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

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

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

> I assume I ought to check it out after my discrete math class? Or does CLRS teach the proofs as if the reader has no background knowledge about proofs?

Sadly it does not teach proofs. You will need to substitute this on your own. You don't need deep proof knowledge, but just the ability to follow a proof, even if it means you have to sit there for 2-3 minutes on one sentence just to understand it (which becomes much easier as you do more of this).

> We didn't do proof by induction, though I have learned a small (very small) amount of it through reading a book called Essentials of Computer Programs by Haynes, Wand, and Friedman. But I don't really count that as "learning it," more so being exposed to the idea of it.

This is better than nothing, however I recommend you get very comfortable with it because it's a cornerstone of proofs. For example, can you prove that there are less than 2 ^ (h+1) nodes in any perfect binary tree of height h? Things like that.

> We did go over Delta Epsilon, but nothing in great detail (unless you count things like finding the delta or epsilon in a certain equation). If it helps give you a better understanding, the curriculum consisted of things like derivatives, integrals, optimization, related rates, rotating a graph around the x/y-axis or a line, linearization, Newton's Method, and a few others I'm forgetting right now. Though we never proved why any of it could work, we were just taught the material. Which I don't disagree with since, given the fact that it's a general Calc 1 course, so some if not most students aren't going to be using the proofs for such topics later in life.

That's okay, you will need to be able to do calculations too. There are people who spend all their time doing proofs and then for some odd reason can't even do basic integration. Being able to do both is important. Plus this knowledge will make dealing with other math concepts easier. It's good.

> I can completely understand that. I myself want to be as prepared as possible, even if it means going out and learning about proofs of Calc 1 topics if it helps me become a better computer scientist. I just hope that's a last resort, and my uni can at least provide foundation for such areas.

In my honest opinion, a lot of people put too much weight on calculus. Computer science is very much in line with discrete math. The areas where it gets more 'real numbery' is when you get into numerical methods, machine learning, graphics, etc. Anything related to theory of computation will probably be discrete math. If your goal is to get good at data structures and algorithms, most of your time will be spent on discrete topics. You don't need to be a discrete math genius to do this stuff, all you need is some discrete math, some calc (which you already have), induction, and the rest you can pick up as you go.

If you want to be the best you can be, I recommend trying that book I linked first to get your feet wet. After that, try CLRS. Then try TAOCP.

Do not however throw away the practical side of CS if you want to get into industry. Reading TAOCP would make you really good but it doesn't mean shit if you can't program. Even the author of TAOCP, Knuth, says being polarized completely one way (all theory, or all programming, and none of the other) is not good.

> From reading ahead in your post, is Skiena's Manual something worth investing to hone my skills in topics like proof skills? I'll probably pick it up eventually since I've heard nothing but good things about it, but still. Does Skiena's Manual teach proofing skills to those without them/are not good at them? Or is there a separate book for that?

You could, at worst you will get a deeper understanding of the data structure and how to implement them if the proof goes over your head... which is okay, no one on this planet starts off good at this stuff. After you do this for a year you will be able to probably sit down and casually read the proofs in these books (or that is how long it took me).

Overall his book is the best because it's the most fun to read (CLRS is sadly dry), and TAOCP may be overkill right now. There are probably other good books too.

> I guess going off of that, does one need a certain background to be able to do proofs correctly/successfully, such as having completed a certain level of math or having a certain mindset?

This is developed over time. You will struggle... trust me. There will be days where you feel like you're useless but it continues growing over a month. Try to do a proof a day and give yourself 20-30 minutes to think about things. Don't try insane stuff cause you'll only demoralize yourself. If you want a good start, this is a book a lot of myself and my classmates started on. If you've never done formal proofs before, you will experience exactly what I said about choking on these problems. Don't give up. I don't know anyone who had never done proofs before and didn't struggle like mad for the first and second chapter.

> I mean, I like the material I'm learning and doing programming, and I think I'd like to do at least be above average (as evident by the fact that I'm going out of my way to study ahead and read in my free time). But I have no clue if I'll like discrete math/proving things, or if TAOCP will be right for me.

Most people end up having to do proofs and are forced to because of their curriculum. They would struggle and quit otherwise, but because they have to know it they go ahead with it anyways. After their hard work they realize how important it is, but this is not something you can experience until you get there.

I would say if you have classes coming up that deal with proofs, let them teach you it and enjoy the vacation. If you really want to get a head start, learning proofs will put you on par with top university courses. For example at mine, you were doing proofs from the very beginning, and pretty much all the core courses are proofs. I realized you can tell the quality of a a university by how much proofs are in their curriculum. Any that is about programming or just doing number crunching is literally missing the whole point of Computer Science.

Because of all the proofs I have done, eventually you learn forever how a data structure works and why, and can use it to solve other problems. This is something that my non-CS programmers do not understand and I will always absolutely crush them on (novel thinking) because its what a proper CS degree teaches you how to do.

There is a lot I could talk about here, but maybe such discussions are better left for PM.

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.

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.

Your mileage with certifications may vary depending on your geographical area and type of IT work you want to get into. No idea about Phoenix specifically.

For programming work, generally certifications aren't looked at highly, and so you should think about how much actual programming you want to do vs. something else, before investing in training that employers may not give a shit about at all.

The more your goals align with programming, the more you'll want to acquire practical skills and be able to demonstrate them.

I'd suggest reading the FAQ first, and then doing some digging to figure out what's out there that interests you. Then, consider trying to get in touch with professionals in the specific domain you're interested in, and/or ask more specific questions on here or elsewhere that pertain to what you're interested in. Then figure out a plan of attack and get to it.

A lot of programming work boils down to:

• Using appropriate data structures, and algorithms (often hidden behind standard libraries/frameworks as black boxes), that help you solve whatever problems you run into, or tasks you need to complete. Knowing when to use a Map vs. a List/Array, for example, is fundamental.
  
• Integrating 3rd party APIs. (e.g. a company might Stripe APIs for abstracting away payment processing... or Salesforce for interacting with business CRM... countless 3rd party APIs out there).
  
• Working with some development framework. (e.g. a web app might use React for an easier time producing rich HTML/JS-driven sites... or a cross-platform mobile app developer might use React-Native, or Xamarin to leverage C# skills, etc.).
  
• Working with some sort of platform SDKs/APIs. (e.g. native iOS apps must use 1st party frameworks like UIKit, and Foundation, etc.)
  
• Turning high-level descriptions of business goals ("requirements") into code. Basic logic, as well as systems design and OOD (and a sprinkle of FP for perspective on how to write code with reliable data flows and cohesion), is essential.
  
• Testing and debugging. It's a good idea to write code with testing in mind, even if you don't go whole hog on something like TDD - the idea being that you want it to be easy to ask your code questions in a nimble, precise way. Professional devs often set up test suites that examine inputs and expected outputs for particular pieces of code. As you gain confidence learning a language, take a look at simple assertion statements, and eventually try dabbling with a tdd/bdd testing library (e.g. Jest for JS, or JUnit for Java, ...). With debugging, you want to know how to do it, but you also want to minimize having to do it whenever possible. As you get further into projects and get into situations where you have acquired "technical debt" and have had to sacrifice clarity and simplicity for complexity and possibly bugs, then debugging skills can be useful.
  
  As a basic primer, you might want to look at Code for a big picture view of what's going with computers.
  
  For basic logic skills, the first two chapters of How to Prove It are great. Being able to think about conditional expressions symbolically (and not get confused by your own code) is a useful skill. Sometimes business requirements change and require you to modify conditional statements. With an understanding of Boolean Algebra, you will make fewer mistakes and get past this common hurdle sooner. Lots of beginners struggle with logic early on while also learning a language, framework, and whatever else. Luckily, Boolean Algebra is a tiny topic. Those first two chapters pretty much cover the core concepts of logic that I saw over and over again in various courses in college (programming courses, algorithms, digital circuits, etc.)
  
  Once you figure out a domain/industry you're interested in, I highly recommend focusing on one general purpose programming language that is popular in that domain. Learn about data structures and learn how to use the language to solve problems using data structures. Try not to spread yourself too thin with learning languages. It's more important to focus on learning how to get the computer to do your bidding via one set of tools - later on, once you have that context, you can experiment with other things. It's not a bad idea to learn multiple languages, since in some cases they push drastically different philosophies and practices, but give it time and stay focused early on.
  
  As you gain confidence there, identify a simple project you can take on that uses that general purpose language, and perhaps a development framework that is popular in your target industry. Read up on best practices, and stick to a small set of features that helps you complete your mini project.
  
  When learning, try to avoid haplessly jumping from tutorial to tutorial if it means that it's an opportunity to better understand something you really should understand from the ground up. Don't try to understand everything under the sun from the ground up, but don't shy away from 1st party sources of information when you need them. E.g. for iOS development, Apple has a lot of development guides that aren't too terrible. Sometimes these guides will clue you into patterns, best practices, pitfalls.
  
  Imperfect solutions are fine while learning via small projects. Focus on completing tiny projects that are just barely outside your skill level. It can be hard to gauge this yourself, but if you ever went to college then you probably have an idea of what this means.
  
  The feedback cycle in software development is long, so you want to be unafraid to make mistakes, and prioritize finishing stuff so that you can reflect on what to improve.

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

Sounds like you are about 4 years behind me (Future physics PhD candidate). Glad to know you have discovered Dover books, they really are great and so cheap. It also sounds like you know what you're doing so good job, keep at it and you might make a good case for graduate school (if that's your destination). But I will warn you that upper division mathematics courses are different. I have seen so many people who think they are really great at mathematics up to vector calculus and then get completely shit on by more abstract courses like real analysis, abstract algebra and topology. The reason for this is that it requires more formalism and is very rigorous as far as proofs go. You'll eventually learn that math is all about making sure you have checked every possible condition in order to move on. I think something you will need is mathematical logic before you tackle abstract courses. If you do collect textbooks (like I do) then I would also recommend this textbook. It teaches you how to think like a mathematician and the logic behind proofs. I think a mathematics logic course is essential to students and it's a shame many mathematics students don't go through a formal logic course before they tackle advanced courses. Of course, some don't need it but unless you are brilliant, I would recommend it (Even if you are brilliant it would be a easy read). Just dig deep and focus and good luck with your future work. Mathematics and Physics are two beautiful subjects and it's always great to talk to future mathematicians or physicists(or any aspiring scientist in that case!) and help them get inspired or motivated!

