Best abstract algebra books according to redditors
We found 246 Reddit comments discussing the best abstract algebra books. We ranked the 56 resulting products by number of redditors who mentioned them. Here are the top 20.
We found 246 Reddit comments discussing the best abstract algebra books. We ranked the 56 resulting products by number of redditors who mentioned them. Here are the top 20.
The answer is "virtually all of mathematics." :D
Although lots of math degrees are fairly linear, calculus is really the first big branch point for your learning. Broadly speaking, the three main pillars of contemporary mathematics are:
You might also think of these as the three main "mathematical mindsets" — mathematicians often talk about "thinking like an algebraist" and so on.
Calculus is the first tiny sliver of analysis and Spivak's Calculus is IMO the best introduction to calculus-as-analysis out there. If you thought Spivak's textbook was amazing, well, that's bread-n-butter analysis. I always thought of Spivak as "one-dimensional analysis" rather than calculus.
Spivak also introduces a bit of algebra, BTW. The first few chapters are really about abstract algebra and you might notice they feel very different from the latter chapters, especially after he introduces the least-upper-bound property. Spivak's "properties of numbers" (P1-P9) are actually the 9 axioms which define an algebraic object called a field. So if you thought those first few chapters were a lot of fun, well, that's algebra!
There isn't that much topology in Spivak, although I'm sure he hides some topology exercises throughout the book. Topology is sometimes called the study of "shape" and is where our most general notions of "continuous function" and "open set" live.
Here are my recommendations.
Analysis If you want to keep learning analysis, check out Introductory Real Analysis by Kolmogorov & Fomin, Principles of Mathematical Analysis by Rudin, and/or Advanced Calculus of Several Variables by Edwards.
Algebra If you want to check out abstract algebra, check out Dummit & Foote's Abstract Algebra and/or Pinter's A Book of Abstract Algebra.
Topology There's really only one thing to recommend here and that's Topology by Munkres.
If you're a high-school student who has read through Spivak in your own, you should be fine with any of these books. These are exactly the books you'd get in a more advanced undergraduate mathematics degree.
I might also check out the Chicago undergraduate mathematics bibliography, which contains all my recommendations above and more. I disagree with their elementary/intermediate/advanced categorization in many cases, e.g., Rudin's Principles of Mathematical Analysis is categorized as "elementary" but it's only "elementary" if your idea of doing math is pursuing a PhD. Baby Rudin (as it's called) is to first-year graduate analysis as Spivak is to first-year undergraduate calculus — Rudin says as much right in the introduction.
A Book of Abstract Algebra by Charles C. Pinter
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
Calculus
Linear Algebra
Differential Equations
Number Theory
Proof-Writing
Analysis
Complex Analysis
Functional Analysis
Partial Differential Equations
Higher-dimensional Calculus and Differential Geometry
Abstract Algebra
Geometry
Topology
Set Theory and Logic
Combinatorics / Discrete Math
Graph Theory
P. S., if you Google search any of the topics above, you are likely to find many resources. You can find a lot of lecture notes by searching, say, "real analysis lecture notes filetype:pdf site:.edu"
I started from scratch on the formal CS side, with an emphasis on program analysis, and taught myself the following starting from 2007. If you're in the United States, I recommend BookFinder to save money buying these things used.
On the CS side:
On the math side, I was advantaged in that I did my undergraduate degree in the subject. Here's what I can recommend, given five years' worth of hindsight studying program analysis:
Final bit of advice: you'll notice that I heavily stuck to textbooks and Ph.D. theses in the above list. I find that jumping straight into the research literature without a foundational grounding is perhaps the most ill-advised mistake one can make intellectually. To whatever extent that what you're interested in is systematized -- that is, covered in a textbook or thesis already, you should read it before digging into the research literature. Otherwise, you'll be the proverbial blind man with the elephant, groping around in the dark, getting bits and pieces of the picture without understanding how it all forms a cohesive whole. I made that mistake and it cost me a lot of time; don't do the same.
For real analysis I really enjoyed Understanding Analysis for how clear the material was presented for a first course. For abstract algebra I found A book of abstract algebra to be very concise and easy to read for a first course. Those two textbooks were a lifesaver for me since I had a hard time with those two courses using the notes and textbook for the class. We were taught out of rudin and dummit and foote as mainly a reference book and had to rely on notes primarily but those two texts were incredibly helpful to understand the material.
​
If any undergrads are struggling with those two courses I would highly recommend you check out those two textbooks. They are by far the easiest introduction to those two fields I have found. I also like that you can find solutions to all the exercises so it makes them very valuable for self study also. Both books also have a reasonable amount of excises so that you can in theory do nearly every problem in the book which is also nice compared to standard texts with way too many exercises to realistically go through.
