calculus which also covers linear algebra. Knowledge of proofs is recommended.

Number Theory: Hardy and Wright's An Introduction to the Theory of Numbers. As he explains in a foreword to the sixth edition, Andrew Wiles received this book from his teacher in high school and was a starting point for him. It also covers the zeta function. However, it may be too difficult for absolute beginners as it doesn't contain any problems. Another book is Stark's An Introduction to Number Theory which has a great section on continued fractions. You should have familiarity with proof before learning number theory.

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.

Sure, there are lots of cool websites that don't ask for crazy prerequisites. One which I share with all of my friends who are starting out in math is the Fun Facts site, hosted by Harvey Mudd College.

As far as learning specific materials, you can try Khan Academy for what are perhaps some of the more elementary topics (it goes up to differential equations and linear algebra). If you want to learn more about number systems and algebra I think that either picking up a good, cheap book on number theory, or even checking out the University of Reddit's Group Theory course (presented by Math Doctor Bob) are both very strong options. Otherwise, you can check out YouTube for other lecture series that people are more and more frequently putting up.

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.

If you're looking for other texts, I would suggest Spivak's Calculus and Calculus on Manifolds. At first the text may seem terse, and the exercises difficult, but it will give you a huge advantage for later (intermediate-advanced) undergraduate college math.

It may be a bit obtuse to recommend you start with these texts, so maybe your regular calculus texts, supplemented with linear algebra and differential equations, should be approached first. When you start taking analysis and beyond, though, these books are probably the best way to return to basics.

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

Mathematical Proofs: A Transition to Advanced Mathematics was the book for my college proofs class. I found it to be a good resource and easy to follow. It covers some introductory set theory as well. Just be prepared to work through the proof exercises if you really want a good intuition on the topic.

I've found Alfred Tarski's Introduction to Logic: and to the Methodology of Deductive Sciences to be a great primer on sentential (predicate) logic. Tarski was a good jumping off point from mathematics to mathematical logic and analytical philosophy. Discovering this book was a turning point in my life; it galvanized my interest in mathematics and lead me to study the foundations of mathematics and philosophy of language in my free time.

Yeah, definitely the best book I've read on differential forms was Spivaks Calculus on Manifolds. Its very readable once you have a solid foundational calculus background and is pretty small given what it covers (160pp). If you need to know this stuff then this is definitely the right place to learn it.

Two great introductions are:

• Armstrong - Groups and Symmetry
  
  
• Halmos - Naive Set Theory
  
  
  both are short and very well written. Some books particularly close to my heart are:
  
  
  
• Sipser - Theory of Computation
  
  
• Lothaire Combinatorics on Words
  
  
• Davey and Priestley Lattices and Order

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.

In a math course I recently took that was basically an introduction to math proofs we used Mathematical Proofs: A Transition to Advanced Mathematics which I found to be a great text. It begins by going through the language and syntax used in proofs and slowly progresses through theory, different types of proofs, and eventually proofs from advanced calculus. There are so many examples that are very well laid out and explained. I would highly recommend it for learning proofs from scratch.

For single variable calculus, like everyone else I would recommend Calculus - Spivak. If you have already seen mechanical caluculus, mechanical meaning plug and chug type problems, this is a great book. It will teach you some analysis on the real line and get your proof writing chops up to speed.

For multivariable calculus, I have three books that I like. Despite the bad reviews on amazon, I think Vector Calculus - Marsden & Tromba is a good text. Lots of it is plug and chug, but the problems are nice.

One book which is proofed based, but still full of examples is Advanced Calculus of Several Variables - Edwards Jr.. This is a nice book and is very cheap.

Lastly, I would like to give a bump to Calculus on Manifolds - Spivak. This book is very proofed based, so if you are not comfortable with this, I would sit back and learn from of the others first.

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..."

They may not be the best books for complete self-learning, but I have a whole bookshelf of the small introductory topic books published by Dover- books like An Introduction to Graph Theory, Number Theory, An Introduction to Information Theory, etc. The book are very cheap, usually $4-$14. The books are written in various ways, for instance the Number Theory book is highly proof and problem based if I remember correctly... whereas the Information Theory book is more of a straightforward natural-language summary of work by Claude Shannon et al. I still find them all great value and great to blast through in a weekend to brush up to a new topic. I'd pair each one with a real learning text with problem sets etc, and read the Dover book first quickly which introduces the reader to any unfamiliar terminology that may be needed before jumping into other step by step learning texts.

