Reddit mentions: The best combinatorics books

> 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.

Sure, there are a few directions you could go:

Algorithms: A basic understanding of how to think about and analyze algorithms is pretty necessary if you were to go into combinatorial optimization and is a generally useful topic to know in general. CLRS is the most famous introductory book on algorithms, and it gets the job done. It's long, but I thought it was decent enough. There are also plenty of video lectures on algorithms online; I liked the MIT OpenCourseWare of this class.

Graph Theory: Many combinatorial optimization problems involve graphs, so you would definitely want to know some graph theory. It's also super interesting, and definitely worth learning regardless! West is a good book with lots of exercises. Bondy and Murty and Diestel also have good books, which are freely available in PDF if you do a google search. Since you're doing a project on traffic optimization, you might find network flows interesting. Networks are directed graphs, where you think about moving "flow" across the edges of the graph, so they are useful for modelling a lot of real-life problems, including traffic. Ahuja is the best book I know on network flows.

Linear and Integer Programming: Many optimization problems can be described as maximizing (or minimizing) some linear function subject to a set of linear constraints. These are linear programs (LPs). If the variables need to take on integer values, then you have an integer program (IP). Most combinatorial optimization problems can be formulated as integer programs. Integer programming is NP-hard, but in practice there are methods that can solve most IPs , even very large ones, relatively quickly. So, if you actually want to optimize things in real-life this is a very useful thing to know. There's also a mathematically rich field of developing methods to solve IPs. It's a bit of a different flavor than the rest of this stuff, but it's definitely a fertile area of research. Bertsimas is good for learning linear programming. Unfortunately, I don't have a good recommendation for learning integer programming from scratch. Perhaps the chapters in Papadimitriou - Combinatorial Optimization would be a good introduction.

Approximation Algorithms: This is about algorithms which quickly (in polynomial time) find provably good but not necessarily optimal solutions to NP-hard problems. Williamson and Shmoys have a great book that is freely available here.

The last book I'd recommend is Schrijver. This is the bible for the field. I put it here at the end because it's more of a reference book rather than something you could read cover to cover, but it's REALLY good.

Lastly, if you like traffic optimization, maybe look up what people are doing in operations research departments. A lot of OR is about modelling real problems with math and analyzing the models, so this would include things like traffic optimization, vehicle routing problems, designing smart electric grids, financial engineering, etc.

Edit: Not sure why my links aren't all formatting correctly... sorry!

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.

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!

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

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

I 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.
  

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.

> 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).

I would say that it's difficult for most students. Getting an A is certainly not impossible, but it is going to take a lot of work. I took it with Truman and I sat in on some of Manning's office hours so I can only really speak to those two professors.

If you've never seen proofs before, then I would suggest getting an introductory textbook and working out of it over the summer. A really good one that I recommend is How to Prove It by Daniel Velleman. Since 310 is an introductory course, you don't necessarily need to be familiar with proofs beforehand. But it will definitely make the transition much easier. Even a basic understanding of elementary logic and basic proof strategies will be helpful. The book I recommended goes at a much slower pace than I think is warranted, but it can be very helpful nonetheless.

If you have experience writing proofs, then the transition into the class shouldn't be too bad. But be warned that the class may be more rigorous than you're used to. If you've taken a class like CMSC250, you should keep in mind that my opinion, as well as that of everyone I know who's taken both classes, is that MATH310 requires much more time and patience.

Kate is an awesome professor. She does a good job of conveying the material and she makes expectations very clear. The homeworks are definitely on the long side and I often found myself using other resources, both online and from textbooks, to get an intuition for how to solve problems. With that said, the exams were easier than the homework assignments. That's not to say that the exams were easy, just easier than other assignments.

While I can't speak to how good Dr. Manning is as an educator, I can say that, without a doubt, his exams were much more difficult. I got a glimpse of one of his midterms and I tried to use his finals from the testbank to practice for our own final. It always seemed that his problems were a step up in difficulty from Kate's.

All I know about Dr. Williams and Dr. Halperin is what is readily accessible from the usual sources.

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.

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

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!

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

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

Oh, I'm terrible at calculus, haha. I teach discrete maths and logic, and never have to touch calculus at all, thank goodness :)

But a younger friend of mine is doing calculus just now, so I'll find out what he found useful and PM you. He did say that some of the books I'd recommended him were immensely useful for maths generally (not necessarily calculus in particular). In roughly ascending order of difficulty:

• Gowers, Mathematics: A Very Short Introduction (<http://amzn.to/1TQNEWY>)
  
• Devlin, The Language of Mathematics (<https://www.amazon.com/Language-Mathematics-Making-Invisible-Visible/dp/0805072543>)
  
• Stewart, Concepts of Modern Mathematics (<http://bit.ly/1L7rY4i>)
  
• Liebeck, A Concise Introduction to Pure Mathematics (<http://amzn.com/1439835985>)
  
  So you could pick one of those, and see if it meets the level you're after. There are also some standard books on "how to write proofs" which often get recommended: Velleman, How to Prove It: A Structured Approach <https://www.amazon.com/How-Prove-Structured-Daniel-Velleman-ebook/dp/B009XBOBL6>, and
  Solow, How to Read and Do Proofs: An Introduction to Mathematical Thought Processes <https://www.amazon.com/How-Read-Proofs-Introduction-Mathematical/dp/1118164024>, and they might be useful. I've never read either, shamefully, tho I probably should.
  
  Also: something that can be handy once you're past the basics is the amazing (but large and expensive) Princeton Companion to Mathematics. Which has something useful and interesting to say about pretty much every major area of modern mathematics.
  
  I hope that helps!

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.