P.S. Funny story, I had a friend who thought it would be funny to make people believe that Euler is pronounce "you-ler" with the argument that Euclid is pronounced "you-clid". It was pretty funny seeing people believe him.

Learning proofs can mean different things in different contexts. First, a few questions:

1. What's your current academic level? (Assuming, of course, you're still a student, rather than trying to learn mathematical proofs as an autodidact.)
   
   The sort of recommendations for a pre-university student are likely to be very different from those for a university student. For example, high school students have a number of mathematics competitions that you could consider (at least in The United States; the structure of opportunities is likely different in other countries). At the university level, you might want to look for something like a weekly problem solving seminar. These often have as their nominal goal preparing for the Putnam, which can often feel like a VERY ambitious way to learn proofs, akin to learning to swim by being thrown into a lake.
   
   As a general rule, I'd say that working on proof-based contest questions that are just beyond the scope of what you think you can solve is probably a good initial source of problems. You don't want something so difficult that it's simply discouraging. Further, contest questions typically have solutions available, either in printed books or available somewhere online.
   
   
2. What's your current mathematical background?
   
   This may be especially true for things like logic and very elementary set theory.
   
   
3. What sort of access do you have to "formal" mathematical resources like textbooks, online materials, etc.?
   
   Some recommendations will make a lot more sense if, for example, you have access to a quality university-level library, since you won't have to spend lots of money out-of-pocket to get copies of certain textbooks. (I'm limiting my recommendations to legally-obtained copies of textbooks and such.)
   
   
4. What resources are available to you for vetting your work?
   
   Imagine trying to learn a foreign language without being able to practice it with a fluent speaker, and without being able to get any feedback on how to improve things. You may well be able to learn how to do proofs on your own, but it's orders of magnitude more effective when you have someone who can guide you.
   
   
5. Are you trying to learn the basics of mathematical proofs, or genuinely rigorous mathematical proofs?
   
   Put differently, is your current goal to be able to produce a proof that will satisfy yourself, or to produce a proof that will satisfy someone else?
   
   
6. What experience have you already had with proofs in particular?
   
   Have you had at least, for example, a geometry class that's proof-based?
   
   
7. How would you characterize your general writing ability?
   
   Proofs are all about communicating ideas. If you struggle with writing in complete, grammatically-correct sentences, then that will definitely be a bottleneck to your ability to make progress.
   
   ---
   
   With those caveats out of the way, let me make a few suggestions given what I think I can infer about where you in particular are right now.
   
   

• The book How to Prove It: A Structured Approach by Daniel Velleman is a well-respected general introduction to ideas behind mathematical proof, as is How to Solve It: A New Aspect of Mathematical Method by George Pólya.
  
  
• Since you've already taken calculus, it would be worth reviewing the topic using a more abstract, proof-centric text like Calculus by Michael Spivak. This is a challenging textbook, but there's a reason people have been recommending its different editions over many decades.
  
  
• In order to learn how to write mathematically sound proofs, it helps to read as many as you can find (at a level appropriate for your background and such). You can find plenty of examples in certain textbooks and other resources, and being able to work from templates of "good" proofs will help you immeasurably.
  
  
• It's like the old joke about how to get to Carnegie Hall: practice, practice, practice.
  
  Learning proofs is in many ways a skill that requires cultivation. Accordingly, you'll need to be patient and persistent, because proof-writing isn't a skill one typically can acquire passively.
  
  ---
  
  How to improve at proofs is a big question beyond the scope of what I can answer in a single reddit comment. Nonetheless, I hope this helps point you in some useful directions. Good luck!

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

On the CS side:

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

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

Going on the typical

Arithmetic > Algebra > Calculus

****

Arithmetic

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


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

*****

Algebra

Algebra by Chrystal Part I

Algebra by Chrystal Part II

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

chrystal algebra part I filetype:pdf

chrystal algebra part II filetype:pdf

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

Algebra by Gelfland

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

Calculus

Calculus made easy - Thompson

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

Pauls Online Notes (Calculus)

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

Spivak - Calculus

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

**

Geometry

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

****

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

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

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

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

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


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

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

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

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

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

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

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

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

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

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

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

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

A sample of what is on my reference shelf includes:

Real and Complex Analysis by Rudin

Functional Analysis by Rudin

A Book of Abstract Algebra by Pinter

General Topology by Willard

Machine Learning: A Probabilistic Perspective by Murphy

Bayesian Data Analysis Gelman

Probabilistic Graphical Models by Koller

Convex Optimization by Boyd

Combinatorial Optimization by Papadimitriou

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

The Elements of Statistical Learning by Hastie, et al.

Statistical Decision Theory by Liese, et al.

Statistical Decision Theory and Bayesian Analysis by Berger

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

Tensor Calculus by Synge

Anyway, hope that helps.

Yet another lonely data scientist,

Tim.

I study topology and I can give you some tips based on what I've done. If you want extra info please PM me. I'd love to help someone discover the beautiful field of topology. TLDR at bottom.

If you want to study topology or knot theory in the long term (actually knot theory is a pretty complicated application of topology), it would be a great idea to start reading higher math ASAP. Higher math generally refers to anything proof-based, which is pretty much everything you study in college. It's not that much harder than high school math and it's indescribably beneficial to try and get into it as soon as you possibly can. Essentially, your math education really begins when you start getting into higher math.

If you don't know how to do proofs yet, read How to Prove It. This is the best intro to higher math, and is not hard. Absolutely essential going forward. Ask for it for the holidays.

Once you know how to prove things, read 1 or 2 "intro to topology" books (there are hundreds). I read this one and it was pretty good, but most are pretty much the same. They'll go over definitions and basic theorems that give you a rough idea of how topological spaces (what topologists study) work.

After reading an intro book, move on to this book by Sutherland. It is relatively simple and doesn't require a whole lot of knowledge, but it is definitely rigorous and is definitely necessary before moving on.

After that, there are kind of two camps you could subscribe to. Currently there are two "main" topology books, referred to by their author's names: Hatcher and Munkres. Both are available online for free, but the Munkres pdf isn't legally authorized to be. Reading either of these will make you a topology god. Hatcher is all what's called algebraic topology (relating topology and abstract algebra), which is super necessary for further studies. However, Hatcher is hella hard and you can't read it unless you've really paid attention up to this point. Munkres isn't necessarily "easier" but it moves a lot slower. The first half of it is essentially a recap of Sutherland but much more in-depth. The second half is like Hatcher but less in-depth. Both books are outstanding and it all depends on your skill in specific areas of topology.

Once you've read Hatcher or Munkres, you shouldn't have much trouble going forward into any more specified subfield of topology (be it knot theory or whatever).

If you actually do end up studying topology, please save my username as a resource for when you feel stuck. It really helps to have someone advanced in the subject to talk about tough topics. Good luck going forward. My biggest advice whatsoever, regardless of what you study, is read How to Prove It ASAP!!!

TLDR: How to Prove It (!!!) -> Mendelson -> Sutherland -> Hatcher or Munkres

I double majored in math and CS as an undergrad and I enjoyed math more than CS. I'm a graduate student right now planning on doing research in a mathy area of CS. Everything I write below comes from that perspective.

• In my experience Wikipedia has some pretty good math articles. Many articles do a decent job of explaining the intuition behind of various concepts, not just the formalism.
  
  
• Math.StackExchange.com is similar to stackoverflow and I've found it to be quite helpful on occasion. Example of a question with some great answers
  
  
• /r/math is pretty active and has a very knowledgeable user base.
  
  
• One of the best known living mathematicians is Terrence Tao. He has a math blog but you might not have the background necessary to understand much of the material; I would guess that you need knowledge covering at least the standard undergraduate math major coursework to understand many of the posts.
  
  But if you're interested in really digging in and understanding some math at an advanced undergraduate level (analysis, abstract algebra, topology, etc.) then I don't think there is any substitute for books.
  
  
• A personal favorite is The Princeton Companion to Math. It has expository articles that provide high level overviews of different branches of math, important theorems, biographies of mathematicians, articles about the historical development of math, and more. It has some top notch contributors and was designed to be approachable by anyone with a good knowledge of calculus. This would be a great place to get a sense of the areas of study in math. I bought this book right after it came out after graduating high school and have loved it ever since. Everyone with a love of math should own this book.
  
  
• How to Prove It does a great job of introducing proofs and set theory which are both fundamental to higher math.
  
  
• Dover is a well loved publisher among math folks because they offer extremely cheap books on math that are of fairly high quality if a little old. You can find textbooks on any topic in the undergraduate math curriculum for less than $20 from Dover.
  

> 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.

It's less math intensive in the sense that you won't be solving calculus problems very often (or at all), but there are classes where a (basic) understanding of calculus will be helpful. For instance, I just completed algorithms and was pretty glad that I had taken Calculus. Knowing a lot about limits and knowing L'Hopital's rule made parts of asymptotic analysis a lot more intuitive than it otherwise would have been.

With that said, discrete math (which you'll cover in CS 225) is a pretty big part of the program and computer science as a whole. You'll serve yourself well by getting a solid understanding of discrete math--even in classes where it's not an explicit requirement.