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.

First and foremost discard the idea of contributing to an area of research mathematics. It's not that it's impossible for you to do so, but it's not a good goal to set. It's best for you to try and explore a field of mathematics that interests you to learn more about it. After all this is what mathematical research actually is, we have questions that we would like to know the answers to so we figure them out. It is also a much more attainable goal, whether the material is new to the mathematical community or not you will have learned something new.

Second, if you really want to try and get to the forefront of mathematical understanding, expect to put in about a year or two at minimum to get there, and that's only if you pick a new or obscure field whose frontier is not as far removed from where you are. Fields like combinatorics and graph theory also have frontiers that are easily approachable for beginners.

If you're really dead set on algebra I would put forth two different fields. The first is combinatorial group theory, which is a bit older, but a lot of people have vacated the field. The classic text on that is Combinatorial Group Theory by Magnus. I don't know much about the status of open questions in the field, but I do know that combinatorial methods crop up in solutions to open problems in group theory all the time. You might be able to get the background needed to understand and work though most of that book. You'd need at minimum a solid understanding of presentations of groups and a bit of knowledge about combinatorics.

The field that most mathematicians have moved into after working in CGT is geometric group theory. It's a relatively new field with lots of interesting accessible open questions, but requires a bit of background knowledge of metric topology. There aren't any English language classical texts that are approachable at your level, but these notes by Alessandro Sisto are a quite good introduction. (The ending of the notes tapers off with fewer and fewer details, I wouldn't read past page 64 or 65). There are also some errors you have to catch in his proofs, and statements of theorems throughout.

This is all assuming that you've read an introductory book to abstract algebra and done all of the problems, such as Contemporary Abstract Algebra and actually know all the basics solidly.

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"
  

You should check out Spivaks Calculus on Manifolds.

http://www.amazon.com/Calculus-Manifolds-Approach-Classical-Theorems/dp/0805390219

Read the first chapter or 2 and see how you like it, if you feel overwhelmed check some of the other recommendations out.
It is however a good book, and you should read it sooner or later.

Rudins principles of mathematical analysis is also excellent, however it
is not strictly multi-dimensional analysis.
Read at least chapter 2 and 3, they lay a very important groundwork.

The Haskell Road to Logic, Maths and Programming

http://amzn.com/0954300696

I read only the first chapter or two a long time ago. I don't remember much, but I do remember I was able to progress through the book and learn new things about both math and Haskell from the text.

I didn't have any trouble getting the outdated examples to work. I had read LYAH previously though, so I wasn't a complete beginner.

I would really enjoy hearing what others have thought about this book.

Daniel J Velleman's How to Prove It : A Structured Approach


This book is a pretty dang good intro to proofs, I highly reccommend it. This is the first edition, so you'll be able to find a used copy for super cheap.

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

Number theory is pretty cool. I enjoyed Dudley's book for a number of reasons.

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.

------------

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 can recommend two books which I have read recently.

1. An Introduction to Mathematical Logic is more structured and formal description of logic.
   
2. [Introduction to Logic] (http://www.amazon.com/Introduction-Logic-Methodology-Deductive-Mathematics/dp/048628462X/ref=sr_1_7?ie=UTF8&qid=1449702263&sr=8-7&keywords=mathematical+logic) gives more insights and helps to get a big picture of logic.
   I enjoyed both of them a lot and going to read them again.

Also pick a book on Number Theory, pretty useful in computer science. I would recommend: https://www.amazon.com/Elementary-Number-Theory-Second-Mathematics/dp/048646931X/ref=sr_1_1?ie=UTF8&qid=1500074415&sr=8-1&keywords=elementary+number+theory+dudley

I don't know of any video lectures that covers these topics, but I do know of a couple of good books that should be good resources to reference if you find Rudin a bit too terse in some places:

1. "Understanding Analysis" by Stephen Abbott - This should cover the first half of Rudin, plus the sequences/series of functions. I would really recommend, when you have the time, that you go back over Analysis with this book.
   
   
2. "Analysis on Manifolds" by James Munkres - Covers the stuff on Differential Forms. In fact, I would say that Rudin's main area of weakness in his Principles of Mathematical Analysis is precisely his coverage of differential forms, and so I would definitely pick up this book or the next.
   