It should also be noted that simpler subjects and introductory texts tend to be common knowledge to the point that citation is often not needed. You don't need to cite that water is wet, not even on Wikipedia.
Journals and modern texts about modern subjects tend to be very well cited, because they're building heavily on other sources of information.
When you don't do this, you have to have an enormous amount of backgrounding. For instance, check out this algebra text. It's 944 pages because it doesn't tend to cite much of exposition and instead states it all directly. It includes an enormous amount of information -- it's meant to be used as fundamental education material. It's not just high level conclusions that could fit in 20-50 pages.
So the amount of citation depends a great deal on the purpose of the text and how close it is to common knowledge. However, Anita's criticism is clearly not common knowledge because nobody but her sees it the way she does. Therefore, she should be explaining how she comes to her conclusions, and citing information. She should also be citing the direct quotes she uses, because it's plagiarism otherwise (and we have huge volumes of evidence that she outright plagiarizes a great deal). Plagiarism in academia is something that ends your career.
I was in the same position as you in high school (and am finishing my math major this semester). Calculus is not "math" in the sense you're referring to it, which is pure mathematics, without application, just theory and logic. Calculus, as it is taught in high school, is taught as a tool, not as a theory. It is boring, tedious, and has no aesthetic appeal because it is largely taught as rote memorization.
Don't let this bad experience kill your enthusiasm. I'm not sure what specifically to recommend to you to perk your enthusiasm, but what I did in high school was just click around Wikipedia entries. A lot of them are written in layman enough terms to give you a glimpse and you inspire your interest. For example, I remember being intrigued by the Fibonacci series and how, regardless of the starting terms, the ratio between the (n-1)th and nth terms approaches the golden ratio; maybe look at the proof of that to get an idea of what math is beyond high school calculus. I remember the Riemann hypothesis was something that intrigued me, as well as Fermat's Last Theorem, which was finally proved in the 90s by Andrew Wiles (~350 years after Fermat suggested the theorem). (Note: you won't be able to understand the math behind either, but, again, you can get a glimpse of what math is and find a direction you'd like to work in).
Another thing that I wish someone had told me when I was in your position is that there is a lot of legwork to do before you start reaching the level of mathematics that is truly aesthetically appealing. Mathematics, being purely based on logic, requires very stringent fundamental definitions and techniques to be developed first, and early. Take a look at axiomatic set theory as an example of this. Axiomatic set theory may bore you, or it may become one of your interests. The concept and definition of a set is the foundation for mathematics, but even something that seems as simple as this (at first glance) is difficult to do. Take a look at Russell's paradox. Incidentally, that is another subject that captured my interest before college. (Another is Godel's incompleteness theorem, again, beyond your or my understanding at the moment, but so interesting!)
In brief, accept that math is taught terribly in high school, grunt through the semester, and try to read farther ahead, on your own time, to kindle further interest.
As an undergrad, I don't believe I yet have the hindsight to recommend good books for an aspiring math major (there are plenty of more knowledgeable and experienced Redditors who could do that for you), but here is a list of topics that are required for my undergrad math degree, with links to the books that my school uses:
And a couple electives:
And a couple books I invested in that are more advanced than the undergrad level, which I am working through and enjoy:
Lastly, if you don't want to spend hundreds of dollars on books that you might not end up using in college, take a look at Dover publications (just search "Dover" on Amazon). They tend to publish good books in paperback for very cheap ($5-$20, sometimes up to $40 but not often) that I read on my own time while trying to bear high school calculus. They are still on my shelf and still get use.
My main hobby is reading textbooks, so I decided to go beyond the scope of the question posed. I took a look at what I have on my shelves in order to recommend particularly good or standard books that I think could characterize large portions of an undergraduate degree and perhaps the beginnings of a graduate degree in the main fields that interest me, plus some personal favorites.
Neuroscience: Theoretical Neuroscience is a good book for the field of that name, though it does require background knowledge in neuroscience (for which, as others mentioned, Kandel's text is excellent, not to mention that it alone can cover the majority of an undergraduate degree in neuroscience if corequisite classes such as biology and chemistry are momentarily ignored) and in differential equations. Neurobiology of Learning and Memory and Cognitive Neuroscience and Neuropsychology were used in my classes on cognition and learning/memory and I enjoyed both; though they tend to choose breadth over depth, all references are research papers and thus one can easily choose to go more in depth in any relevant topics by consulting these books' bibliographies.