To give an example, in CS 344 (operating systems), there was an assignment where we had to build a pretty simple dungeon-crawler game where a player moved through a series of rooms. Each time the player played the game, there needed to be a new random dungeon, and the connections between rooms needed to be two-way. Calculus isn't really going to help you solve this problem, but if you're good with discrete math, you'll quickly realize that this sort of problem can easily be solved with a graph. Further, you can represent the graph as a 2D array, and at that point the implementation becomes pretty easy.

So, there is math in the program, but not the type that you've probably been doing throughout your academic career. Discrete math comes naturally to some, and it's really difficult for others. I'd recommend picking up this book (which is used in the program) whenever you get a chance:

https://www.amazon.com/Discrete-Mathematics-Applications-Susanna-Epp/dp/0495391328

I'm almost done with the program, but I've been returning to that a lot to review concepts we covered in class and to learn new stuff that we didn't have time for in the term. It's a great book.

Good luck!

http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?ie=UTF8&qid=1342068971&sr=8-1&keywords=spivak%27s+calculus

This book starts with basic properties of numbers (associativity, commutativity, etc), then moves onto some proof concepts followed by a very good foundation (functions, vectors, polar coordinate). Be forewarned that the content is VERY challenging in this book, and will definitely require a determined effort, but it will certainly be good if you can get through it.

A more gentle introduction to Calculus is http://www.amazon.com/Thomas-Calculus-12th-George-B/dp/0321587995/ref=sr_1_1?s=books&ie=UTF8&qid=1342069166&sr=1-1&keywords=thomas%27+calculus and it is a much easier book, but you don't prove much in this one. Both of these can likely be found online for free. Also, if you want to get a decent understanding I recommend, http://www.amazon.com/How-Prove-Structured-Daniel-Velleman/dp/0521675995/ref=sr_1_1?s=books&ie=UTF8&qid=1342069253&sr=1-1&keywords=how+to+prove+it or http://www.people.vcu.edu/~rhammack/BookOfProof/index.html the latter is definitely free.

You may also need a more introductory text for trig and functions. I can't find the book my school used for precalc, hopefully someone else can offer a good recommendation.

Also, getting a dummies book to read alongside was pretty helpful for me, and Paul's online notes(website) is very nice.

Depends on what you are looking for. You might not be aware that the concepts in that book are literally the foundations of math. All math is (or can be) essentially expressed in set theory, which is based on logic.

You want to improve math reasoning, you should study reasoning, which is logic. It's really not that hard. I mean, ok its hard sometimes but its not rocket science, its doable if you dedicate real time to it and go slowly.

Two other books you may be interested in instead, that teach the same kinds of things:

Introduction to Mathematical Thinking which he wrote to use in his Coursera course.

How to Prove It which is often given as the gold standard for exactly your question. I have it, it is fantastic, though I only got partway through it before starting my current class. Quite easy to follow.

Both books are very conversational -- I know the second one is and I'm pretty sure the first is as well.

What books like this do is teach you the fundamental logical reasoning and math structures used to do things like construct the real number system, define operations on the numbers, and then build up to algebra step by step. You literally start at the 1+1=2 type level and build up from there by following a few rules.

Also, I just googled "basic logic" and stumbled across this, it looks like a fantastic resource that teaches the basics without any freaky looking symbols, it uses nothing but plain-English sentences. But scanning over it, it teaches everything you get in the first chapter or two of books like those above. http://courses.umass.edu/phil110-gmh/text/c01_3-99.pdf

Honestly if I were starting out I would love that last link, it looks fantastic actually.

There are a lot of good classics on /u/thebenson's list. I want to highlight the books that are good for what you'll be learning, and give you a sense of how the sequence works. And I'll add a few.


Calculus books: Thomas' Calculus, Calculus by James Stewart (not multivariable), and this cheap easy read by Morris Kline.

Have you learned calculus in the past? It sounds like you'll need it for at least one of those courses, but either way, it will definitely help you conceptually for the others. You should really try to get solid on this before you need to use it.



Intro physics books: Fundamentals of Physics (Halliday & Resnick), Physics for Scientists and Engineers (Serway & Jewett), Physics for Scientists and Engineers (Tipler & Mosca), University Physics (Young), and Physics for Scientists and Engineers (Knight) are all good. Gee, they get really unoriginal with the names, huh?

Each of these books assumes no background in physics, but you do need to use calculus. If you're going to take a class in basic mechanics that doesn't involve any calculus, you may find it more useful to get a book at that level. The only such book that I'm familiar with is Physics: Principles with Applications by Giancoli. I know there are many others, but I can't speak for them.



Mathematical methods: Greenberg is way more than you need here. I think you would find Engineering Mathematics by Stroud & Booth more useful as a reference, since it covers a lot of the less advanced stuff that you may need a refresher on.



Sequence: it's typical to start learning physics by learning about Newtonian mechanics, with or without calculus. After that, one often goes on to thermodynamics or to electricity and magnetism. It sounds like this is roughly how your program is going to work.

If you are learning mechanics with calculus, you can expect E&M to be even heavier on the calculus and thermodynamics to be less so. More calculus is not a bad thing. People often get scared of it, but it actually makes things easier to understand.

It is very typical that you will use only one book (from the intro books above) for all of these topics. You shouldn't need to get any books on specific topics.

**

The other books on /u/thebenson's list are all great textbooks, but I think you should avoid them for now. They generally assume a healthy background in basic physics, and they may not be very relevant to the physics you'll be studying.

But I do want to give some mention to Spacetime Physics* by Taylor and Wheeler, since I don't want to imply that this is a background-heavy book. On the contrary, this is one of the most beginner-friendly physics books ever written, and it is my favorite introduction to special relativity. Special relativity is probably not something you need to learn about right now, but if you have any interest, I seriously recommend finding an old used copy of this book—it's a fun read aside from any other uses!

/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

> Are the deep mathematical answers to things usually very complex or insanely elegant and simple when you get down to it?

I would say that the deep mathematical answers to questions tend to be very complex and insanely elegant at the same time. The best questions that mathematicians ask tend to be the ones that are very hard but still within reach (in terms of solving them). The solutions to these types of questions often have beautiful answers, but they will generally require lots of theory, technical detail, and/or very clever solutions all of which can be very complex. If they didn't require something tricky, technical, or the development of new theory, they wouldn't be difficult to solve and would be uninteresting.

For any experts that happen to stumble by, my favorite example of this is the classification of semi-stable vector bundles on the complex projective plane by LePotier and Drezet. At the top of page 7 of this paper you'll see a picture representing the fractal structure that arises in this classification. Of course, this required a lot of hard math and complex technical detail to come up with this, but the answer is beautiful and elegant.

> How hard would it be for a non mathematician to go to a pro? Is there just some brain bending that cannot be handled by some? How hard are the concepts to grasp?

I would say that it's difficult to become a professional mathematician. I don't think it has anything to do with an individual's ability to think about it. The concepts are difficult, certainly, but given time and resources (someone to talk to, good books, etc) you can certainly overcome that issue. The majority of the difficulty is that there is so much math! If you're an average person, you've probably taken at most Calculus. The average mathematics PhD (i.e., someone who is just getting their mathematical career going) has probably taken two years of undergraduate mathematics courses, another two years of graduate mathematics courses, and two to three years of research level study beyond calculus to begin to be able tackle the current theory and solve the problems we are interested in today. That's a lot of knowledge to acquire, and it takes a very long time. That doesn't mean you can't start solving problems earlier, however. If you're interested in this type of thing, you might want to consider picking up this book and see if you like it.

Copying my answer from another post:


I was literally in the bottom 14th percentile in math ability when i was 12.

One day, by pure chance, i stumbled across this (free and open) book written by Carl Stitz and Jeff Zeager, of Lakeland Community College

Precalculus

It covers everything from elementary algebra (think grade 5), all the way up to concepts used in Calculus and Linear Algebra (Partial fractions and matrix algebra, respectively.) The book is extremely well organized. Every sections starts with a dozen or so pages of proofs and derivations that show you the logic of why and how the formulas you'll be using work. This book, more than any other resource (and i've tried a lot of them), helped me build my math intuition from basically nothing.


Math is really, really intimidating when you've spent your whole life sucking at it. This book addresses that very well. The proofs are all really well explained, and are very long. You'll basically never go from one step to the next and be completely confused as to how they got there.


Also, there is a metric shitload of exercises, ranging from trivial, to pretty difficult, to "it will literally take your entire class working together to solve this". Many of the questions follow sort of an "arc" through the chapters, where you revisit a previous problem in a new context, and solve it with different means (Also, Sasquatches. You'll understand when you read it.)


I spent 8 months reading this book an hour a day when i got home from work, and by the end of it i was ready for college. I'm now in my second year of computer science and holding my own (although it's hard as fuck) against Calculus II. I credit Stitz and Zeager entirely. Without this book, i would never have made it to college.


Edit: other resources

Khan Academy is good, and it definitely complements Stitz/Zeager, but Khan also lacks depth. Like, a lot of depth. Khan Academy is best used for the practice problems and the videos do a good job of walking you through application of math, but it doesn't teach you enough to really build off of it. I know this from experience, as i completed all of Khan's precalculus content. Trust me, Rely on the Stitz book, and use Khan to fill in the gaps.


Paul's Online Math Notes

This website is so good it's ridiculous. It has a ton of depth, and amazing reference sheets. Use this for when you need that little extra detail to understand a concept. It's still saving my ass even today (Damned integral trig substitutions...)

Stuff that's more important than you think (if you're interested in higher math after your GED)

Trigonometric functions: very basic in Algebra, but you gotta know the common values of all 6 trig functions, their domains and ranges, and all of their identities for calculus. This one bit me in the ass.

Matrix algebra: Linear algebra is p. cool. It's used extensively in computer science, particularly in graphics programming. It's relatively "easy", but there's more conceptual stuff to understand.