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

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

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

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

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

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

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

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

http://www-math.mit.edu/~rstan/ec/
I'll give you a brief about the book: It's really dense and probably will take you a while to get through just a couple of pages, however, the book introduces a lot of interesting and difficult concepts that you'd definitely see if you pursue the field.

https://math.dartmouth.edu/news-resources/electronic/kpbogart/ComboNoteswHints11-06-04.pdf
Is a Free book available online and is for a real beginner, basically, if you have little to no mathematical background. I will however say something, in Chapter 6, when he talks about group theory, he doesn't really explain it at all (at that point, it would be wise to branch into some good pure math text on group and ring theory).

https://www.amazon.ca/Combinatorics-Techniques-Algorithms-Peter-Cameron/dp/0521457610
This is a fantastic book when it comes to self studying, afaik, the first 12 chapters are a good base for combinatorics and counting in general.

https://www.amazon.ca/Concrete-Mathematics-Foundation-Computer-Science/dp/0201558025
I've heard fantastic reviews about the book and how the topics relate to Math 2 3/4 9. Although I've never actually used the book myself, from the Table of Contents, it appears like it's a basic introduction to counting (a lot lighter than the other books).

Regarding whether or not you can find them online, you certainly can for all of them, the question is whether legally or not. These are all fairly famous books and you shouldn't have trouble getting any one of them. I'm certain you can study Combinatorics without statistics (at least, at a basic level), however, I'm not sure if you can study it without at least a little probability knowledge. I'd recommend going through at least the first couple of chapters of Feller's introduction to Probability Theory and it's Applications. He writes really well and it's fun to read his books.

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 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.

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 is a link to John Baez's overview of what Topos theory is.

As /u/ziggurism has already said, you don't need to understand algebraic geometry to understand the theory of elementary toposes but some of the early motivating examples, the category of sheaves on a grothendieck site, are heavily steeped in the language of modern algebraic geometry.

A good, non algebro-geometric introduction to topos theory for those seeking to understand its place in logic is the book 'Topoi: The Categorial Analysis of Logic by Goldblatt. It also serves as a great introduction to the ideas of category theory.

Another fantastic book is Lawvere/Roserugh's Sets for Mathematics. This book seeks to explain the axioms of set theory using the language of category theory. It's not a book on arbitrary topos but it does serve to give you an idea of how topos theory axioms serve to build the logical system that every mathematician is familiar with, the logic in the category of sets. It's a good idea to have this 'concrete' application of topos axioms in Set under your belt before you tackling a book that seeks to explain how an arbitrary topos gives you a more abstract and unfamiliar logical system.

Edit: Also worth looking into is how topos theory can be used in the foundations of physics

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.

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.

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.

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!

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.

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. :)

Do you know of similar guides or lists? Please contribute resources or books in a list here.

Remember, this is for teachers that also intend to become middle school teachers, and for this, a mathematics major is not necessary (there are many educated professionals changing their career to teach), so reconsider if you think this list is missing such advanced materials that a teacher may never even need mention in a classroom, let alone master. I'm not suggesting such advanced topics are inappropriate, but don't forget the audience, new teachers preparing for a successful beginning and beyond!

I'll begin:

Basic Mathematics:

• Precalculus by Sheldon Axler
  
  Calculus
  
  
• Calculus by Gilbert Strang. The book, the solutions manual, the instructor's manual, and a study guide are free to download from the author.
  
  
• Basic Calculus: From Archimedes to Newton to its Role in Science
  
  Mathematics Education
  (not your own, but as the subject)
  
  
• MATH! Encounters with High School Students
  
  
• On the Shoulders of Giants – New Approaches to Numeracy
  
  
• What's Math Got to Do with It?
  
  
• Math Made Visual
  
  
• Proofs Without Words and Volume II
  
  Education
  
  
• Nonviolent Communication
  
  
• Fires in the Bathroom, there is also a version for middle school but I haven't read it.
  
  Abstract Algebra
  
  
• Classical Algebra I'm still reading this, but it may be more appropriate for non-mathematics majors to understand that connection between the apparent chasm of school algebra and "modern" algebra.
  
  Number Theory
  
  
• The Higher Arithmetic This too may be more appropriate for someone not intending to master number theory.
  
  Mathematics History
  
  
• Men of Mathematics though I know there are better or more accurate histories, such as the one on the list by Morris Kline, but this one really is written well, though it doesn't have much mathematics, as it's more about the mathematicians.
  

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.

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.

I liked Richard Brualdi's Introductory Combinatorics (Amazon Link) due to the broad nature of topics presented. There are also good solutions guides too, which I find appealing when first getting in to a topic and doing the exercises.

If you pursue graph theory from there, there are plenty of books that are great, including free online editions like Diestel's book. http://diestel-graph-theory.com/.

Neither of these are the end all, be all for these topics, and they give you a taste of a lot of nice topics. Combinatorics is quite diverse, and I have yet to see one book that can come close to encompassing every high point. From there, you can usually look for "TOPIC NAME" + lectures notes and find something worthwhile.

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!

First of all, I am glad to hear that your attitude towards math has changed so drastically! Welcome to the dark side. As for where to start relearning math, that probably depends on where you're starting from. Have you taken any classes on discrete math or linear algebra for your IT major? How about calculus?

If you want to work in mathematics, then you probably want a light introduction to proofs. Most undergraduates in math and computer science at my university use
Discrete Mathematics: Mathematical Reasoning and Proof with Puzzles, Patterns, and Games. I personally recommend How to Prove It by Velleman, which promotes a structured approach to proving that jives well with the step-wise refinement approach used by people in comp prog and comp sci.

If you have had some calculus, another good place to start is Calculus by Spivak. It will start you over from the basics, but with the rigor one should expect from a mathematics course.

Of course, I'm just making guesses about your background and your interests. What do you think you'd want to do with math or physics, or both? Perhaps you might enjoy scientific computing? If physics (or maybe even engineering) is your thing, then skip the first two books I recommended and try out Spivak. From there go on to books on linear algebra and differential equations -- the necessary math background -- while also checking out some physics books (I have a few to recommend, but that's off-topic. Let me know if that's something in which you're interested).