General chemistry, organic chemistry/synthesis: I liked Linus Pauling's General Chemistry more than whatever my school gave us for general chemistry. I liked this undergraduate organic chemistry book, though I should say that I have little exposure to other organic chemistry books, and I found Protective Groups in Organic Synthesis to be very informative and useful. Unfortunately, I didn't have time to take instrumental/analytical/inorganic/physical chemistry and so have no idea what to recommend there.
Biochemistry: Lehninger is the standard text, though it's rather expensive. I have limited exposure here.
Mathematics: When I was younger (i.e. before having learned calculus), I found the four-volume The World of Mathematics great for introducing me to a lot of new concepts and branches of mathematics and for inspiring interest; I would strongly recommend this collection to anyone interested in mathematics and especially to people considering choosing to major in math as an undergrad. I found the trio of Spivak's Calculus (which Amazon says is now unfortunately out of print), Stewart's Calculus (standard text), and Kline's Calculus: An Intuitive and Physical Approach to be a good combination of rigor, practical application, and physical intuition, respectively, for calculus. My school used Marsden and Hoffman's Elementary Classical Analysis for introductory analysis (which is the field that develops and proves the calculus taught in high school), but I liked Rudin's Principles of Mathematical Analysis (nicknamed "Baby Rudin") better. I haven't worked my way though Munkres' Topology yet, but it's great so far and is often recommended as a standard beginning toplogy text. I haven't found books on differential equations or on linear algebra that I've really liked. I randomly came across Quine's Set Theory and its Logic, which I thought was an excellent introduction to set theory. Russell and Whitehead's Principia Mathematica is a very famous text, but I haven't gotten hold of a copy yet. Lang's Algebra is an excellent abstract algebra textbook, though it's rather sophisticated and I've gotten through only a small portion of it as I don't plan on getting a PhD in that subject.
Computer Science: For artificial intelligence and related areas, Russell and Norvig's Artificial Intelligence: A Modern Approach's text is a standard and good text, and I also liked Introduction to Information Retrieval (which is available online by chapter and entirely). For processor design, I found Computer Organization and Design to be a good introduction. I don't have any recommendations for specific programming languages as I find self-teaching to be most important there, nor do I know of any data structures books that I found to be memorable (not that I've really looked, given the wealth of information online). Knuth's The Art of Computer Programming is considered to be a gold standard text for algorithms, but I haven't secured a copy yet.
Physics: For basic undergraduate physics (mechanics, e&m, and a smattering of other subjects), I liked Fundamentals of Physics. I liked Rindler's Essential Relativity and Messiah's Quantum Mechanics much better than whatever books my school used. I appreciated the exposition and style of Rindler's text. I understand that some of the later chapters of Messiah's text are now obsolete, but the rest of the book is good enough for you to not need to reference many other books. I have little exposure to books on other areas of physics and am sure that there are many others in this subreddit that can give excellent recommendations.
Other: I liked Early Theories of the Universe to be good light historical reading. I also think that everyone should read Kuhn's The Structure of Scientific Revolutions.
This is from Paolo Aluffi's excellent Algebra: Chapter 0, which uses categories as a unifying theme.
A groupoid is a small category in which every morphism is an isomorphism. An automorphism of an object A of a category C is an isomorphism from A to itself. The set of automorphisms of A is denoted Aut_C(A).
Edit: added that groupoids are small categories (thank you cromonolith)
Types and Programming Languages by Benjamin Pierce covers type theory, and systems of type inference that we can have, and the ones we can't and why.
Pearls of Functional Algorithm Design by Richard Bird covers how to think equationally about code. It is probably the best guide out there on how to "think" like a Haskeller. Not directly about a discipline of math you can apply, but the mindset is invaluable.
Wadler's original papers on monads are probably when they finally clicked for me.
The original Idiom paper is also a golden resource for understanding the motivation behind applicatives.
Jeremy Gibbons' The Essence of the Iterator Pattern motivates
Traversable, which so nicely encapsulates what folks meant bymapMover the years.Uustalu and Vene's The Essence of Dataflow Programming captures a first glimmer of how and why you might want to use a comonad, but it can be fairly hard reading.
Awodey's Category Theory is probably the best general purpose category theory text book.
For folks weak on the math side Lawvere and Schanuel's Conceptual Mathematics can be used to bootstrap up to Awodey and provides a lot of drill for the areas it covers.
Dan Piponi's blog is excellent and largely set the tone for my own explorations into Haskell.
For lenses the material is a bit more sparse. The best theoretical work in this space I can point you to is by Mike Johnson and Bob Rosebrugh. (Pretty much anything in the last few papers linked at Michael's publication page at Macquarie will do to get started). I have a video out there as well from New York Haskell. SPJ has a much more gentle introduction on Skills Matter's website. You need to signup there to watch it though.