   
3. "Calculus on Manifolds" by Spivak - This covers basically the same material as Munkres, but is more concise in the exposition. This is a classic, by the well-known differential geometer Michael Spivak. One warning, though: Spivak uses superscripts to index elements, so x = ( x^1 , x^2 , ... , x^n ) is how he writes points in R^n .
   
   I would recommend a combination of 2 and 3 for the differential forms and stuff from Rudin, and 1 for single variable real analysis.

I know a few people who highly recommend How to Prove It by Velleman. I've never read it so I can't say for sure. The first book I used to learn mathematical logic was Lay's Analysis with an Intro to Proof. I can't recommend that book enough. The first quarter of the book or so is a pretty gentle introduction to mathematical logic, sets, functions, and proof techniques. I imagine it will get you where you need to be pretty quickly.

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.

The Haskell Road to Logic, Maths and Programming. I had already fallen in love with programming, and with Haskell, and this book showed me how well math, logic, and computer science play together. Shoutout to my aunt Trisha for giving me this book as a Christmas present in my junior year of high school

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.

I would highly recommend spending some time learning number theory first. Much of crypto relies on understanding a fair amount of number theory in order to understand what and why stuff works.

The book antiantiall linked is fantastic (I have a copy), however if you don't have a strong foundation in number theory will likely be a bit over your head.

Here is the textbook that was used in my number theory course. It isn't necessarily the best out there, but is cheap and does a good job covering the basics.

I took a long, long break between undergrad and grad school (think decades). I found this GRE math prep book very helpful. (The GRE math section tests high school math knowledge), I'd take the sample tests, see where I fell short, and focus on understanding why. I also found Practical Algebra to be a good review-and-practice guide, for the fundamentals. I boned up on discrete math by buying an old copy of Rosen and the matching solutions guide. And, I watched a bunch of videos of this guy explaining various facets of the math you need for computer science.

Awesome, I will take a look at that. Here is the book I have to teach myself with (used it for Calculus 2 a year ago). It seems like a solid book.

As someone that didn't start off in math, I've always heard that Godel and Tarski are formalizing Russell's set paradox. Have I got it all wrong?

P.S.

I always love reading about famous people in philosophy that are also immensely important in other fields. Tarski's T-schema is an excellent correspondence theory of truth in philosophy; he's even bigger in logic (by the way, his Introduction To Logic is a great read).

The same goes for Kant - I sat in on a sociology class years ago that started off discussing Kant. Later, when discussing the class with the professor he admitted that he didn't know Kant was Serious Business outside of sociology.

P.P.S.

You said, "I spent lots of time as a youth thinking about the "deeper" meaning to the world we inhabit of the theorems (which ultimately is very little)." If you deny that we discover a deeper meaning to the world we inhabit when we discover the connection between the falling of an apple and the rotation of the planets, or between table salt and sodium, we've got a serious dispute.

We only need Pythagorean theorem to understand special relativity. Consider two dudes X and Y. Suppose X is on a flying carpet holding up two mirrors distance of h apart. Also assume there's a light particle bouncing between these mirrors vertically like this here. So we see that h = ct_x where c is the speed of light and t_x is the amount of time it takes for the light to go from one mirror to the other. Now have Y stand on the ground and observe the behavior of the light particle as the carpet flies horizontally. From the perspective of Y, the particle flies in a sawtooth pattern like this. The distance the particle travels diagonally depends on the speed of light c and so it is ct_y where t_y is time taken by light to bounce from one mirror to the other as seen by Y. The distance the particle travels horizontally depends on the speed s of carpet and so it is st_y. By Pythagorean theorem, we have h^2 + (st_y)^2 = (ct_y)^2 which implies (t_y)^2 = h^2 / (c^2 (1 - s^2 / c^2 )) which further implies t_y = t_x / (sqrt(1 - s^2 / c^2 )). Thus if s = 0, then t_x = t_y and so time is universal. But as s approaches the speed of light c, the clocks desynchronize.

@ OP, if you want to get into high-falutin physics, you want to know the basics of real, functional (covers linear algebra), complex analyses; some probability and statistics; a bit of group theory.