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).

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. :)

You're probably right.

For algorithms, the usual undergraduate text is Introduction to Algorithms by Cormen, et al. I do recommend it, too. It's comprehensive and good.

Besides Cormen, I've also made good use of Analysis of Algorithms and Analytic Combinatorics by Sedgewick and Flajolet. They're a bit more advanced, but I don't know algorithms analysis well enough as a field to know where they would usually be taught or how central their material is. They've been very worthwhile purchases for me, though, and I do recommend them as well.

Edit: It's also worth pointing out that, at my institution, the graduate intro algorithms course used Kozen's other textbook, The Design and Analysis of Algorithms. You might want to take a look.

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.

You could consider starting with a book like Velleman's How to Prove It. It doesn't have to be that book, there are also free options online, but learning some logic and set theory from a book like that is a good way to figure out how to work with the other subjects you're working on.

Then, you could find a rigorous treatment of the subjects you want to learn. Something like Axler's Linear Algebra Done Right or Spivak's Calculus.

Learning math from textbooks like this is harder, but you end up with a better understanding of the math.

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.

Depends on your background. Mac Lane is the standard text and he is a phenomenal author in general, but it builds off knowledge of concepts such as modules, tensor products and homotopy (I still don't have a sufficient background in AT to be honest though). For a more modest background, I would recommend the book "Sets for Mathematics" by Lawvere and Rosebrugh. The book is entirely on category theory, the title is because there is a focus on the category of sets. The first chapter or so is deceptively simple, it gets very difficult as it goes on, but still doesn't require much specific background.

I'll also note that I first got into the subject through a whim purchase in a local Borders of a cheap dover book Topoi by Robert Goldblatt when I was very into mathematical logic. It's 500 pages and requires pretty much no background (I'd know what a topological space is, but I can't think of anything else). It gets very challenging though, and I never got more than 250 pages in before getting overwhelmed, but the first hundred pages really sparked my interest in category theory. Functors (and especially adjoint functors) are postponed much later than you will see in many other sources though. You can find a link to an online version free from the author's webpage too.

If you happen to have the UCLA edition of Friedberg's Linear Algebra (the one you'll likely use for 115A) already, there's a section at the end with an intro to proofs. This book is pretty popular at universities with a dedicated intro to proofs class, so it might be worth checking out; I read a bit of it before taking the upper divs. Hope that helps!

I also tried to learn calculus through spivak and found it very difficult; I stopped at then 4th chapter and switched to an easier textbook. If it's your first time learning calculus choosing an easier and verbose text like Stewart may suite you better. It's important to remember Spivak's Calculus is more like a textbook on Analysis (the theory of calculus), which is what often comes junior or senior year for math majors/minors.

If you have already learned calculus I'd suggest the bookHow to Prove It which helps think of math in a more concrete way that can help with proofs, even though no calculus is presented. Also, remember that Spivak likely didn't intend for people to find his questions easy, so don't feel like you are unprepared if it takes a while to do a single question.

Sorry, the solution is to do lots of proofs.

There's more to it, but honestly it's more of a thing that you have to read a book about rather than a message on reddit. How are you learning about this right now? Is it part of a course or self-study? I personally found How to Prove It to be a very useful textbook. Doesn't require any particular knowledge, and it builds out a nice foundation in logic and set theory.

I don't know if category theory is your thing but maybe you'd be interested in categorical logic?
http://www.amazon.com/Introduction-Higher-Order-Categorical-Cambridge-Mathematics/dp/0521356539

that book goes a lot into various ways to study logics inside categories and indeed categories as certain kinds of deduction systems. It also has a nice introduction to toposes which are kinds of categories which have a kind of inner logic (usually multivalued and always intuitionistic). It links nicely to some newer branches of math too.

If you're unfamiliar with categories they are abstractions of collections of mathematical structures and the functions between them(eg. sets and functions, or vector spaces and linear functions).

I know the answer to this.

First, though: arithmetic and all that, through calculus, is not math.

True math is the discovery of properties of ideas. One interesting example is the fact that there is a hypothetical machine that is proven to be able to do everything a (real) computer can do, but that there are many things that it can never do. Therefore, there are questions that can never be answered by a computer, no matter how powerful.

If you actually want to know about the beauty, you need to see it for yourself. As I recall, How to Prove it is pretty decent.

Since you're studying comp sci, combinatorics might be a really good topic--most of the difficulty students have studying algorithms comes from not having a strong enough background in combinatorics. It's also something that you can delve right into. I don't really have any book recommendations, but my undergrad class used Roberts and Tesman. I don't remember enough about this book to say much about it.

Alternatively, logic is super cool! A book on computability or mathematical logic would work. Herbert Enderton has written a book on each. I find his writing a little too informal, but they're good for getting a grip on the basic notions.

Specifically Graph Theory? Pearls In Graph Theory was the one that I used in my undergraduate course. It covers some algorithms and the theory behind completeness, infinite lattices, etc. Fun stuff!

If you want a more combinatorial view of things (ie: not so specialized with regards to Graph Theory), I highly recommend Brualdi's Introductory Conbinatorics as bucket1004 said. I also used that one for my combinatorics course and I loved it.

Disclaimer: There's some errata in the edition that I used. I'm not so sure how much better the new editions are.

I haven't read the book you say you are working through, inducing, but I can tell you that based on the description on Amazon, it sounds like OP would be better served by picking up Velleman's How to Prove It which is, as far as I know, currently considered the gold standard for learning basic mathematical logic and proof techniques. It is also much, much cheaper.

Also, I can vouch for it. I used it in undergrad several years ago. It sounds like it does everything inducing's book does based on his/her post and the amazon description.