Edit 2: Electric Boogaloo

Other good, cheap math textbooks

/u/ismann has pointed out to me that Dover Publications has a metric shitload of good, cheap texts (~$25CAD on Amazon, as low as a few bucks USD from what i hear).

Search up Dover Mathematics on Amazon for a deluge of good, cheap math textbooks. Many are quite old, but i'm sure most will agree that math is a fairly mature discipline, so it's not like it makes a huge difference at the intro level. Here is a Math Overflow Exchange list of the creme de la creme of Dover math texts, all of which can be had for under $30, often much less. I just bought ~1,000 pages of Linear Algebra, Graph Theory, and Discrete Math text for $50. If you prefer paper to .pdf, this is probably a good route to go.

Also, How to Prove it is a very highly rated (and easy to read!) introduction to mathematical proofs. It introduces the basic logical constructs that mathematicians use to write rigorous proofs. It's very approachable, fairly short, and ~$30 new.

I've always enjoyed all types of math but all throughout (engineering) undergrad and grad school all I ever got to do was computational-based math, i.e. solving problems. This was enjoyable but it wasn't until I learned how to read and write proofs (by self-studying How to Prove It) that I really fell in love with it. Proofs are much more interesting because each one is like a logic puzzle, which I have always greatly enjoyed. I also love the duality of intuition and rigorous reasoning, both of which are often necessary to create a solid proof. Right now I'm going back and self-studying Control Theory (need it for my EE PhD candidacy but never took it because I was a CEG undergrad) and working those problems is just so mechanical and uninteresting relative to the real analysis I study for fun.

EDIT: I also love how math is like a giant logical structure resting on a small number of axioms and you can study various parts of it at various levels. I liken it to how a computer works, which levels with each higher level resting on those below it. There's the transistor level (loosely analogous to the axioms), the logic gate level, (loosely analogous set theory), and finally the high level programming language level (loosely analogous to pretty much everything else in math like analysis or algebra).

aha, first congratulations if you've gotten in, if not good luck.

Econ grad student here.

I'm going to make suggestions more specific to the course of study given there. I'm going to assume you're going into undergrad and from the UK, and I won't say anything about PolySci.

You can start with an introduction to logic like this. I unfortunately don't remember the name of my textbook, but I think they were similar, save money by emailing the professor who teaches that course and asking him what book he uses. If you want to take the 'classics' approach, maybe Aristotle's Organon. Definitely start with the logic because it's the one thing that will help you in any of the other classes.

I don't see any reason not to read Descartes' Discourse on Method, Plato’s Republic; or Aristotle’s Ethics from your second year core courses. Kant I remember as being kind of dense and hard to read, but I wasn't reading it for a purpose so I gave up easily.

As far as economics, sooner or later you will want to get a jumpstart on the mathematical methods. Hoy's Mathematics for Economists is my favorite. It's very readable and has answers in the back. Kahnacademy will help with any parts you don't remember, or need additional work to ease into. How to Prove it may come in handy, if not it's just really neat to see how mathematics evolves out of logical philosophy.

For your micro and macro series. If you want to go with a classical approach, Adam Smith's Wealth of Nations is very readable (skim over the parts about corn sales). Nudge is an interesting popular economics book. Again, you can email the professor's who teach the first year Micro and Macro and ask them what textbooks they use. (this will also allow you to buy the textbooks used online, rather than paying through the nose for them at the bookstore).

Sorry for the wall!

Edit 1: Kahnacademy also has several sections of videos on economics, and I think you would find the documentary series The Ascent of Money interesting.

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

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

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

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

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

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

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

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

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

Thanks for the answer!

Glad to hear about Spivak! I've heard good things about that textbook and am looking forward to going through it soon :). Are the course notes for advanced algebra available online? If so, could you link them?

Is SICP used only in the advanced CS course or the general stream one, too? (last year I actually worked my way through the first two chapters before getting distracted by something else - loved it though!) Also, am I correct in thinking that the two first year CS courses cover functional programming/abstraction/recursion in the first term and then data structures/algorithms in the second?

That's awesome to know about 3rd year math courses! I was under the impression that prerequisites were enforced very strongly at Waterloo, guess I was wrong :).

As for graduate studies in pure math, that's the plan, but I in no way have my heart set on anything. I've had a little exposure to graph theory and I loved it, I'm sure that with even more exposure I'd find it even more interesting. Right now I think the reason I'm leaning towards pure math is 'cause the book I'm going through deals with mathematical logic / set theory and I think it's really fascinating, but I realize that I've got 4/5 years before I will even start grad school so I'm not worrying about it too much!

Anyways, thanks a lot for your answer! I feel like I'm leaning a lot towards Waterloo now :)

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.

This is just my perspective, but . . .

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

​

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

​

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

​

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

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

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

​

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

​

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

​

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

​

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

​

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

What this expressions says

First of all let's specify that the domain over which these statements operate is the set of all people say.
Let us give the two place predicate P(x,y) a concrete meaning. Let us say that P(x,y) signifies the relation x loves y.

This allows us to translate the statement:
∀x∀yP(x,y) -> ∀xP(x,x)

What does ∀x∀yP(x,y) mean?

This is saying that For all x, it is the case that For all y, x loves y.
So you can interpret it as saying something like everyone loves everyone.

What does ∀xP(x,x) mean?

This is saying that For all x it is the case that x loves x. So you can interpret this as saying something like everyone loves themselves.

So the statement is basically saying:
Given that it is the case that Everyone loves Everyone, this implies that everyone loves themselves.
This translation gives us the impression that the statement is true. But how to prove it?

Proof by contradiction

We can prove this statement with a technique called proof by contradiction. That is, let us assume that the conclusion is false, and show that this leads to a contradiction, which implies that the conclusion must be true.

So let's assume:
∀x∀yP(x,y) -> not ∀xP(x,x)

not ∀xP(x,x) is equivalent to ∃x not P(x,x).
In words this means It is not the case that For all x P(x,x) is true, is equivalent to saying there exists x such P(x,x) is false.

So let's instantiate this expression with something from the domain, let's call it a. Basically let's pick a person for whom we are saying a loves a is false.

not P(a,a)

Using the fact that ∀x∀yP(x,y) we can show a contradiction exists.

Let's instantiate the expression with the object a we have used previously (as a For all statement applies to all objects by definition) ∀x∀yP(x,y)

This happens in two stages:

First we instantiate y
∀xP(x,a)

Then we instantiate x
P(a,a)

The statements P(a,a) and not P(a,a) are contradictory, therefore we have shown that the statement:

∀x∀yP(x,y) -> not ∀xP(x,x) leads to a contradiction, which implies that
∀x∀yP(x,y) -> ∀xP(x,x) is true.

Hopefully that makes sense.

Recommended Resources

Wilfred Hodges - Logic

Peter Smith - An Introduction to Formal Logic

Chiswell and Hodges - Mathematical Logic

Velleman - How to Prove It

Solow - How to Read and Do Proofs

Chartand, Polimeni and Zhang - Mathematical Proofs: A Transition to Advanced Mathematics

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

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

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

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

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

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

You should absolutely not give up.

• Axler is fairly advanced for a freshman course in linear algebra. The fact that it's making more sense the second time you go over it is much more important than failing to understand it the first time.
  
• Nobody can learn sophisticated math from a lecture if they haven't seen it before. Well, maybe geniuses can, but my guess is that the majority of successful mathematicians reach a point where the lecture medium becomes much less important. You have to read the textbook with a pencil in hand, proving lemmas yourself. Digest proofs at your own pace, there's nothing wrong or unusual with not understanding it the way your Professor presented it.
  
• About talking math with people - this just takes time. Hold off on judging yourself. You can also get practice by getting involved with math subreddits or math.stackexchange.
  
• It's pretty unlikely that you are "too stupid" to study math. I've seen people with a variety of natural ability learn a tremendous amount about math and related disciplines, just by working hard.
  
  None of this is groundbreaking, and a lot of it is pretty cliché, but it's true. Everyone struggles with math at some point. Einstein said something like "whatever your struggles with math are, I assure you that mine are greater."
  
  As for specific recommendations, make the most of this summer. The most important factor in learning math in my experience is "time spent actively doing math." My favorite math quote is "you don't learn math, you get used to it." I might recommend a book like How to Prove It. I read it the summer before I entered college, and it helped immensely with proofs in real analysis and abstract algebra. Give that a read, and I bet you will be able to prove most lemmas in undergraduate algebra and topology books, and solve many of their problems. Just keep at it!

You do realize that there is guesswork but the extremes of the confidence interval are strictly positive right? In other words, no one is certain but what we are certain about is that optimum homework amount is positive. Maybe it's 4 hours, maybe it's 50 hours. But it's definitely not 0.

I don't like homework either when I was young. I dreaded it, and I skipped so many assignments, and I regularly skipped school. I hated school. In my senior year I had such severe senioritis that after I got accepted my grades basically crashed to D-ish levels. (By the way this isn't a good thing. It makes you lazy and trying to jumpstart again in your undergrad freshman year will feel like a huge, huge chore)

Now that I'm older I clearly see the benefits of homework. My advice to you is not to agree with me that homework is useful. My advice is to pursue your dreams, but when doing so be keenly aware of the pragmatical considerations. Theoretical physics demands a high level of understanding of theoretical mathematics: Lie groups, manifolds and differential algebraic topology, grad-level analysis, and so on. So get your arse and start studying math; you don't have to like your math homework, but you'd better be reading Spivak if you're truly serious about becoming a theoretical physicist. It's not easy. Life isn't easy. You want to be a theoretical physicist? Guess what, top PhD graduate programs often have acceptance rates lower than Harvard, Yale, Stanford etc. You want to stand out? Well everyone wants to stand out. But for every 100 wannabe 15-year-old theoretical physicists out there, only 1 has actually started on that route, started studying first year theoretical mathematics (analysis, vector space), started reading research papers, started really knowing what it takes. Do you want to be that 1? If you don't want to do homework, fine; but you need to be doing work that allows you to reach your dreams.