For comonads you may get some benefit out of my site comonad.com and the stuff I have up on FP Complete, but you'll need to dig back a ways.
Besides the Napkin Project I mentioned, which is a genuinely good resource? I got a coordinate-free treatment of linear algebra in my school's prelim. abstract algebra course. We used Dummit and Foote, which must be prescribed by law somewhere because I haven't yet seen a single department not use it. However, in reviewing abstract algebra I instead used Hungerford, which I definitely prefer for its brevity. But really, you can pick any graduate intro algebra text and it should teach this stuff.
> Mathematical Logic
It's not exactly Math Logic, just a bunch of techniques mathematicians use. Math Logic is an actual area of study. Similarly, actual Set Theory and Proof Theory are different from the small set of techniques that most mathematicians use.
Also, looks like you have chosen mostly old, but very popular books. While studying out of these books, keep looking for other books. Just because the book was once popular at a school, doesn't mean it is appropriate for your situation. Every year there are new (and quite frankly) pedagogically better books published. Look through them.
Here's how you find newer books. Go to Amazon. In the search field, choose "Books" and enter whatever term that interests you. Say, "mathematical proofs". Amazon will come up with a bunch of books. First, sort by relevance. That will give you an idea of what's currently popular. Check every single one of them. You'll find hidden jewels no one talks about. Then sort by publication date. That way you'll find newer books - some that haven't even been published yet. If you change the search term even slightly Amazon will come up with completely different batch of books. Also, search for books on Springer, Cambridge Press, MIT Press, MAA and the like. They usually house really cool new titles. Here are a couple of upcoming titles that might be of interest to you: An Illustrative Introduction to Modern Analysis by Katzourakis/Varvarouka, Understanding Topology by Shaun Ault. I bet these books will be far more pedagogically sound as compared to the dry-ass, boring compendium of facts like the books by Rudin.
If you want to learn how to do routine proofs, there are about one million titles out there. Also, note books titled Discrete Math are the best for learning how to do proofs. You get to learn techniques that are not covered in, say, How to Prove It by Velleman. My favorites are the books by Susanna Epp, Edward Scheinerman and Ralph Grimaldi. Also, note a lot of intro to proofs books cover much more than the bare minimum of How to Prove It by Velleman. For example, Math Proofs by Chartrand et al has sections about doing Analysis, Group Theory, Topology, Number Theory proofs. A lot of proof books do not cover proofs from Analysis, so lately a glut of new books that cover that area hit the market. For example, Intro to Proof Through Real Analysis by Madden/Aubrey, Analysis Lifesaver by Grinberg(Some of the reviewers are complaining that this book doesn't have enough material which is ridiculous because this book tackles some ugly topological stuff like compactness in the most general way head-on as opposed to most into Real Analysis books that simply shy away from it), Writing Proofs in Analysis by Kane, How to Think About Analysis by Alcock etc.
Here is a list of extremely gentle titles: Discovering Group Theory by Barnard/Neil, A Friendly Introduction to Group Theory by Nash, Abstract Algebra: A Student-Friendly Approach by the Dos Reis, Elementary Number Theory by Koshy, Undergraduate Topology: A Working Textbook by McClusckey/McMaster, Linear Algebra: Step by Step by Singh (This one is every bit as good as Axler, just a bit less pretentious, contains more examples and much more accessible), Analysis: With an Introduction to Proof by Lay, Vector Calculus, Linear Algebra, and Differential Forms by Hubbard & Hubbard, etc
This only scratches the surface of what's out there. For example, there are books dedicated to doing proofs in Computer Science(for example, Fundamental Proof Methods in Computer Science by Arkoudas/Musser, Practical Analysis of Algorithms by Vrajitorou/Knight, Probability and Computing by Mizenmacher/Upfal), Category Theory etc. The point is to keep looking. There's always something better just around the corner. You don't have to confine yourself to books someone(some people) declared the "it" book at some point in time.
Last, but not least, if you are poor, peruse Libgen.
I would guess that career prospects are a little worse than CS for undergrad degrees, but since my main concern is where a phd in math will take me, you should get a second opinion on that.
Something to keep in mind is that "higher" math (the kind most students start to see around junior level) is in many ways very different from the stuff before. I hated calculus and doing calculations in general, and was pursuing a math minor because I thought it might help with job prospects, but when I got to the more abstract stuff, I loved it. It's easily possible that you'll enjoy both, I'm just pointing out that enjoying one doesn't necessarily imply enjoying the other. It's also worth noting that making the transition is not easy for most of us, and that if you struggle a lot when you first have to focus a lot of time on proving things, it shouldn't be taken as a signal to give up if you enjoy the material.