It's the other way around: mathematics is logic, and has been described that way, again, ever since Frege's Basic Laws of Arithmetic, although the most famous work in attempting to put mathematics on a solid logical foundation remains the three volumes of Russell and Whitehead's Principia Mathematica. And now we come to our own history, because Alonzo Church came up with a simpler, more powerful logic than Russell and Whitehead's, but then his grad students Kleene and Rosser proved that it was inconsistent. So Church added a variant of Russell's theory of types to his logic, and here we are today.

Update: This isn't to say everyone who does math does so explicitly in terms of logic, of course. But when we say "you can't divide by zero," or "this series diverges," or we do "real" or "complex analysis," or when we use "set theory," we are explicitly making logical statements or employing the logical descriptions of the mathematics, possibly even going so far as to talk about the "real" numbers in terms of Cauchy completion or Dedekind cuts. Any good, modern text on mathematical foundations will define the natural numbers in terms of the Peano axioms, then the integers in terms of the natural numbers, then the rationals in terms of the integers, and finally the reals in terms of the rationals. And so on.

After this coming semester, all or almost all of the math courses that you take will be heavily proof-based. Analysis, topology, and algebra all involve a lot of reading and writing proofs. So I would recommend using this time to become acquainted with mathematical proof techniques. Hammack's Book of Proof is freely available online. I haven't read it myself but I've heard good things about it. Eccles's An Introduction to Mathematical Reasoning is another good choice and it is the intro-to-proof book that I learned from.

Whatever books or resources you choose, just remember that it is very important to do the exercises.

I would recommend the following two books:

1. "How to Prove It" by Daniel Velleman.
   
   
2. "Understanding Analysis" by Stephen Abbott.
   
   The first book introduces most of the topics in the book that you linked, and was what was used in my Foundations of Mathematics class (essentially the same thing as your class).
   
   Understanding Analysis, on the other hand, is probably the perfect book to follow up with, since it is such a well-motivated, yet rigorous book on the analysis of one real variable, that you may, in fact, become too accustomed to such lucid and entertaining prose for your own good.

The exercises in Spivak’s Calculus (amzn) are the best part of the book.

• • 
    
    /u/WelpMathFanatic You’ll probably have a better (more efficient, more enjoyable) time if you take a course, or otherwise find someone to help you face to face. But if you’re studying by yourself you might want to look at a book about writing proofs, such as Velleman’s [How to Prove It](https://amzn.com/0521675995) or Hammack’s [Book of Proof*](https://amzn.com/0989472108). (Disclaimer: I haven’t read either of these.)
    
    

As you said, the first step is to figure out what you know. You don't need a tutor for that per se. See if you know the rules of basic algebra and trigonometry, as in Lang's Basic Mathematics. If not, that book is a great source to learn from. After that, if you're interested in higher math you should learn some basic logic and how to write proofs. Velleman's How to Prove It is a good starting point. From there you can look into studying calculus, linear algebra, abstract algebra, or discrete math. Those are probably the most useful starting subjects.

You received A's in your math classes at a major public university, so I think you're in pretty good shape. That being said, have you done proof-based math? That may help tremendously in giving intuition because with proofs, you are giving rigor to all the logic/theorems/ formulas, etc that you've seen in your previous math classes.

Statistics will become very important in machine learning. So, a proof-based statistics book, that has been frequently recommended by /r/math and /r/statistics is Statistical Inference by Casella & Berger: https://www.amazon.com/Statistical-Inference-George-Casella/dp/0534243126

I've never read it myself, but skimming through some of the beginning chapters, it seems pretty solid. That being said, you should have an intro to proof-course if you haven't had that. A good book for starting proofs is How to Prove It: https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995

Book of proof is a more gentle introduction to proofs then How to Prove it.

​

No bullshit guide to linear algebra is a gentle introduction to linear algebra when compared to the popular Linear Algebra Done Right.

​

An Illustrated Theory of Numbers is a fantastic introduction book to number theory in a similar style to the popular Visual Complex Analysis.

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

Are you proficient in reading and writing proofs?

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

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

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

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

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

I'll be that guy. There are two types of Calculus: the Micky Mouse calculus and Real Analysis. If you go to Khan Academy you're gonna study the first version. It's by far the most popular one and has nothing to do with higher math.

The foundations of higher math are Linear Algebra(again, different from what's on Khan Academy), Abstract Algebra, Real Analysis etc.

You could, probably, skip all the micky mouse classes and start immediately with rigorous(proof-based) Linear Algebra.

But it's probably best to get a good foundation before embarking on Real Analysis and the like:

Discrete Mathematics with Applications by Susanna Epp

How to Prove It: A Structured Approach Daniel Velleman

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

Book of Proof by Richard Hammock

That way you get to skip all the plug-and-chug courses and start from the very beginning in a rigorous way.

My two cents

1. Maths is difficult. There isn't one of us who at some point has not struggled with it
   
2. Maths should be difficult. The moment you find it easy you are not pushing yourself!
   
   If you want to improve your skills you can do two things in the short term -- read and practice.
   
   I would recommend Basic Mathematics by Lang (it gets mentioned a lot around here). Or if you are interested in higher math look at How to Prove It by Velleman
   
   The great thing is that both include exercises.

There are a ton here just to get you started. Honestly, googling "proofs without words" or "visual proofs" should get you a lot that you'll like :)

If you're hungry for more, there are entire books that collect them(2nd and 3rd volumes, too)

)

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

Amazon search for Dover Books on mathematics

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

Pre-Calculus / Problem-Solving

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

The first step is to become comfortable with proofs. It is extremely different than the type of math you likely did in calculus, linear, and diffeq. There is very little "carry out this set of steps until you have computed the answer". This is not what proofs are like, and it is not what mathematics is really about. I've heard this is a very good book for learning about proofs and proving: http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995

After that, you can begin studying specific topics. I would recommend starting with abstract algebra or analysis. To do this, just get the textbooks. There are lots of resources to answer the question "which textbook should I use" (see sidebar).