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

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

Everyone has to take MATH 150--MATH 152's prerequisite isn't Calculus 12. So after 150, you're at the same level as everyone else.

A tip: make sure when studying, you understand every part of what's being taught. You won't be able to just memorize this stuff. If you don't get something, spend a bit of time trying to figure it out, move forward if the following information doesn't rely on what you're passing, but come back to it later and try again and again till you understand what that thing is, how it works, and why. YouTube the name of what you're having trouble with, cause there are going to be several tutorials from people on there per topic.


You'll have to put in the hours, though, and study smart. Remember: being a student is your job, and 3 courses is full time (equivalent to 9-5 Mon-Fri). SFU uses the "flipped classroom" where you're supposed to read the sections of the textbook before class, the lecture reinforces and clarifies the most important stuff, then you self-study till you understand it 100%.


The rule of thumb for all classes is 2-3 hours of study for every hour in lecture. That means for MATH 150 you should expect to spend 8-12 hours studying on your own outside of class.


Engineering requires 12 credits/semester, so you'd have at least 13 in the semester you take 150--That means 26-39 hours of studying on your own outside class i.e. 6 hours a day 7 days a week, 6.5 hours every day but Sat/Sun, or 8 hours a day Mon-Fri.


Here are a couple useful resources:

• http://kl1p.com/studytips a sort of summary, lots of the below info is covered, but not everything.
  
• Study less study smart overview -- this guy wrote out notes for
  
• http://www.lbcc.edu/LAR/studyskills.cfm (the videos) -- This guy is awesome. It's a full course on how to study. When I get a job after grad I'm definitely getting him a gift.
  
• http://altair.pw/pub/lib/How%20to%20Become%20a%20Straight-A%20Student.pdf
  
• http://www.cse.buffalo.edu/~rapaport/howtostudy.html -- note: has some repeated information from the first 2 links
  
• https://tomato-timer.com -- makes using the pomodoro technique a bit easier.
  
• http://khanacademy.org
  
• http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx -- apparently this guy's got really good, concise, notes. Also applies for MATH 152 and 153.
  
• How to Ace Calculus: The Streetwise Guide (recommended by the math profs, and I've seen tons of people say it helped a lot. It doesn't go in-depth, but gives you a simple overview of the concepts so you get a good idea of how and why they work)
  
  
  Also, FYI for YouTube, if you hit shift+> on YouTube, it speeds up the video 1.25x 1.5x 2.0x -- which has really saved my ass, cause I don't have patience for slow talkers. (Conversely, shift+< slows the video back down)

This will give you some solid theory on ODEs (less so on PDEs), and a bunch of great methods of solving both ODEs and PDEs. I work a lot with differential equations and this is one of my principal reference books.

This is an amazing book, but it mostly covers ODEs sadly. Both the style and the material covered are great. It might not be exactly what you're looking for, but it's a great read nonetheless.

This covers PDEs from a very basic level. It assumes no previous knowledge of PDEs, explains some of the theory, and then goes into a bunch of elementary methods of solving the equations. It's a small book and a fairly easy read. It also has a lot of examples and exercises.

This is THE book on PDEs. It assumes quite a bit of knowledge about them though, so if you're not feeling too confident, I suggest you start with the previous link. It's something great to have around either way, just for reference.

Hope this helped, and good luck with your postgrad!

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

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

How to prove it

Number theory

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

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

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

I may be in the minority here, but I think that high school students should be exposed to statistics and probability. I don't think that it would be possible to exposed them to full mathematical statistics (like the CLT, regression, multivariate etc) but they should have a basic understanding of descriptive statistics. I would emphasize things like the normal distribution, random variables, chance, averages and standard deviations. This could improve numerical literacy, and help people evaluate news reports and polls critically. It could also cut down on some issues like the gambler's fallacy, or causation vs correlation.

It would be nice if we could teach everyone mathematical statistics, the CLT, and programming in R. But for the majority of the population a basic understanding of the key concepts would be an improvement, and would be useful.

Edit At the other end of the spectrum, I would like to see more access to an elective class that covers the basics of mathematical thinking. I would target this at upperclassmen who are sincerely interested in mathematics, and feel that the standard trig-precalculus-calculus is not enough. It would be based off of a freshman math course at my university, that strives to teach the basics of proofs and mathematical thinking using examples from different fields of math, but mostly set theory and discrete math. Maybe use Velleman's book or something similar as a text.

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

To illustrate my point:

Linear Algebra:

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

Depends what kind of math you're interested in. If you're looking for an introduction to higher (college) math, then How to Prove It is probably your best bet. It generally goes over how proofs work, different ways of proving stuff, and then some.

If you already know about proofs (i.e. you are comfortable with at least direct proofs, induction, and contradiction) then the world is kind of your oyster. Almost anything you pick up is at least accessible. I don't really know what to recommend in this case since it's highly dependent on what you like.

If you don't really know the basics about proofs and don't care enough to yet, then anything by Dover is around your speed. My favorites are Excursions in Number Theory and Excursions in Geometry. Those two books use pretty simple high school math to give a relatively broad look at each of those fields (both are very interesting, but the number theory one is much easier to understand).

If you're looking for high school math, then /u/ben1996123 is probably right that /r/learnmath is best for that.

If you want more specific suggestions, tell me what you have enjoyed learning about the most and I'd be happy to oblige.

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

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

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

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

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

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

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

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

I didn't take any lower division classes here but the upper division classes are pretty great. I haven't really had any bad professors and they seem to be a lot better at teaching than the professors in my upper division physics courses were.

The quarter system isn't bad. I think it's actually a good pace and the courses that have more than 10 weeks of content are 2 or 3 quarters long, which is great because it means you're not stuck in a class that you hate for very long.

The difficulty depends entirely on the professor, but I haven't had a class that was super difficult and uncurved. Curves always seem fair for the difficulty of the class. Finals are usually fair but midterms really suck because they're only 50 minutes long. You will probably do horrible on a few of them before you figure out a way to make it work. We have a much cushier path to upper division than most schools. Instead of being dumped into linear algebra or real analysis and having to learning how to do proofs, we have an intro to proofs and logic prerequisite and another class where you essentially just practice proof techniques that you will use in analysis later. I loved it because it let me focus on the material in my more challenging classes without having to figure out the mechanics and techniques of general proof writing.

One thing to keep in mind is that upper division math is nothing whatsoever like the math that you're probably used to. You essentially start over and learn things correctly, and you usually have to pretend that you don't know anything that you've learned over the past 14 years of math classes outside of basic arithmetic and algebra. You will be writing paragraphs in plain English with occasional math symbols. It's all about taking definitions and theorems that you know and using them to argue that other theorems are true. It's a lot more fun than it sounds. If you want to get a feel for what it's going to be like, check out this book:

http://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995/

It's easy to find elsewhere. You don't need to know anything to get started and it's actually really fun to work through. This was the textbook for my intro to proofs class.

The foundation of probability is measure theory, not nonstandard analysis (the topic that includes the hyperreals). So, when dealing with statements about probability, we deal with probability measures, which assign numbers in the real interval [0, 1] to subsets of the space of possible events. (Perhaps someone has studied a variant of measure theory that substitutes the hyperreals for the reals, but if so, it's sufficiently obscure that I've never heard of it.)

Also, nowhere in all this is anyone "raising a real number to the power of infinity". There are formal statements of the following sort:

>(*) The limit of 1/2^N as N approaches infinity is zero.

However, this is a statement about the limit of a sequence of real numbers, which is most definitely a real number, and is also formally defined in a way that makes no reference to "infinity". The expression "as N approaches infinity" is just a mildly informal (but much more readable) way of expressing that formal definition.

If you care to parse the formal statement, here it is:

>For any real number ε > 0, there exists a natural number N such that for all natural numbers n ≥ N, we have |1/2^(N) - 0| < ε.

This is how we precisely formalize statement (*).

For more information on limits of sequences, I recommend reading a book on mathematical analysis. Spivak's Calculus has a good chapter on this; it's an excellent book, so it's worth reading anyway.

The only previous knowledge I really used when I took intro to proofs were some factoring methods that were helpful with proofs by induction, although they weren't necessary. That said, reviewing exponent/log laws, and certain methods of factoring couldn't hurt.

An intro to proofs course should be fairly self contained, meaning any necessary axioms and definitions should be covered in the course. Those examples that you gave are exactly the type of things that should be proven and not knowing them beforehand should be fine. The important thing is being able to understand and reproduce the proofs on your own, and with a bit of experience you will be able to intuitively reason whether a statement is true or false. This intuitive reasoning will also become much more important than memorizing later in the course when you come across statements you've never seen before that aren't immediately obvious.

I would recommend getting very comfortable with logic and basic set theory. I also highly recommend this book if you want some extra reading material (pdf). It's still one of my favorite math books. Hope that helps.

Well the good news is that we have more resources available now than even 5 years ago. :) I'm in calc 1 right now, and was having trouble putting the pieces together into a whole that made sense. A few of my resources are classroom specific but many would be great for anyone not currently in a class.

Free:
www.khanacademy.org

free video lectures and practice problems on all manner of topics, starting with elementary algebra. You can start at the beginning and work your way through, or just start wherever.

http://ocw.mit.edu/index.htm

free online courses and lessons from MIT (!!) where you can watch lectures on a subject, do practice problems, etc. Use just for review or treat it like a course, it's up to you.

Cheap $$

http://www.amazon.com/How-Ace-Calculus-Streetwise-Guide/dp/0716731606/ref=sr_1_1?ie=UTF8&qid=1331675661&sr=8-1

$10ish shipped for a book that translates calculus from math-professor to plain english, and is funny too.

http://www.amazon.com/Calculus-Lifesaver-Tools-Excel-Princeton/dp/0691130884/ref=pd_cp_b_1

$15 for a book that is 2-3x as thick as the previous one, a bit drier, but still very readable. And it covers Calc 1-3.