This wouldn't be necessary, but if you like, here are some books on abstract math topics that are aimed towards beginners you could look into to get a basic idea of what more abstract math is like:
Different mathematicians gravitate towards different subjects, so it's not easy to predict which you would enjoy more. I'm recommending these five because they were personally helpful to me a few years ago and I've read them in full, not because I don't think anyone can suggest better. And of course, you could just jump right into coursework like how most of us start. Best of luck!
(edit: can't count and thought five was four)
You are in a very special position right now where many interesing fields of mathematics are suddenly accessible to you. There are many directions you could head. If your experience is limited to calculus, some of these may look very strange indeed, and perhaps that is enticing. That was certainly the case for me.
Here are a few subject areas in which you may be interested. I'll link you to Dover books on the topics, which are always cheap and generally good.
Basically, don't limit yourself to the track you see before you. Explore and enjoy.
"Analysis" just means proof-based calc, at least at the undergraduate level.
It sounds like the main thing you're missing is abstract algebra, i.e. the contents of this book. If you feel like you remember how to prove things, you could try to either self-study the material (potentially difficult) or enroll in a two-semester abstract algebra course through Ohio State.
If you're not comfortable with proofs, you might want to start by taking a introductory proofs course to refresh yourself, or maybe something like linear algebra or another relatively accessible proofs-based math course such as number theory, graph theory, or combinatorics.
It might also be worth learning a little bit about point-set topology, though not all Masters programs will assume this.
I found this in Galois Theory by Rotman
Rather than try to explain the proof here, I scanned in three pages from the book - I think this is allowed under the fair use clause of the copyright act. You can find the relevant pages here.
The pages are numbers 43, 66 and 67.
Page 43 has two exercises - the first and last - that give you hints on how to prove x^4 - 10x^2 +1 is irreducible over Q. (I don't think this part is difficult.)
The next two pages give a proof due to Gerald Janusz of the actual reducibility over Zp. The proof is not conceptually difficult but it is, in my not very humble opinion, messy.
Aleph_not is correct.
In the end, the original polynomial factors into two quadratics in Zp.
The hen-scratching on the outside is mine.
I hope this helps.
Please, simply disregard everything below if the info is old news to you.
------------
Algebraic geometry requires the knowledge of commutative algebra which requires the knowledge of some basic abstract algebra (consists of vector spaces, groups, rings, modules and the whole nine yards). There are many books written on abstract algebra like those of Dummit&Foote, Artin, Herstein, Aluffi, Lang, Jacobson, Hungerford, MacLane/Birkhoff etc. There are a million much more elementary intros out there, though. Some of them are:
Discovering Group Theory: A Transition to Advanced Mathematics by Barnard/Neil
A Friendly Introduction to Group Theory by Nash
Abstract Algebra: A Student-Friendly Approach by the Dos Reis
Numbers and Symmetry: An Introduction to Algebra by Johnston/Richman
Rings and Factorization by Sharpe
Linear Algebra: Step by Step by Singh
As far as DE go, you probably want to see them done rigorously first. I think the books you are looking for are titled something along the lines of "Analysis on Manifolds". There are famous books on the subject by Sternberg, Spivak, Munkres etc. If you don't know basic real analysis, these books will be brutal. Some elementary analysis and topology books are:
Understanding Analysis by Abbot
The Real Analysis Lifesaver by Grinberg
A Course in Real Analysis by Mcdonald/Weiss
Analysis by Its History by Hirer/Wanner
Introductory Topology: Exercises and Solutions by Mortad
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:
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:
There are tons more books on abstract/modern algebra. Just search them on Amazon. Some of the famous, but less accessible ones are
Intro to Real Analysis:
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
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.
You can try something like Artin or Dummit & Foote.
Harvard’s Benedict Gross gave a course using Artin’s textbook with lectures available on youtube.
Whenever I explain algebra (rings or groups) I start with the issue of defining what a "number" is. This leads us to the ideas of addition, multiplication, etc. The basic idea that I try to convey is that a "number" is a very general concept and that instead of trying to define a "number," we instead define how certain kinds of "numbers" behave and interact. This gives us an algebra. I give examples of groups and rings to point out the similarities between seemingly different algebraic structures.
A lot of people have pointed out that groups can be understood in the context of symmetries. My first course in algebra actually took a different route. We were taught groups after we were taught rings. It isn't traditional, but Hungerford does a pretty good job and it appeals to an early math student's intuition.
ps: I use the term "number" (with quotes) very loosely. I know that to many people the word number is synonymous with integer but to a layman a "number" is really just something with which we "do math."