These two are great in that they kind of take a CS approach to proofs by using the idea of a pattern:

Proof Patterns https://www.amazon.com/dp/3319162497/ref=cm_sw_r_cp_apa_i_0Y44BbDC5HWYB

How to Prove It: A Structured Approach, 2nd Edition https://www.amazon.com/dp/0521675995/ref=cm_sw_r_cp_apa_i_C244Bb8QY01F2


This is great for understanding limitations and the history of the development of proof techniques:

Reverse Mathematics: Proofs from the Inside Out https://www.amazon.com/dp/0691177171/ref=cm_sw_r_cp_apa_i_M044BbM0BMAE3


As has been suggested, you should look into mathematical logic, modern symbolic logic, which includes propositional and quantificational logic, relational logic, and perhaps even modal logic. Copi is great for a classical treatment of modern and Aristotelian logic. Gensler is great for learning translations, and Smullyan is great for mathematical logic.

You might also look into set theory and hott or type theory, but only after you've approached some of the other stuff. I can't emphasize logic enough if you want to really understand proof theory. Perhaps even check out computability and complexity theory. A lot of these topics in theoretical computer science run into the limitations and possibilities of these and other ideas. You'll then realize as others have suggested, that yes there are some fundamentals taken for granted, but real math is difficult because of the intuition needed in approaching a proof, and the possibility that you could be working on a problem that is unanswerable.

Hey mate, if you're still looking for introductory logic books, I just came across a book which looks great for that. It looks like an introduction to thinking about mathematical proofs, and introduces basic logic and set theory to do so. I'm sure I've covered all the material in the book in courses I've taken and other books I've read already, but I'm tempted to get it anyway because it looks great!

Linear programming:

• Vanderbei, Linear Programming: Foundations and Extensions
  
  Combinatorial optimization:
  
  
• Papadimitriou and Steiglitz, Combinatorial Optimization: Algorithms and Complexity
  
• Cook, Cunningham, Pulleyblank, and Schrijver, Combinatorial Optimization
  
  Game theory:
  
  
• Fudenberg and Tirole, Game Theory
  
• Osborne and Rubinstein, A Course in Game Theory
  
  Combinatorial game theory:
  
  
• Berlekamp, Conway, and Guy, Winning Ways for Your Mathematical Plays (volume 1, volume 2, volume 3, volume 4)
  
• Berlekamp, The Dots and Boxes Game: Sophisticated Child's Play
  
• Albert, Nowakowski, and Wolfe, Lessons in Play: An Introduction to Combinatorial Game Theory
  
• Conway, On Numbers and Games

Get used to proof based mathematics. How to Prove It: A Structured Approach, by Daniel J. Velleman, would be a great start.

EDIT: Ok math that's useful for a STEM major, maybe forget about the proof based math unless you're considering mathematical physics. It's still a good book though.

I'm going to recommend the book How to Prove It. Its all about learning the logic for proofs and strategies for writing proofs. Its one of those books that you work through slowly and complete all the exercises. Its recommended around here a-lot. I'd also suggest using the search feature if you ever want to look for other recommended books because those threads come up often around here.

Best wishes.

I always liked C. L. Liu, Introduction to Combinatorial Mathematics, McGraw-Hill, 1968. It's old but i think still one of the best introductions on that subject.

Peter J. Cameron, Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press, 1994 is imho also decent. More information on the book is available here.

I've heard the book How To Prove it is pretty good. Also I'd recommend the Art of Problem Solving books as well for algebra and the likes. (It seems to go over stuff you'd learn in 7th grade, but written at a level adequate for adults).

I would also recommend sites like www.expii.com and www.brilliant.org

Khan academy also has a problem generator iirc.

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


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

I picked up a book a couple years ago called How to Prove It.

It has helped me develop a greater appreciation for logic and proofs. I wish I took this stuff more seriously when I started programming. A little bit of knowledge of boolean algebra can help tremendously.

You need a good foundation: a little logic, intro to proofs, a taste of sets, a bit on relations and functions, some counting(combinatorics/graph theory) etc. The best way to get started with all this is an introductory discrete math course. Check these books out:

Mathematics: A Discrete Introduction by Edward A. Scheinerman

Discrete Mathematics with Applications by Susanna S. Epp

How to Prove It: A Structured Approach Daniel J. Velleman

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

Combinatorics: A Guided Tour by David R. Mazur

If your intent is to take a class like analysis, you really should look into something like logic.

Daniel Velleman wrote an excellent little book called How to Prove It: A Structured Approach. It's actually designed for High School level students, but it works through the subject incredibly well.

Here's an Amazon link to the book:

http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995/ref=sr_1_1?s=books&ie=UTF8&qid=1333383091&sr=1-1

Like one of the other's suggested, learning more about proofs is probably what you're interested in since that's where these rules and equations come from. I've seen this book recommended a few times. It should give you a better understanding of how math is formed.

The deck corresponding to the intellectual property book has ~325 cards.

The deck corresponding to IEA has ~400 cards.

The deck corresponding to linear algebra has ~1000 cards. That seems weird to me, since I feel that I make fewer cards for math books; most of the extra time comes from doing a lot of scratch work. Weird. In addition to timing, I've more recently started keeping track of how many Ankis I've added each section, so maybe I'll have more insight there later. We'll see.

And please message me when you start doing math! If you're looking towards advanced mathematics (beyond calculus/linear algebra-for-engineers), I recommend starting with either Mathematics for Computer Science (review) or, if you really have no interest in doing that, How to Prove It.

Here's my radical idea that might feel over-the-top and some here might disagree but I feel strongly about it:

In order to be a grad student in any 'mathematical science', it's highly recommended (by me) that you have the mathematical maturity of a graduated math major. That also means you have to think of yourself as two people, a mathematician, and a mathematical-scientist (machine-learner in your case).