So here are some options I recommend:

• (Advanced) Go through a few chapters of Spivak's Calculus. This is the MAT157 textbook and will over prepare you for the course and you will probably do very well. This will require a lot of self motivation, but I think is worth it (I went through a bit of Spivak's after 137). Keep in mind that this material is more rigorous than what you will see in MAT137
  
  
• (Computer Science) If you're a CS student, grab How to Prove It. You will be dealing with a lot of proofs in MAT137, CSC165, 236/240, etc. This is a more broad approach and is not directly calculus, though what you learn will help for 137. Also, get familiar with epsilon-delta proofs.
  
  
• (At your own pace: videos) Khan Academy tries to build an intuitive knowledge of calculus, which is something that MAT137 also tries to do. The videos are well done and you get points and achievements for watching them (gamification is great), you can watch the videos in your free time and it's fun(?).
  
  
• (At your own pace: reading) One of the (previous?) instructors for MAT137 has some really good lecture notes, which you can read/download here. This is essentially the exact content of the course, if you go through it, you will do well. Try to read at least up to page 50 (the end of limits chapter), and do the exercises.
  
  You can find all the textbooks I mentioned online, if you know what I mean. All of these assume you haven't seen math in a while, and they all start from the very basics. Take your time with the material, play around with it a bit, and enjoy your summer :D
  
  EditL this article describes one way you can go about your studies

Here's my list of the classics:

General Computing

• But How Do It Know? - The Basic Principles of Computers for Everyone
  
  
• The Elements of Computing Systems: Building a Modern Computer from First Principles
  
  
• Code: The Hidden Language of Computer Hardware and Software
  
  
• Gödel, Escher, Bach: An Eternal Golden Braid
  
  Computer Science
  
  
• The New Turing Omnibus: Sixty-Six Excursions in Computer Science
  
  
• Compilers: Principles, Techniques, and Tools, 2nd Edition (aka The Dragon Book)
  
  
• The Art of Computer Programming
  
  
• Algorithms, 4th Edition
  
  
• Introduction to Algorithms, 3rd Edition (aka CLRS)
  
  
• Essential Algorithms: A Practical Approach to Computer Algorithms Using Python and C#, 2nd Edition
  
  
• Grokking Algorithms: An Illustrated Guide for Programmers and Other Curious People
  
  
• Structure and Interpretation of Computer Programs, 2nd Edition (aka SICP, available for free on the MIT website)
  
  
• Discrete Mathematics with Applications, 4th Edition
  
  Software Development
  
  
• Designing Data-Intensive Applications: The Big Ideas Behind Reliable, Scalable, and Maintainable Systems by Martin Kleppmann
  
  
• Design Patterns: Elements of Reusable Object-Oriented Software (aka The Gang of Four)
  
  
• Think Like a Programmer: An Introduction to Creative Problem Solving
  
  
• The Pragmatic Programmer: your journey to mastery, 20th Anniversary Edition, 2nd Edition
  
  
• The Mythical Man-Month: Essays on Software Engineering
  
  
• Code Complete: A Practical Handbook of Software Construction, 2nd Edition
  
  
• The Design of Everyday Things
  
  
• Clean Code: A Handbook of Agile Software Craftsmanship
  
  
• Programming Pearls, 2nd Edition
  
  
• Soft Skills: The Software Developer's Life Manual
  
  
• Hello, Startup: A Programmer's Guide to Building Products, Technologies, and Teams
  
  
• Hacker's Delight, 2nd Edition
  
  
• Peopleware: Productive Projects and Teams, 3rd Edition
  
  Case Studies
  
  
• The Soul of a New Machine
  
  
• Masters of Doom: How Two Guys Created an Empire and Transformed Pop Culture
  
  
• Hackers: Heroes of the Computer Revolution
  
  
• Where Wizards Stay Up Late: The Origins Of The Internet
  
  
• Fire in the Valley: The Birth and Death of the Personal Computer, 3rd Edition
  
  Employment
  
  
• Cracking the Coding Interview: 189 Programming Questions and Solutions, 6th Edition
  
  
• The Complete Software Developer's Career Guide: How to Learn Programming Languages Quickly, Ace Your Programming Interview, and Land Your Software Developer Dream Job
  
  
• The Self-Taught Programmer: The Definitive Guide to Programming Professionally
  
  
• The Passionate Programmer: Creating a Remarkable Career in Software Development
  
  Language-Specific
  
  C
  
  
• The C Programming Language, 2nd Edition
  
  Python
  
  
• Automate the Boring Stuff with Python: Practical Programming for Total Beginners, 2nd Edition
  
  
• Python Crash Course: A Hands-On, Project-Based Introduction to Programming
  
  
• Python Programming: An Introduction to Computer Science, 3rd Edition
  
  
• Effective Python: 59 Specific Ways to Write Better Python
  
  C#
  
  
• The C# Player's Guide, 3rd Edition
  
  C++
  
  
• The C++ Programming Language, 4th Edition
  
  
• Starting Out with C++: from Control Structures to Objects, 9th Edition
  
  Java
  
  
• Introduction to Java Programming and Data Structures: Comprehensive Version, 11th Edition
  
  
• [Java: A Beginner's Guide](https://www.amazon.com/Java-Beginners-Seventh-Herbert-Schildt/dp/1259589315/, 7th Edition)
  
  
• Java: The Complete Reference, 11th Edition
  
  
• Core Java Volume I - Fundamentals, 11th Edition
  
  
• Starting Out with Java: From Control Structures through Objects, 6th Edition
  
  
• Effective Java, 3rd Edition
  
  Linux Shell Scripts
  
  
• The Linux Command Line: A Complete Introduction, 2nd Edition
  
  
• Wicked Cool Shell Scripts: 101 Scripts for Linux, OS X, and UNIX Systems, 2nd Edition
  
  
• Linux Command Line and Shell Scripting Bible
  
  
• Mastering Linux Shell Scripting
  
  Web Development
  
  
• Learning Web Design: A Beginner's Guide to HTML, CSS, JavaScript, and Web Graphics, 5th Edition
  
  
• Eloquent JavaScript: A Modern Introduction to Programming, 3rd Edition (also available for free on the author's website)
  
  
• JavaScript and JQuery: Interactive Front-End Web Development
  
  
• HTML and CSS: Design and Build Websites
  
  
• You Don't Know JS: Up & Going
  
  Ruby and Rails
  
  
• Ruby on Rails Tutorial: Learn Web Development with Rails, 4th Edition (Note: this book is somewhat out of date compared to the live version on the author's website.)
  
  
• The Well Grounded Rubyist, 3rd Edition
  
  
• Practical Object-Oriented Design in Ruby: An Agile Primer
  
  
• Agile Web Development with Rails 5.1
  
  
• Programming Ruby 1.9 & 2.0: The Pragmatic Programmers' Guide, 4th Edition
  
  
• Eloquent Ruby
  
  Assembly
  
  
• Assembly Language for x86 Processors, 8th Edition

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

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

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

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

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

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

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

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

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

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

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

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

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

But that’s what I’m working on! :)

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

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

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

For compsci you need to study tons and tons and tons of discrete math. That means you don't need much of analysis business(too continuous). Instead you want to study combinatorics, graph theory, number theory, abstract algebra and the like.

Intro to math language(several of several million existing books on the topic). You want to study several books because what's overlooked by one author will be covered by another:

Discrete Mathematics with Applications by Susanna Epp

Mathematical Proofs: A Transition to Advanced Mathematics by Gary Chartrand, Albert D. Polimeni, Ping Zhang

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

Numbers and Proofs by Allenby

Mathematics: A Discrete Introduction by Edward Scheinerman

How to Prove It: A Structured Approach by Daniel Velleman

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

Some special topics(elementary treatment):

Rings, Fields and Groups: An Introduction to Abstract Algebra by R. B. J. T. Allenby

A Friendly Introduction to Number Theory Joseph Silverman

Elements of Number Theory by John Stillwell

A Primer in Combinatorics by Kheyfits

Counting by Khee Meng Koh

Combinatorics: A Guided Tour by David Mazur


Just a nice bunch of related books great to have read:

generatingfunctionology by Herbert Wilf

The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates by by Manuel Kauers, Peter Paule

A = B by Marko Petkovsek, Herbert S Wilf, Doron Zeilberger

If you wanna do graphics stuff, you wanna do some applied Linear Algebra:

Linear Algebra by Allenby

Linear Algebra Through Geometry by Thomas Banchoff, John Wermer

Linear Algebra by Richard Bronson, Gabriel B. Costa, John T. Saccoman

Best of Luck.

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

Some user friendly books on Real Analysis:

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

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

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

A couple good books:

An Illustrated Theory of Numbers

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

A Book of Abstract Algebra

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

For whichever professor you have for Math 42, I highly recommend you get this book: https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995
It definitely saved me a ton. It’s straight to the point, and not as dry as most textbooks can be. Math 32 will be a bit more work, but in my experience just start homework early and don’t be afraid to go to professor office hours and ask questions. Even if they seem distant during class, most professors do appreciate students who make the effort to ask questions. If you need free tutoring in any of your classes, contact Peer Connections. Specifically for math, I believe MacQuarrie Hall room 221 offers drop-in tutoring for free as well! And for physics, Science building room 319 has free drop-in tutoring.

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

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

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

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

Traditionally, a mathematical proof has one and only one job: convince other people that your proof is correct. (In this day and age, there is such a thing as a computer proof, but if you don't understand traditional proofs, you can't handle computer proofs either.)

Notice what I just said: "convince other people that your proof is correct." A proof is, in some sense, always an interactive undertaking, even if the interaction takes place across gulfs of space and time.