Most of the books on this list that I'm familiar with are really good. And I may use the list's recommendations for a few areas that I'm unfamiliar with.
But what do people think of the abstract algebra recommendations? I've tried to work with both Gallian, and Dummit & Foote and I found both of them to be boring (each in their own unique way). I'd be happy to elaborate on this if people strongly disagree.
For abstract algebra, my favorite introductory book is Elements of Abstract Algebra by Allan Clark. You can't beat the price, and it's just fun to read. It's organized into manageable sections, with just the right amount of relevant problems for each section. The problems are pitched just right for a person who's had some--but not a lot--of experience with proofs (I think this is the case for most students of intro abstract algebra).
First and foremost discard the idea of contributing to an area of research mathematics. It's not that it's impossible for you to do so, but it's not a good goal to set. It's best for you to try and explore a field of mathematics that interests you to learn more about it. After all this is what mathematical research actually is, we have questions that we would like to know the answers to so we figure them out. It is also a much more attainable goal, whether the material is new to the mathematical community or not you will have learned something new.
Second, if you really want to try and get to the forefront of mathematical understanding, expect to put in about a year or two at minimum to get there, and that's only if you pick a new or obscure field whose frontier is not as far removed from where you are. Fields like combinatorics and graph theory also have frontiers that are easily approachable for beginners.
If you're really dead set on algebra I would put forth two different fields. The first is combinatorial group theory, which is a bit older, but a lot of people have vacated the field. The classic text on that is Combinatorial Group Theory by Magnus. I don't know much about the status of open questions in the field, but I do know that combinatorial methods crop up in solutions to open problems in group theory all the time. You might be able to get the background needed to understand and work though most of that book. You'd need at minimum a solid understanding of presentations of groups and a bit of knowledge about combinatorics.
The field that most mathematicians have moved into after working in CGT is geometric group theory. It's a relatively new field with lots of interesting accessible open questions, but requires a bit of background knowledge of metric topology. There aren't any English language classical texts that are approachable at your level, but these notes by Alessandro Sisto are a quite good introduction. (The ending of the notes tapers off with fewer and fewer details, I wouldn't read past page 64 or 65). There are also some errors you have to catch in his proofs, and statements of theorems throughout.
This is all assuming that you've read an introductory book to abstract algebra and done all of the problems, such as Contemporary Abstract Algebra and actually know all the basics solidly.
I don't know that I can help because everyone learns so differently, but I'll say a couple of things. (I'm going to warn you right now that I'm kind of tired and I didn't proofread very much.)
I think my best advice about (1) is to play with small examples. If you're asked to prove something about all real vector spaces then look at R, R\^2, R\^3, and see if you can understand what's going on in those situations. If you're asked to prove something about all differential functions, pick one and play with the statement in that case. Once you get the small examples, move onto bigger examples. What is the craziest, wildest differential function of which you can think? What vector space really pushes the boundaries of the hypotheses that you're given? It's not a fool-proof plan, but I find that working with examples is the single most helpful thing I can do when I'm working on a hard proof.
Also, about number (1), this isn't very concrete advice, but I like to tell my students to try to understand a statement before attempting a proof. Proofs are a linear line of logic between some assumptions and a conclusion. Our brains, however, aren't always compatible with the rigor and linearity of a proof; it can be hard to see everything that's going on from step seven in a proof. If you can really, *really* convince yourself of something and understand it then your intuition will guide you much better during the proof. It can be easy to fall into a trap of saying "this is on my homework so it must be true" and then diving into a proof but it's better to think critically about the statement first. Along the lines of the previous paragraph, an attempt to construct a counterexample (even if you know something is definitely true) can be really helpful.
As to part (2), all I can really say it that it's very frustrating and challenging. I'm about to graduate with my PhD and I still struggle with learning new math from a textbook or paper. You're not alone in that. It really helps to find the right book but it's also very hard. Some books are excellent because they are packed full of content and make a good reference (sort of like an encyclopedia) and other books are excellent because they have great exercises and make a good companion to a course. Rudin's analysis book is notorious for this; the exercises are fantastic which means it gets used in the classroom a lot, but the exposition is pretty brief (and, in my opinion, quite poor at times) which makes it difficult to use when self-teaching. Unfortunately, the best way to sell the most books is to write a book that is meant to be used in a class. It can be really hard to find books that are good for learning on your own. (The best example I know is Gallian's Abstract Algebra book.)