For analysis the books by Lara Alcock, Amol Sasane, Paul Zorn, Robert Strichartz, Jonathan Kane, Steven Lay, Stephen Abbot, K.G Binmore, Charles Pugh, Mary Hart and many others are very user-friendly. And taking into account your background, Linear Algebra: Step by Step by Kuldeep Singh is perfect for you.

OK..

http://www.amazon.com/Contemporary-Abstract-Algebra-Joseph-Gallian/dp/0547165099/

http://www.amazon.com/Algorithm-Design-Jon-Kleinberg/dp/0321295358/

http://www.amazon.com/Introduction-Algorithms-Thomas-H-Cormen/dp/0262033844/

http://www.amazon.com/Introductory-Combinatorics-5th-Richard-Brualdi/dp/0136020402/

http://www.amazon.com/Artificial-Intelligence-Modern-Approach-3rd/dp/0136042597/

And I guarantee you those nice amazon prices aren't that steeply discounted around the start of the semester, and certainly not in any brick-and-mortar bookstore.

Hardy and Wright, An Introduction to the Theory of Numbers. Awesome book.

http://www.amazon.com/An-Introduction-Theory-Numbers-Hardy/dp/0199219869

I second Michael Spivak's Calculus if you haven't done a proper analysis course before. It's a 100% rigorous treatment of calculus from first principles and is probably better thought of as "analysis in one dimension." I post on a subreddit of folks working through the book pretty frequently: /r/calculusstudygroup

After that, I like Kolmogorov and Fomin's Introductory Real Analysis and Walter Rudin's Principles of Mathematical Analysis.

There's also Michael Spivak's Calculus on Manifolds, which focuses purely on multi-variable calculus on manifolds. Torus calculus!

I think Linear Algebra by Kuldeep Singh is the best fit for newcomers to LA. It's unpretentious and meant to be actually read by students (can you imagine?). This book will take you from someone who just discovered there exists such a thing as LA to someone who solves problems in Linear Algebra Done Right By Axler cold. After Kuldeep Singh you can pick up Advanced Linear Algebra by Steven Roman which is an extreme overkill even for mathematicians.

Basically, once you get the basics of LA down, you can simply read up on the newest matrix algos for machine learning on ArXiv or something. BTW, if your goal is working with data you need to learn some probability.

I've been taking it this year, and we've been using Velleman's "How to Prove It." Unfortunately, there aren't answers for all the problems, but I've found it to be a pretty good book. Amazon

For Calculus:

Calculus Early Transcendentals by James Stewart

^ Link to Amazon

Khan Academy Calculus Youtube Playlist

For Physics:

Introductory Physics by Giancoli

^ Link to Amazon

Crash Course Physics Youtube Playlist

Here are additional reading materials when you're a bit farther along:

Mathematical Methods in the Physical Sciences by Mary Boas

Modern Physics by Randy Harris

Classical Mechanics by John Taylor

Introduction to Electrodynamics by Griffiths

Introduction to Quantum Mechanics by Griffiths

Introduction to Particle Physics by Griffiths

The Feynman Lectures

With most of these you will be able to find PDFs of the book and the solutions. Otherwise if you prefer hardcopies you can get them on Amazon. I used to be adigital guy but have switched to physical copies because they are easier to reference in my opinion. Let me know if this helps and if you need more.

My university used George Andrews's book, which is Dover and really cheap. It was a pretty good book.

If you're not afraid of math there are some cheap introductory textbooks on topics that might be accessible:
For abstract algebra: http://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178/ref=sr_1_1?ie=UTF8&qid=1459224709&sr=8-1&keywords=book+of+abstract+algebra+edition+2nd

For Number Theory: http://www.amazon.com/Number-Theory-Dover-Books-Mathematics/dp/0486682528/ref=sr_1_1?ie=UTF8&qid=1459224741&sr=8-1&keywords=number+theory

These books have complimentary material and are accessible introductions to abstract proof based mathematics. The algebra book has all the material you need to understand why quintic equations can't be solved in general with a "quintic" formula the way quadratic equations are all solved with the quadratic formula.

The number theory book proves many classic results without hard algebra, like which numbers are the sum of two squares, etc, and has some of the identities ramanujan discovered.