AFAICT, your weekends, winter break and next summer are jam-packed if you prefer self-study. Or if you prefer classes then you get things done in fall, and spring.

Step 0 (prereqs): You should be comfortable with high-school math, plus calculus. Keep a calculus text handy (Stewart, old edition okay, or Thomas-Finney 9th edition) and read it, and solve some problem sets, if you need to review.

Step 0b: when you're doing this, forget about machine learning, and don't rush through this stuff. If you get stuck, seek help/discussion instead of moving on (I mean move on, attempt other problems, but don't forget to get unstuck). As a reminder, math is learnt by doing, not just reading. Resources:

• math subreddit
  
• math.stackexchange.com
  

• math on irc.freenode.net
  

• the math department of your college (don't forget that!)
  
  
  Here are two possible routes, one minimal, one less-minimal:
  
  Minimal
  
  
• Get good with proofs/math-thinking. Texts: One of Velleman or Houston (followed by Polya if you get a chance).
  
• Elementary real analysis. Texts: One of Spivak (3rd edition is more popular), Ross, Burkill, Abbott. (If you're up for two texts, then Spivak plus one of the other three).
  
  
  Less-minimal:
  
  
• Two algebras (linear, abstract)
  
• Two analyses (real, complex)
  
• One or both of geometry, and topology.
  
  
  NOTE: this is pure math. I'm not aware of what additional material you'd need for machine-learning/statistical math. Therefore I'd suggest to skip the less-minimal route.

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

If your Calculus is rusty before Rudin read Spivak Calculus it is great intro to analysis and you will get your calculus in order. Rudin is going to be overkill for you. Also before trying to do proofs read How to prove it It is a great crash course to naive set theory and proof strategies. And i promise i won't bore you with math any more.:D

It's common to have some difficulty adjusting from lower-level courses with a computational emphasis to upper-level courses with an emphasis on proof. Fortunately, this phenomenon is well known, and there are a number of books aimed at bridging the gap between the two types of courses. A few such books are listed below.

• How to Prove It (Velleman)
  
• Book of Proof (Hammack)
  
• Mathematical Proofs: A Transition to Advanced Mathematics (Chartrand et al)
  
  I hope that helps!

Oh man 2011 was probably the hardest MATA31 revision. Don't worry, about that midterm though, the course content is really different now, that was when CSC/MATA67 used to be merged with MATA31, so they did a lot more set theory/number theory in MATA31 than they do now. I doubt most people who took MATA31 (and did well) could even pass that midterm just because we don't learn that stuff in MATA31 anymore. If you're trying to get started on studying for MATA31 now, I actually recommend you don't learn MATA31 material. Instead, improve on your critical thinking skills which your high school has definitely not given you. "Find" a book called how to prove it and go through maybe the first two or so chapters which just introduce proofs, and start to build up your proof skills. Becoming comfortable with proofs will come in handy immensely for CSCA67, MATA37, and in a big chunk of MATA31.

I recommend proofs. How To Prove It is a good introduction, but there are also free resources available online.

Math major courses after the first year consist of proof based courses. If you already have interest in proof writing and argumentation then I suggest you go for it. If not then you try it out. Firstly, you should brush up on basic logic. Check out the quick introduction at berkeley. Once that's cleared, go complete a book such as How To Prove It. Now, you should have the most basic tools needed to tackle and solve problems in future courses in analysis, algebra, topology, etc. The time required to learn and write good proofs is steep. It takes constant feedback and solving numerous problem to get a knack of. I would say that you should get used to proof writing in a semester. Overall, learning to write proofs isn't very difficult but learning the material for the courses and developing a solid theoretical understanding is.
In regard to your concerns about time commitment with courses, it depends on how far you are willing to go to obtain a career in mathematics and to do deep research.

How to Prove It: A Structured Approach by Velleman is good for developing general proof writing skills.

How to Think About Analysis by Lara Alcock beautifully deconstructs all the major points of Analysis(proofs included).

Not sure what sort of thing you're trying to prove, but there are a few good books on techniques for proof that you'll end up using if you go into higher math. I like How to Prove It by Velleman. It's geared towards students finishing high school math who are planning to do math at the university level, so it might be the sort of thing you're looking for.

You will first want to learn fundamental logic and set theory before diving into topics like analysis, algebra, and discrete topics. You will need an understanding of a rigorous proof -- not the hand-wavey kind of proof we've seen in our introductory calculus courses. This book is very readable and will prepare you for advanced mathematics. I've seen it work for many students.

After you're finished with it, you may want to study analysis which will build up the Calculus for you. If you don't care for calculus anymore, consider reading an abstract algebra text. Algebra is pretty fun. You can also pick a discrete topic like graph theory or combinatorics whose applications are very easy to see.

There are many ways to go, but in all of them you will absolutely need a a basic understanding of the use of logic in a mathematical proof.

Don't pay too much attention to the other replies - if you really want to take Math 145/146 it's possible, it will just be a lot of work.

My marks were good in high school (but not 95+) and my score on the Euclid was terrible (in order to enrol without an override you need 80+ on the Euclid). The thing to know is these courses have heavy emphasis on proofs, so the summer before coming I worked my way through the first half of a book on proofs and ended up doing relatively well in these courses.

You can certainly do it, but you have to be really dedicated.

I've been studying How To Prove It by Daniel Velleman for a few months now and I don't know if it's the best book, but it's really good and it has opened my mind in so many ways. Plus, it's really cheap for a textbook.

OK..

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

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

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

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

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

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

Linear programming isn't actually programming, the quick dirty description is linear algebra, but with inequalities. This book is a good introduction, plus it's cheap. It does include pseudo code if you want to get into coding.