My only real advice for finding good references is to ask your teachers. I'll also say that it's usually easier to learn math in small chunks. It can be really daunting to decide "I'm going to learn all of Real Analysis" but deciding "I want to learn how the real numbers are constructed" is much more doable and it likely won't feel quite as discouraging when you get stuck. Pick a specific topic that interests you (ideally somewhat close to what you already know), and try to understand that one topic. Find videos, find websites, ask a teacher, and look into references. A complaint I often get about this technique is "I don't really know much about Topology so I don't know what interests me" but a good place to start can be "I'd like to understand what the area of Topology is really trying to study."
So, anyway, I've rambled long enough now. Good luck and try to stick with it!
I think Dummit and Foote would be too difficult. It's almost always used for a second course in algebra, not a first. I'd try http://www.amazon.com/Abstract-Algebra-Introduction-Thomas-Hungerford/dp/0030105595 first, or maybe Artin's "Algebra."
My first introduction to group theory/abstract algebra came from this book by Fraleigh (for God's sake don't pay $120 for it). As I remember, it started pretty basic so take a look.
I'm also a big fan of Dummit & Foote as AngelTC mentioned.
I studied with this book on abstract. It's authoritative and brutal.
It's hard to give an objective answer, because any sufficiently advanced book will be bound to not appeal to everyone.
You probably want Daddy Rudin for real analysis and Dummit & Foote for general abstract algebra.
Mac Lane for category theory, of course.
I think people would agree on Hartshorne as the algebraic geometry reference.
Spanier used to be the definitive algebraic topology reference. It's hard to actually use it as a reference because of the density and generality with which it's written.
Spivak for differential geometry.
Rotman is the group theory book for people who like group theory.
As a physics person, I must have a copy of Fulton & Harris.
You need some grounding in foundational topics like Propositional Logic, Proofs, Sets and Functions for higher math. If you've seen some of that in your Discrete Math class, you can jump straight into Abstract Algebra, Rigorous Linear Algebra (if you know some LA) and even Real Analysis. If thats not the case, the most expository and clearly written book on the above topics I have ever seen is Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers.
Some user friendly books on Real Analysis:
Some user friendly books on Linear/Abstract Algebra:
Topology(even high school students can manage the first two titles):
Some transitional books:
Plus many more- just scour your local library and the internet.
Good Luck, Dude/Dudette.
I actually bought http://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178
On chapter three now :D
Have you read A Book of Abstract Algebra by Charles Pinter? https://www.amazon.co.uk/d/cka/Abstract-Algebra-Dover-Books-Mathematics-Charles-Pinter/0486474178
the "vanilla" books are IMO quite boring to read - especially when you don't know more than Set/Functions.
but I really enjoy P. Aluffi; Algebra: Chapter 0 that builds up algebra using CT from the go instead of after all the work
----
remark I don't know if this will really help you understanding Haskell (I doubt it a bit) but it's a worthy intellectual endeavor all in itself and you can put on a knowing smile whenever you hear those horrible words after
Any introductory abstract algebra book will have the basics of of rings, ideals, and quotient rings, as well as a few other things.
My class on intro to group theory used Gallian's Contemporary Abstract Algebra, which I'm a pretty big fan of as an introduction. It's gentle and doesn't rush into things, but has a large amount of exercises, some of which will really stretch your understanding.
If you want something a little harder, but a little deeper, Artin's Algebra is very popular, and for very good reason. It'll help you develop your group theory knowledge as well.
> Hungerford
I assume you're talking about the GTM book that everyone thinks of when they say Hungerford. I'm currently teaching a class out of his other, more elementary abstract algebra textbook:
https://www.amazon.com/Abstract-Algebra-Introduction-Thomas-Hungerford/dp/1111569622
That might be more what you're looking for. Polynomial rings are discussed in chapters 4 and 5. Chapter 4 does the basic division algorithm/unique factorization/irreduciblity criteria from a very hands on perspective. Chapter 5 is polynomial modular arithmetic and F[x]/(p(x)).
Ideals and quotient rings in general aren't introduced until Chapter 6, Euclidean domains/UFDs/PIDs don't come up until chapter 10. Homomorphisms show up briefly in Section 3.3, but they aren't really used much in chapters 4 and 5.
When I first took abstract algebra a couple years ago, we worked out of Fraleigh's A First Course in Abstract Algebra. My classmates and I thoroughly enjoyed it. Well written, well paced, and all around an enlightening introductory read about my most favorite field of math :)
I think it's perfectly tractable for any interested student with a good command of algebra.
Edit: Oh I misread the question. If he's already gone through these elementary parts of abstract algebra, that's about the entire undergraduate coursework I had. The one quarter of graduate algebra I ended up taking went over the orbit-stabilizer theorem, free groups, then dove right into module theory and homologies. We worked out of Artin and Rotman.