Because interaction is so central to the notion of a proof, it is rare for students to successfully self-study how to write proofs. That seems like what you're asking. Don't get me wrong. Self-study helps. But it is not the only thing you need. You need, at some point, to go through the process of presenting your proofs to others, answering questions about your proof, adjusting your proof to take into account new feedback, and using this experience to anticipate likely issues in future proofs.

What you're proposing to do, in most cases, is the wrong strategy. You need more interactive experience, not less. You should be beating down the doors of your professor or TA in your class during their office hours, asking for feedback on your proofs. (This implies that you should be preparing your proofs in advance for them to read before going to their office hours.) If your school has a tutorial center, that's a wonderful resource as well. A math tutor who knows math proofs is a viable source of help, but if you don't know how to do proofs, it's hard for you to judge whether or not your tutor knows how to do proofs.

If you do self-study anything, you should not be self-studying calculus, linear algebra, real analysis, or abstract algebra. You should be self-studying how to do proofs. Some people here say that How to Prove It is a useful resource. My own position is that while self-studying can be helpful, it needs to be balanced with some amount of external interactive feedback in order to really stick.

Sure! There is a lot of math involved in the WHY component of Computer Science, for the basics, its Discrete Mathematics, so any introduction to that will help as well.
http://www.amazon.com/Discrete-Mathematics-Applications-Susanna-Epp/dp/0495391328/ref=sr_sp-atf_title_1_1?s=books&ie=UTF8&qid=1368125024&sr=1-1&keywords=discrete+mathematics

This next book is a great theoretical overview of CS as well.
http://mitpress.mit.edu/sicp/full-text/book/book.html

That's a great book on computer programming, complexity, data types etc... If you want to get into more detail, check out: http://www.amazon.com/Introduction-Theory-Computation-Michael-Sipser/dp/0534950973

I would also look at Coursera.org's Algorithm lectures by Robert Sedgewick, thats essential learning for any computer science student.
His textbook: http://www.amazon.com/Algorithms-4th-Robert-Sedgewick/dp/032157351X/ref=sr_sp-atf_title_1_1?s=books&ie=UTF8&qid=1368124871&sr=1-1&keywords=Algorithms

another Algorithms textbook bible: http://www.amazon.com/Introduction-Algorithms-Thomas-H-Cormen/dp/0262033844/ref=sr_sp-atf_title_1_2?s=books&ie=UTF8&qid=1368124871&sr=1-2&keywords=Algorithms




I'm just like you as well, I'm pivoting, I graduated law school specializing in technology law and patents in 2012, but I love comp sci too much, so i went back into school for Comp Sci + jumped into the tech field and got a job at a tech company.

These books are theoretical, and they help you understand why you should use x versus y, those kind of things are essential, especially on larger applications (like Google's PageRank algorithm). Once you know the theoretical info, applying it is just a matter of picking the right tool, like Ruby on Rails, or .NET, Java etc...

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.

You can start with Calculus by Spivak. If you're going to buy it then wait until after the Fall semester begins; the price is inflated right now because students need it for school.

This is a PDF of the third edition of the above book.

This is an excellent introduction to logic and proofs. You will want a strong understanding of how mathematicians communicate via proof and that book will really help.

The math subreddit is primarily undergrads talking about various topics. Make a point of just hanging out and reading stuff. If you don't understand something just tell us and they'll do their best to help out.

Hang out on the math stack exchange and ask questions about things you do not understand while trying to help with things you do understand.

Hope that helps!

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

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

Geometry and solutions

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

A First Course in Calculus

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

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

The Method of Coordinates

Functions and Graphs

Algebra

Trigonometry

Lines and Curves: A Practical Geometry Handbook

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

​

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

You could read Timothy Gowers' welcome to the math students at Oxford, which is filled with great advice and helpful links at the bottom.

You could read this collection of links on efficient study habits.

You could read this thread about what it takes to succeed at MIT (which really should apply everywhere). Tons of great discussion in the lower comments.

You could read How to Solve It and/or How to Prove It.

If you can work your way through these two books over the summer, you'll be better prepared than 90% of the incoming math majors (conservatively). They'll make your foundation rock solid.

You might want to check out Stein and Shakarchi's book Complex Analysis http://press.princeton.edu/titles/7563.html. This book is a bit hard but iirc doesn't require you to have had real analysis before hand. I would highly recommend that you work through a proof based book before hand though. Often times this will be a course book but something like https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995?ie=UTF8&*Version*=1&*entries*=0 that should also get the job done.

Or you can go the traditional route like other people mentioned of getting about a semester's worth of real analysis under your belt. The reason why this is usually the suggested path is because it's not expected that you are 100% competent at writing proofs in the beginning of real but you are in complex.

It's really smart to be playing to your strengths: if you excel at language and writing, then read a book that talks about math in more detail. Textbooks are good for problems and for reference, but I find them very hard to read. They use equations where they should be using words.

Go to your local library, and look in the math section until you find something interesting. I found this book when I was struggling with calculus: How to Ace Calculus: The Streetwise guide. It was smart, funny, and really explained topics in ways I could relate to.

That's the kind of thing I would look for if I were you. Good luck! I hope you see post in all the ~430 comments!

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!

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

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

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

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

Here are some books I'd recommend.

General Books

These are general books that are more focused on proving things per se. They'll use examples from basic set theory, geometry, and so on.

1. How to Prove It: A Structured Approach by Daniel Velleman
   
2. How to Solve It: A New Aspect of Mathematical Method by George Pólya
   
   Topical Books
   
   For learning topically, I'd suggest starting with a topic you're already familiar with or can become easily familiar with, and try to develop more rigor around it. For example, discrete math is a nice playground to learn about proving things because the topic is both deep and approachable by a beginning math student. Similarly, if you've taken AP or IB-level calculus then you'll get a lot of out a more rigorous treatment of calculus.
   
   

• An Invitation to Discrete Mathematics by Jiří Matoušek and Jaroslav Nešetřil
  
• Discrete Mathematics: Elementary and Beyond by László Lovász and Jaroslav Pelikan
  
• Proofs from THE BOOK by Martin Aigner and Günter Ziegler
  
• Calculus by Michael Spivak
  
  I have a special place in my hear for Spivak's Calculus, which I think is probably the best introduction out there to math-as-she-is-spoke. I used it for my first-year undergraduate calculus course and realized within the first week that the "math" I learned in high school — which I found tedious and rote — was not really math at all. The folks over at /r/calculusstudygroup are slowly working their way through it if you want to work alongside similarly motivated people.
  
  General Advice
  
  One way to get accustomed to "proof" is to go back to, say, your Algebra II course in high school. Let's take something I'm sure you've memorized inside and out like the quadratic formula. Can you prove it?
  
  I don't even mean derive it, necessarily. It's easy to check that the quadratic formula gives you two roots for the polynomial, but how do you know there aren't other roots? You're told that a quadratic polynomial has at most two distinct roots, a cubic polynomial has a most three, a quartic as most four, and perhaps even told that in general an n^(th) degree polynomial has at most n distinct roots.
  
  But how do you know? How do you know there's not a third root lurking out there somewhere?
  
  To answer this you'll have to develop a deeper understanding of what polynomials really are, how you can manipulate them, how different properties of polynomials are affected by those manipulations, and so on.
  
  Anyways, you can revisit pretty much any topic you want from high school and ask yourself, "But how do I really know?" That way rigor (and proofs) lie. :)

You could try Book of Proof by Richard Hammack. I've never read Velleman so I can't directly compare, but it's free for pdf (link to author's site above) and quite cheap in paperback (~$15). I found the explanations quite clear, the examples well worked and the exercises plentiful and helpful. Amazon reviewers seem to like it as well.

I graduated w/ degree in Math n' Physics but have been doing programming for startup for last 5+ years so many of my math skills got rusty.

While trying to get back into it went through several books and have found this to be the best if you're interested in more advanced mathematics: https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094. It's not only been an excellent review but has fleshed out some areas I was weak (in higher level courses like complex analysis, topology, group theory the methodology of proofs was assumed and often not taught).

The explanations are solid, varied, and they go through each proof they present (often w/ exhaustive step-by-step details).

From there pick a domain you're interested in and pickup the relevant undergraduate (and maybe some graduate) level books/textbooks and see if you can pick it up.

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

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

Here is the ooh page on Statisticians:
http://www.bls.gov/oco/ocos045.htm

A job straight out of college might see you as a research assistant. I could see you getting a job at Mathematica perhaps. Try to get a SAS certificate before you graduate, a working knowledge of R, and if you feel like tackling it a programming language good for numerical analysis.

Have you taken a course on Regression? I'd consider that, and perhaps even trying to take a Mathematical Statistics Course, if it is offered. You can try to see if you university would allow you to take a class online, or try a Semester Abroad at a university that has that class.

My background: I am an Economist that uses Statistics heavily, and works with Statistical methods often (ie: econometrics). I love it.

Your plans on studying Calc 2 and Linear Algebra are great. That is perfect.

My pay after 10 years is likely to be 100k-150k.


Before you start your first semester at the graduate level know the following things really well: Set theory, integration, matrix algebra, and proofs.

Get this book: http://www.amazon.com/How-Prove-Structured-Daniel-Velleman/dp/0521675995 -- read it before you study linear algebra, and maybe even some Calculus. It doesn't require a heavy Math background and will save you a lot of frustration later on.

>My first goal is to understand the beauty that is calculus.

There are two "types" of Calculus. The one for engineers - the plug-and-chug type and the theory of Calculus called Real Analysis. If you want to see the actual beauty of the subject you might want to settle for the latter. It's rigorous and proof-based.

There are some great intros for RA:

Numbers and Functions: Steps to Analysis by Burn

A First Course in Mathematical Analysis by Brannan

Inside Calculus by Exner

Mathematical Analysis and Proof by Stirling

Yet Another Introduction to Analysis by Bryant

Mathematical Analysis: A Straightforward Approach by Binmore

Introduction to Calculus and Classical Analysis by Hijab

Analysis I by Tao

Real Analysis: A Constructive Approach by Bridger

Understanding Analysis by Abbot.