For an introduction to analytic number theory, a hybrid pop/historical/textbook is : http://www.amazon.com/Gamma-Exploring-Constant-Princeton-Science/dp/0691141339/ref=sr_1_1?ie=UTF8&qid=1459225065&sr=8-1&keywords=havil+gamma

This book guides you through some deep territory in number theory and has many proofs accessible to people who remember calculus 2.

If you want a gentler introduction to calculus, with many examples, lots of intuition, diagrams, and nicer explanations, take any edition of James Stewart's Calculus - Early Transcendentals.

If you feel up to a serious challenge and want to study it as a mathematician would, get Michael Spivak's Calculus.

I can't think of anything that's more important in math than proofs. Study a subject that involves a lot of proofs (any advanced math, really) such as linear algebra or analysis, and practice. How to Prove It by Velleman may help you get started. Writing proofs is just applying logic, definitions, and previously proved theorems.

The Haskell Road to Logic, Maths and Programming takes you through a lot of the basic “essential” math for CS, much of what would be covered in a typical discrete math course, but taught along side Haskell which is fun!

My math skills sucked when I started. Definitely go though a book on math if you can.

There are two books I recommend. One book I found recently and plan to go through once I am done with the program (I am too busy now), just because I want to solidify my math skills is: Mastering Technical Mathematics

I found the book randomly and after skimming through a few pages knew it was a great book. It starts out with basic discrete mathematics concepts like counting and then goes all the way up to some calculus ideas.

The other book I reccomend is one I went through called Practical Algebra: A Self-Teaching Guide, Second Edition. It focuses more on algebra obviously but Algebra is actually the hardest part of CS 225 and CS325!

I posted this in /learnmath but didn't get any response so I'll give it a try here.

I'm a senior high school student and I'm learning linear algebra using Pavel Grinfeld's videos and programming in Haskell with this book.

What can I do to practice and apply concepts of linear algebra and programming?

Any recommended textbooks to complement the LA course?

Is it a good idea to solve project Euler problems in order to acquire programming/math skills?

My first introduction to group theory/abstract algebra came from this book by Fraleigh (for God's sake don't pay $120 for it). As I remember, it started pretty basic so take a look.

I'm also a big fan of Dummit & Foote as AngelTC mentioned.

I think "Mathematical Proofs: A Transition to Advanced Mathematics (2nd Edition)" is a solid book.

It starts off with what I would expect in a discrete math course (which is generally a first proofs course) and ends with a few chapters that would begin a second step writing intensive proofs course: number theory, calculus (real analysis), and group theory (algebra).

There are also many resources online that will help you once you've gotten through the basic notions in the book.

A Pathway into Number Theory by Burns might appeal to you. You might want to put extra effort into digging up a book that approaches elementary number theory from a combinatorial point of view, which is more in line with the stuff you're doing now.

EDIT: This seems perfect for you: https://www.amazon.com/Number-Theory-Dover-Books-Mathematics/dp/0486682528/

I'm really enjoying this book:
Practical Algebra

It starts from scratch and doesn't even assume too much about your knowledge of arithmetic. I was surprised how many gaps in my basic knowledge I had, but it helps explain why teaching myself via Khan or tutors didn't work well.

https://www.amazon.ca/Calculus-Early-Transcendentals-James-Stewart/dp/1285741552

Pretty sure it's this one. You should be able to find a pdf online.

Discrete mathematics with ducks!

Contemporary Abstract Algebra

An Illustrated Theory of Numbers

Visual Complex Analysis

Number-Crunching

The Magic of Math: Solving for x and Figuring Out Why
The Princeton Companion to Mathematics
The Number Devil: A Mathematical Adventure

Professor Stewart's Cabinet of Mathematical Curiosities
Gödel, Escher, Bach: An Eternal Golden Braid

One Two Three . . . Infinity: Facts and Speculations of Science

Recently made a move and these are a few books on my shelf that stand out more.

There are so many beautiful mathematical images that could be chosen though and I do wish more publishers would have more creative book covers also.

If you're planning on learning haskell (you should :D), why not do a book that teaches you both discrete maths and haskell at the same time?

There are atleast two books that do this:

• Haskell Road to Logic, Maths and Programming
  
  
• Discrete Mathematics Using a Computer
  
  Both books use Haskell in an effective way to teach you discrete mathematics. (They don't assume prior knowledge of Haskell).
  