Combinatorial Optimization: Algorithms and Complexity (Dover Books on Computer Science) https://www.amazon.com/dp/0486402584/ref=cm_sw_r_cp_apa_i_5m-4AbFH5EZT8

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

I used Susanna Epp's Discrete Mathematics text and rather enjoyed it. Velleman's How To Prove It is also quite good.

http://www.abebooks.com/servlet/SearchResults?bi=0&bx=off&ds=20&kn=Epp+discrete+applications+3&recentlyadded=all&sortby=17&sts=t

How to Prove It: A Structured Approach by Daniel J. Velleman http://www.amazon.com/dp/0521675995/ref=cm_sw_r_udp_awd_ff3Vtb08QR4FZ

"I'm also sure that due to my limited educational resources, self-directed study will be a huge part. Any suggestions on which books are must reads to gain competency in CS?"

Here are a few good choices for the more theoretical areas of computing:

http://www.amazon.com/Algorithms-4th-Edition-Robert-Sedgewick/dp/032157351X/ref=sr_1_1?ie=UTF8&qid=1408406629&sr=8-1&keywords=algorithms+4th+edition

http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995/ref=sr_1_1?ie=UTF8&qid=1408406673&sr=8-1&keywords=how+to+prove+it

You'll also want to look at a decent discrete mathematics book. Sadly the book I used as an undergrad was rubbish, so I don't have a good recommendation.

Learn math at a more "fundamental" level, and that will test if you love it. For me, I didn't love math until I took a class on proofs and real analysis. One of the books we used was "How to Prove it", and to this day it's my favorite textbook ever. How do we know anything in mathematics? Which rules do we follow and how do we know they are true? This starts from basic logic and truth tables, and works its way up to some really complicated stuff. It's not as fancy as complex integrals and PDE's, but I would say it's a more fundamental form of mathematics and the basis for all other subjects in the field.

Honestly, I think you should be more realistic: doing everything in that imgur link would be insane.

You should try to get a survey of the first 3 semesters of calculus, learn a bit of linear algebra perhaps from this book, and learn about reading and writing proofs with a book like this. If you still have time, Munkres' Topology, Dummit and Foote's Abstract Algebra, and/or Rudin's Principles of Mathematical Analysis would be good places to go.

Roughly speaking, you can theoretically do intro to proofs and linear algebra independently of calculus, and you only need intro to proofs to go into topology (though calculus and analysis would be desirable), and you only need linear algebra and intro to proofs to go into abstract algebra. For analysis, you need both calculus and intro to proofs.

Not specific to calculus, but Daniel Velleman's How to Prove It is an excellent and thorough introduction to the practice/art of proof-writing, and it requires only a high school mathematics education as a prerequisite.

Logic, Number theory, Graph Theory and Algebra are all too much for you to handle on your own without first learning the basics. In fact, most of those books will probably expect you to have some mathematical maturity (that is, reading and writing proofs).

I don't know how theoretical your CS program is going to be, but I would recommend working on your discrete math, basic set theory and logic.

This book will teach you how to write proofs, basic logic and set theory that you will need: http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995


I can't really recommend a good Discrete Math textbook as most of them are "meh", and "How to Prove It" does contain a lot of the material usually taught in a Discrete Math course. The extra topics you will find in discrete maths books is: basic probability, some graph theory, some number theory and combinatorics, and in some books even some basic algebra and algorithm analysis. If I were you I would focus mostly on the combinatorics and probability.


Anyway, here's a list of discrete math books. Pick the one you like the most judging from the reviews:

• http://www.amazon.com/Discrete-Mathematics-Applications-Susanna-Epp/dp/0495391328
  
• http://www.amazon.com/Discrete-Combinatorial-Mathematics-Applied-Introduction/dp/0201726343
  
• http://www.amazon.com/Discrete-Mathematics-Applications-Kenneth-Rosen/dp/0073383090
  
• http://www.amazon.com/Discrete-Mathematics-Computer-Science-Solutions/dp/053449501X
  
• http://www.amazon.com/Mathematics-Discrete-Introduction-Edward-Scheinerman/dp/0840049420 (read the reviews of the previous edition: http://www.amazon.com/Mathematics-Discrete-Introduction-Edward-Scheinerman/dp/0534398987)
  
  
  Don't bother trying to learn too much too soon, as you really do need to let time for the math to sink in.

This is the book I used at university. I thought it was pretty good. Velleman's book is also popular. I've heard good things about this book, but I've not read it.

If you are interested in proofs, you might want to look at Velleman's How To Prove It. Another one people seem to like is The Book of Proof. Do what you want, the world is yours.

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

helps with the first part of the class. the stuff after that I would suggest just having good google-fu.

“How to Prove it”. D. Velleman: Amazon US Link

Probably the best resource on the topic!

Alternately, any introductory book on mathematical analysis will have a section on sentential logic. 'How to Prove It' by Velleman is a great intro, and comes with a link to a web tool to practice!

Start with a book like this:

http://www.amazon.com/books/dp/0521597188

or this:

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

or the one teuthid recommended. When you're doing self-study, it's doubly important to be able to read and follow most of the material.

Not to pile on, but as has been previously stated what you wrote is not a proof. I'm not going to focus on whether or not what you said is true or false because the larger problem is that it's not written as a proof structure-wise. By this I mean, proofs are written using logic. If you're really interested in proof writing and basic analysis I suggest this book: http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995/ref=sr_1_1?s=books&ie=UTF8&qid=1331568877&sr=1-1

I would highly suggest How to Prove it. It's a book that teaches you to think logically about problems and (inevitably) also teaches you a lot about logic. I read it as a freshman coming into an undergrad as a math major and it was super helpful.