Seriously, there are just too many more of these great intros

But you need a good foundation. You need to learn the basics of math like logic, sets, relations, proofs etc.:

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

Discrete Mathematics with Applications by Epp

Mathematics: A Discrete Introduction by Scheinerman

If it helps, here are some free books to go through:

Linear Algebra Done Wrong

Paul's Online Math Notes (fantastic for Calc 1, 2, and 3)

Basic Analysis


Basic Analysis is pretty basic, so I'd recommend going through Rudin's book afterwards, as it's generally considered to be among the best analysis books ever written. If the price tag is too high, you can get the same book much cheaper, although with crappier paper and softcover via methods of questionable legality. Also because Rudin is so popular, you can find solutions online.

If you want something better than online notes for univariate Calculus, get Spivak's Calculus, as it'll walk you through single-variable Calculus using more theory than a standard math class. If you're able to get through that and Rudin, you should be good to go once you get good at linear algebra.

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

Aside from that, try these:

Excursions In Calculus by Robert Young.

Calculus:A Liberal Art by William McGowen Priestley.

Calculus for the Ambitious by T. W. KORNER.

Calculus: Concepts and Methods by Ken Binmore and Joan Davies

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

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

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

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

Analytic Inequalities by Nicholas D. Kazarinoff.

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

My plan would go like this:

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

Linear Algebra Through Geometry by Thomas Banchoff and John Wermer.

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

2. Getting a bit more sophisticated:

Linear Algebra Done Right by Sheldon Axler.

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

Linear Algebra Done Wrong by Sergei Treil.

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

Advanced Linear Algebra by Steven Roman.

Good Luck.

If you have a chance, I recommend checking out some textbooks on real analysis, which will guide you through the derivations and proofs of many theorems in calculus that you've thus far been expected to take for granted.

Some would recommend starting with Rudin's Principles of Mathematical Analysis, and it's certainly a text that I plan to read at some point. For your purposes, I might recommend Spivak's Calculus since it expects you to rigorously derive some of the most important results in calculus through proof-writing exercises. This was my first introduction to calculus during high-school. While it was overwhelming at first, it prepared me for some of my more advanced undergraduate courses (including real analysis and topology), and it seems to be best described as an advanced calculus textbook.

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

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

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

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

Congrats! And sorry about the DNF.

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

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

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

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

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

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 am surprised no one has mentioned M. Spivak's very well known text Calculus. I thought this book was a pleasure to read. His writing was very fun and lighthearted and the book certainly teaches the material very well. In my opinion this is the best introductory calculus text there is.

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

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

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

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

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

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

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

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

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

This book seems silly, but it's honestly great for learning Calculus, especially the second time: https://www.amazon.com/How-Ace-Calculus-Streetwise-Guide/dp/0716731606

(I read it in 1999 when I went from HS -> College, and the college I went into assumed you had already passed calc, and freshmen all had to start with second year calc. The professors recommended all incoming students refresh before the start of class, and I'm glad they did, because that book retaught some things I don't think I learned correctly the first time, made a huge difference).

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

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

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

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

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

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

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

Start With "Foundations Of Analysis" By Edmund Landau

http://www.amazon.com/Foundations-Analysis-AMS-Chelsea-Publishing/dp/082182693X

It's a tiny book, but is very good at explaining basic abstract algebra.

Here is the description from Amazon:

"Why does $2 \times 2 = 4$? What are fractions? Imaginary numbers? Why do the laws of algebra hold? And how do we prove these laws? What are the properties of the numbers on which the Differential and Integral Calculus is based? In other words, What are numbers? And why do they have the properties we attribute to them? Thanks to the genius of Dedekind, Cantor, Peano, Frege and Russell, such questions can now be given a satisfactory answer. This English translation of Landau's famous Grundlagen der Analysis-also available from the AMS-answers these important questions."

With the above book you should then have enough knowledge to move on to calculus.

I recommend the two volume series called "Calculus" by Tom M. Apostol.

The first volume is single variable calculus and the second is multivariate calculus

http://www.amazon.com/Calculus-Vol-One-Variable-Introduction-Algebra/dp/0471000051/ref=sr_1_4?ie=UTF8&s=books&qid=1239384587&sr=1-4

http://www.amazon.com/Calculus-Vol-Multi-Variable-Algebra-Applications/dp/0471000078/ref=sr_1_3?ie=UTF8&s=books&qid=1239384587&sr=1-3

no problem. enjoy the journey! :)

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

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

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

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

Example,

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

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

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

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

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

Books you might like:

Discrete Mathematics with Applications by Susanna Epp

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

Building Proofs: A Practical Guide by Oliveira/Stewart

Book Of Proof by Hammack

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

How to Prove It: A Structured Approach by Velleman

The Nuts and Bolts of Proofs by Antonella Cupillary

How To Think About Analysis by Alcock

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

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

Problems and Proofs in Numbers and Algebra by Millman et al

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

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

Proof Patterns by Joshi

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

Good Luck.

For discrete math I like Discrete Mathematics with Applications by Suzanna Epp.

It's my opinion, but Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers is much better structured and more in depth than How To Prove It by Velleman. If you follow everything she says, proofs will jump out at you. It's all around great intro to proofs, sets, relations.

Also, knowing some Linear Algebra is great for Multivariate Calculus.

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

None of the questions you asked is “silly” or “simple.” There’s a whole lot going on in calculus, most of which is typically explained in a real analysis course. Rigorous proofs of things like the mean value theorem or various forms of integration are challenging, but they will provide the clarity you’re looking for.

I recommend that you check out something like Spivak’s Calculus, which is going to give a more rigorous intro to the subject. Alternately, you can just find a good analysis or intro to proofs class somewhere. It’s a fascinating subject, so good luck!

I'm not sure about PDE's, but ODE's are more than just existence and uniqueness theorems. You could argue that the modern study of ODE's is now dynamical systems.

Strogatz's Nonlinear Dynamics and Chaos is a classic if you want to know what applied dynamical systems is like. A more formal text that still captures some interesting ideas is Hale and Kocak's Dynamics and Bifurcations.

Reading textbooks is, of course, a huge time commitment. So perhaps go talk to the dynamical systems people in your department and ask them what is interesting about ODE's. Hell, even go talk to the numerical analysis and do the same for PDE's. Assuming you haven't taken a numerical analysis class, you might be surprised how "pure" numerical analysis feels.

I dont know much about boot camp, but it sounds like having a physical book will be your best bet.

Personally, my favorite text book to use is Calculus: an Intutitive Approach by Morris Kline, but you might want something more advanced than that.

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

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

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

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

The book that really helped me prepare for CS 6505 this fall was Discrete Mathematics with Applications by Susanna Epp. I found it easy to digest and it seemed to line up well with the needed knowledge to do well in the course.

Richard Hammack's Book of Proof also proved invaluable. Because so much of your success in the class relies on your ability to do proofs, strengthening those skills in advance will help.

I think everyone is on point for the most part, but I'd like to be the devil's advocate and suggest a different route.

Learn logic, proof techniques and set theory as early as possible. It will aid you in further study of all 'types' of math and broaden your mind in a general sense. This book is a perfect place to start.

http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995

The best part is, when you start doing proofs you realize you've been thinking about math all wrong (at least I did). It's an exercise in creativity, not calculation.

In my mind, set theory & calculus are necessary pre-requisites to probability anyway, and linear algebra means much more once you have been introduced to inductive proofs, as well.

Strogatz Nonlinear Dynamics and Chaos covers phase space, phase portraits, and linear stability analysis in great detail with examples from many disciplines including physics. It's probably a good place to start, but I don't think it has very much that's specifically on turbulent fluids. For that, you'll probably want a more focused textbook. Hopefully, someone more knowledgeable can recommend one.

The only "math" you don't learn in secondary education that I would say is truly important to all students of computer science is discrete mathematics. One of my old instructors told me he was trying to get permission to use Discrete Mathematics With Ducks in his curriculum. I flipped through it, it seemed like a pretty good discrete math book, but maybe only if you already had some basic understanding. The more traditional choice might be a better option, but it's much drier.

Every other field of math that has applications in computer science has a narrower breadth of use than discrete mathematics, in my opinion. Learning discrete math well will probably do you more good than learning lots of subjects poorly.

This is good advice. Source: I flunked a private engineering school at age 17, in spite of of being 99th percentile in the ACT. Reason? Besides socialization issues, poor mathematics and academic preparation at my rural high school, where few went to college, let alone out-of-state.

I'm a strong believer in self-education (and self-employment) and am currently rectifying the above-stated issues.

Came here to plug Spivak's Calculus. It's a bit harder and more detailed than most calculus texts used today, but that's because he actually explains all the tricky bits, rather than just using hand-waving to finish those tricky bits. (It was the hand-waving that always left me confused in classroom teaching.) Spivak's Calculus might not be the place to start, but it's where you want to end up, so I want you to know about it.

Peace out, bro, and keep working. We'll make it. ME/EE is a great combo, btw. ME is the first branch of engineering, though it was called something else, when "engines of war", catapaults and whatnot, was the only game in town. But, all machines need sensors, controls, and power, which is the EE bit. Put it together, and you get mechatronics, which is part of the future.

One piece of added advice: stick to one of the main-line branches of engineering: mechanical, electrical, chemical, maybe civil, instead of one of the new, hybrid branches, like biomedical, etc. The jobs are more plentiful, you'll get a sounder foundation in engineering principles, and specializing is still possible.

Ed: Do you already know about MIT's Open Course Ware site? Most MIT courses are online with videoed lectures, recommended textbooks, homework and tests. It's a great resource. They also have edX, a co-operative venture with a bunch of fancy schools.

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

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