  I've been working through the 2nd book and posting my notes and solutions here . You could compare solutions/notes and discuss stuff with me.

For basic Algebra(Linear, Multilinear bla, bla, bla) there exists an amazing book called "Algebra" by Saunders Maclane and Garett Birkhoff

I don't know what second/third semester Calculus means. Is it proof-based or non-proof based? Is it a regular Calculus sequence or is it Analysis?

Fraleigh is a little bit easier to wrap your head around. Get an old edition (or find it at the library), obviously.

Also, I highly recommend Herstein's Topics in Algebra. Again, try to get it from a university library.

Still the best math book cover ever

Algebra by Saunders MacLane and Garret Birkhoff is the best algebra book I have ever encountered.

Concepts of Modern Mathematics by Ian Stewart is an excellent book about modern math. As is Foundations and Fundamental Concepts of Mathematics by Howard Eves I would recommend these two along with the far more expensive Naive Set Theory by Halmos

Introduction to Logic: and to the Methodology of Deductive Sciences by Alfred Tarski really helped me understand the key concepts of mathematical logic when I was young. Dover has republished the book and you can find used copies in great shape for $5 USD.

OCW

Single Variable Calculus

Multivariable Calculus

Differential Equasions

Linear Algebra

Helpful books

Mathematical Proofs: A Transition to Advanced Mathematics

Understanding Probability: Chance Rules in Everyday Life

Linear Algebra

Other Resources

Kahn Academy

Probability (keithpeterb on youtube)

Probability (Kahn)

Linear Algebra (Kahn)


There, Fixed up your links.

For vector calculus, you might enjoy the less formal British text Div, Grad, Curl, and All That by H. M. Schey; for group theory in brief, consider the free textbook Elements of Abstract and Linear Algebra by Edwin H. Connell.

Alternatives to Schey's book include the much more formal Calculus on Manifolds by Michael Spivak, which does have more exercises than Schey but uses most of them to develop the theory, rather than as the mindless drills that fill an ordinary textbook; Michael E. Corral's free textbook Vector Calculus isn't huge but is written closer to an ordinary textbook.

I dunno about “undergraduate”, but you could try Birkhoff & Mac Lane or Greub. Those are both kind of old, so someone else may have a better idea.

I used this book when I was in high school:

Number Theory

Costs $8, explains things beautifully

Friendly info:

"College Algebra" = Elementary Algebra.

College Level Algebra = Abstract Algebra.

Example: Undergrad Algebra book.

Example: Graduate Algebra book.

Don't think of your abilities as fixed. The number of proofs you encounter grows from where you are now. You did not know algebra when you started. You will be increasingly exposed to proofs as you go along. Spend time on them. I recommend you get a tutor, or at least read some extra material. https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321390539/ref=sr_1_69?ie=UTF8&qid=1494805054&sr=8-69&keywords=proofs+math

Algebra by Serge Lang.

It has a good introduction to category theory, as well as being one of the better all around resources for algebra. He can be quite terse sometimes, but it is well-referenced.

I hear that you can also obtain digital copies of this book.

No, it's not. Math-major algebra was typically taught from something like Herstein. These days, Dummit and Foote seems more popular.

Damn, son. That's way bigger than my guesstimate.

The amazon prices I checked out pinned the collection closer to $400, which granted is still really, really impressive.

In case you're curious this was my textbook. It's come down by a lot in price over a couple years. Brand new it was $365 in the shrink wrap from my school's store!

Eh, either way I'm wrong, just by a different amount.

A friend of mine used this book in her undergraduate abstract algebra course. It looks fantasitic: Gallian - Contemporary abstract algebra.

Have you seen The Haskell Road to Logic, Maths, and Programming? It's a pretty decent intro to higher math, and each chapter has a Haskell module.

https://www.amazon.com/Number-Theory-Dover-Books-Mathematics/dp/0486682528/ref=pd_lpo_sbs_14_t_0/130-7956659-8948968?_encoding=UTF8&psc=1&refRID=7N6E11C8GDASPV8X3BAN

This was my first arithmetic book. I found it quite good and covered a wide range of topics.

Set Theory:

Naive Set Theory

Number Theory:

Elementary Number Theory

Introduction to Analytic Number Theory

A Classical Introduction to Modern Number Theory

Topology:

Topology

Introduction to Topological Manifolds

Two things:

1. Khan Academy!
   http://www.youtube.com/user/khanacademy#p/c/19E79A0638C8D449
   (I dunno why more people here don't seem to know about it).
   