To lean from a book like that of Artin's, you need to get a few basics down:

How to Prove It: A Structured Approach by Daniel Velleman

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

A First Course in Graph Theory by Chartrand and Zhang

Combinatorics: A Guided Tour by Mazur

Discrete Math by Epp

For Linear Algebra I like these below:

Lecture Notes by Tao

Linear Algebra: An Introduction to Abstract Mathematics by Robert Valenza

Linear Algebra Done Right by Axler

Linear Algebra by Friedberg, Insel and Spence

How to Prove It by Vellemen is a superb introduction to what proofs are, and how to make them.

Keep in mind certain proof based courses can be frustrating to some students (discrete math and real analysis) as these classes often make formal concepts students may understand intuitively. Abstract Algebra or Topology may give you a more accurate idea of your feelings towards math.

Well if you're going to start at the beginning, you might as well start at the beginning, I suggest you pick up this little number

+1.

OP: You might also want to brush up on your proof skills, in case they're rusty. How to Prove It: A Structured Approach by Velleman is a good place to start.

Combinatorial Optimization, algorithms and complexity. It treats a large amount of graph and combinatorial algorithms, and gives a solid introduction to NP/P/NP-Complete/NP-Hard/Big O etc.

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

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

This is the textbook. Very helpful.

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

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

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

This book was very helpful to me.

ummarycoc has a good point. Snoop around his room and see if he already has How To Prove it: A Structured Approach. Someone bought this book for me and I return to it frequently.

I'd echo what /u/Odnahc has said.

Struggling in Intro the Proofs isn't he end of the world. I struggled in proofs and still ended up with a BS and MS in Math, however, I bought this book and self studied proofs over the Summer and made sure I had a stronger foundation.

The courses normally taken after proofs (Advanced Calculus and Modern Algebra) usually spend the first class reviewing proofs to make sure students have a handle of the material. After that though, you're expected to know the stuff. And honestly, you'll be doing lot of work trying to understand the new material and you're really going to struggle if you're fighting proof writing instead of the new ideas.

Proceed with caution. Definitely speak to your advisor.

How to Prove It is a nice introduction to writing proofs.

Advanced math is subjective. Discrete math is a lot of topics mixed together into one class. A little bit of logic, graph theory, set theory, number theory, modular arithmetic, combinatorics, introduction to proofs, algorithm analysis and some other stuff I might be missing. The only prerequisite for it is pre-calculus. The difficulty of the class is subjective some people find it hard and some people find it easy. If you can remember definitions and theorems and string them together to construct a proof you should be fine. How to prove it is recommended a lot as an intro to writing proofs.

For proof writing techniques I highly recommend Velleman's "How to Prove It" link

Probably something on coursera; however, I really recommend this gem of a book, http://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995/ref=sr_1_1?s=books&ie=UTF8&qid=1458411780&sr=1-1&keywords=how+to+prove+it .

To learn basic proof writing I highly recommend How to Prove It by Velleman.

I recommend Brualdi's Introduction to Combinatorics. I've used the book to teach a course that was an introduction to combinatorics and graph theory and found it readable, informative, and a good source of problems and applications.

If you do decide on it, definitely read this beforehand:

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

How comfortable are you with proofs? If you are not yet comfortable, then read this: How to Prove It: A Structured Approach

When I took 322 with Jing, we mostly used Peter and Gary's course notes, with some content from Cameron's Combinatorics. In many ways, I found 322 to be much different from 222; 222 focuses a lot on counting, generating functions, pigeonhole, etc. while 322 focused a lot more on proving the existence of and enumerating different combinatorial objects.

Get the book [How to Prove It: A Structured Approach by Daniel J. Velleman] (http://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995) it will teach you how to write, and I think more importantly, read proofs.

Perhaps rather than concentrating on these particular proofs you should look at something like How To Prove It.

If I se a movie that I like then I look at other movies with the same director. Have you looked at other books by any of the three? Knuth's Art of Computer Programming comes to mind, for instance.

Another is Analytic Combinatorics (one author is a student of Knuth's).

I'm not doing this for a class. It's just that I have been drawing a blank whenever there was a question "Show that" or "Prove that". So now I'm working through Velleman's How to prove it. There are answers for some problems but not all. Not for this one. This question is in chapter three and before that there has been covered some logic and set theory. Nothing fancy like rings and abstract stuff.

I like your suggestion that a<b<0 implies 0<-b<-a, and squaring a negative number of course gets a positive number. Wouldn't that be enough for a proof?

The combinatorics class I'm in right now is using Richard Brualdi's Introductory Combinatorics. I'm a fan of it.

• Proofs without Words
  
• Proofs without Words II
  
• Math Made Visual: Creating Images for Understanding Mathematics
  

The book Combinatorial Problems and Exercises has a lot of really good problems which range from fairly easy to rather challenging.

You cannot go wrong with How To Prove It: A Structured Approach by Velleman https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995/ref=sr_1_3?keywords=how+to+prove+it&qid=1558195901&s=gateway&sr=8-3

I saw that book highly recommended, and after going through it myself a while ago I highly recommend it as well. When I do proofs I still maintain the mental model and use some of the mechanics that I learned from this book. You don't even have to read the whole thing in my opinion. Pick it up, work through a few pages per day, and stop when you feel like moving onto another subject-specific book, like Understanding Analysis.

Oh, and you might already know this, but do as many practice problems as you can! Learning proofs is all about practice.

Set theory/proof-writing is much more difficult than high school algebra. I'm teaching myself Calc III and Proof Writing right now in preparation for Abstract Algebra - I can say that compared to Calculus, the more advanced set theory is much more difficult. For me, anyway. For the Proof Writing I am using this book - How To Prove It: A Structured Approach. I'd say looking through that before moving on to anything more advanced than Calculus is a good idea. Which.. is why I'm doing it, myself.

I would recommend the book "How To Prove It".

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

It helped me in my transition into proof based mathematics. It will teach common techniques used in proofs and provides a bunch of practice problems as well.