   
2. Books, as mentioned.
   Personally I'd recommend these two:
   -http://www.amazon.com/gp/product/0470073330/ref=pd_lpo_k2_dp_sr_2?pf_rd_p=486539851&pf_rd_s=lpo-top-stripe-1&pf_rd_t=201&pf_rd_i=0471316598&pf_rd_m=ATVPDKIKX0DER&pf_rd_r=1S7Z2QKSMQM70SWRZQFT
   -http://www.amazon.com/Calculus-Stewarts-James-Stewart/dp/0495011606/ref=sr_1_1?ie=UTF8&s=books&qid=1279124596&sr=1-1
   
   You're probably gonna wanna go with earlier editions cuz they're way cheaper and say the same thing.

Naive Set Theory if you want a more textbook approach, the book mentioned in my other response if you're looking for something more like a story with proofs.

Get a logic book. For math majors at my University Sets and Logic is required before Linear Algebra which is the first proof intensive class.

http://www.amazon.com/How-Prove-Structured-Daniel-Velleman/dp/0521446635

This is the textbook. Very helpful.

http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521446635

That's what I started with and it was very helpful. The next semester when I took abstract vector spaces (proof-based linear algebra) I found writing the proofs to generally be straightforward because I'd already learned how to write a proof.

I was once a teacher's aide for an autistic teen. He seemed very bored with the 3rd grade arithmetic the teacher thought was his limit. One day, we had some extra time. I asked him if he wanted to read my set theory book. It's difficult to assess consent and comprehension, but we have our ways. I figured out that not only did he like this book, but he could follow along. It took about 3 months, but he was able to learn basics of sets. What makes me sad is the hard truth that people who know about this kind of math, generally don't find themselves being an educator for special needs students. His higher math education ended when I left. That's not right.

This might be of interest, Spivak's Calculus on Manifolds.

There's a good explanation of introductory quaternion theory in this book, which sets it in the larger context of group and field theory.

If you are a newcomer to abstract algebra, you might consider using a text other than Dummit and Foote. I used baby Herstein (as opposed to big Herstein) in an undergraduate class and found it to be a good introduction.

Quine's Methods of Logic and Mathematical Logic (in that order) have been my favorites, and I've heard good things about Tarski's Introducion to Logic: and the the Methodology of Deductive Sciences but have yet to get around to it.

You may also want to check out The Haskell Road to Logic, Maths and Programming.
This book focusses on logic and how to use it, so you get to learn proofs. It even hits corecursion and combinatorics. If you think math is pretty but you want to use it interactively as source code, this could be the book for you.

This one? How advanced would you say it goes into primes?

For an introductory text, I recommend Herstein's Topics in Algebra. It slowly walks you through groups, rings, vector spaces, modules, fields, linear transformations and other selected topics.

Has plenty of exercises and doesn't skip over any details.

I like Underwood Dudley's book, and it's now available cheap from Dover: https://www.amazon.com/Elementary-Number-Theory-Second-Mathematics/dp/048646931X/ref=sr_1_1?s=books&ie=UTF8&qid=1468275037&sr=1-1&keywords=dudley+number+theory

I recommend this book.

I like this calculus book: https://www.amazon.com/Calculus-7th-James-Stewart/dp/0538497815

And this is Halliday and Resnick https://www.amazon.com/Fundamentals-Physics-David-Halliday/dp/111823071X/ref=sr_1_3?crid=9SK20IQF11Z5&keywords=halliday+and+resnick+fundamentals+of+physics&qid=1567697861&s=books&sprefix=hallida%2Cstripbooks%2C124&sr=1-3

Depends on the market depths, for a lot of books, there may be a couple low priced books where a few purchases will raise the price pretty significantly. I think a lot of booksellers have repricers that don't work very effectively where they'lll lower the price over time until it sells, and there really isnt a market for text books except for at the beginning of semesters.

http://www.amazon.com/gp/offer-listing/1285741552/ref=olp_f_primeEligible?ie=UTF8&f_primeEligible=true

On that book, which is a pretty widely used text, If they sell about 5 books, the price rises almost $70.