Actually now that I think about it, maybe module theory would be a good stepping off point from these parts. I know it gave me a cool new view and appreciation of linear algebra
If you're doing both applied and pure abstract algebra rather than elementary algebra, then you'll probably need to learn to write proofs for the pure side. I recommend Numbers, Groups, and Codes by J. F. Humphreys for an introduction to the basics and to some applied abstract algebra. If you need more work on proofs, the free Book of Proofs can help, and Fraleigh's A First Course in Abstract Algebra is a good book for pure abstract algebra. If you want something more advanced, I recommend the massive Abstract Algebra by Dummit and Foote.
Learn Algebra, arguably the most important math subject (of course, I may be biased). Dummit and Foote is a fantastic intro if you have proof experience etc.
I'm going to shamelessly plug this book which I consider to be one of my favorite books ever. For the price it is definitely worth keeping a copy and reading it on the side if you're learning abstract algebra for the first time and it reads like a novel. It's definitely a small treasure I feel I discovered.
there's a lot going on here, so i'll try to take it a few steps at a time.
> how many REAL operators do we have?
you might be careful about your language here, as the word "real" has implications in the world of mathematics to mean "takes values in the real numbers", i.e., is non-complex. also, "real" in the normal sense of real or fake doesn't have a lot of meaning in mathematics. a better question might be "how many unique operators do we have?", but even that isn't quite good enough. you need to define context. a blanket answer to your question is that there are uncountably infinite amount of operators in mathematics that take all kinds of forms: linear operators, functional operators, binary operators, etc.
> taking a number to the power of another is just defined in terms of multiplication
similar to /u/theowoll's response, how would you define 2^(4.18492) in terms of multiplication? i know you're basing this question off of the interesting fact that 2^1 = 2, 2^2 = 2 2, 2^3 = 2 2 * 2, etc. and similarly for other certain classes of numbers, but how do you multiply 2 by itself 4.18492 times? it gets even more tricky to think of exponents like this if the base and power are non-rational (4.18492=418492/100000 is rational). what about the power of e^X, where e is the normal exponential and X is a matrix? take a look at wikipedia's article on exponentiation to see what a can of worms this discussion opens.
> So am I just plain wrong about all this, or there is some truth to it?
although there is a lot of incorrect things in your description when you consider general classes of "things you can multiply and add", what you are sort of getting at is what the theory of abstract algebra covers. in such a theory, it explores what it means to add, multiply, have inverses, etc. for varying collections of things called groups, rings, fields, vector spaces, modules, etc. and the relationships and properties of such things. you might take a look at a book of abstract algebra by charles pinter. you should be able to follow it, as it is an excellent book.
A book of abstract algebra by Charles Pinter is the best math book I've ever read in terms of readability, I think. The first chapter is an essay on the history of algebra and the book is worth it just for this chapter.
Try Pinter. If you think it is too simple for you go for Aluffi.
Galois Theory
This is a compilation of what I gathered from reading on the internet about self-learning higher maths, I haven't come close to reading all this books or watching all this lectures, still I hope it helps you.
General Stuff:
The books here deal with large parts of mathematics and are good to guide you through it all, but I recommend supplementing them with other books.
Linear Algebra: An extremelly versatile branch of Mathematics that can be applied to almost anything, also the first "real math" class in most universities.
Calculus: The first mathematics course in most Colleges, deals with how functions change and has many applications, besides it's a doorway to Analysis.
Real Analysis: More formalized calculus and math in general, one of the building blocks of modern mathematics.
Abstract Algebra: One of the most important, and in my opinion fun, subjects in mathematics. Deals with algebraic structures, which are roughly sets with operations and properties of this operations.
There are many other beautiful fields in math full of online resources, like Number Theory and Combinatorics, that I would like to put recommendations here, but it is quite late where I live and I learned those in weirder ways (through olympiad classes and problems), so I don't think I can help you with them, still you should do some research on this sub to get good recommendations on this topics and use the General books as guides.
/u/LengthContracted this is a good book, as is Daphne Kollers book on PGMs as well as the associated course http://pgm.stanford.edu
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.
http://www.amazon.com/Book-Abstract-Algebra-Edition-Mathematics/dp/0486474178/ref=sr_1_1?ie=UTF8&qid=1394386195&sr=8-1&keywords=pinter+abstract+algebra)
We used the Dover textbook by Pinter. It's my favorite math textbook ever, the writing was just so clear, and even entertaining and funny. We had a good professor too.
Having used Artin, I can tell you that it's a good reference but a terrible read.
I recommend Gallian for it's overall readability (http://www.amazon.com/Contemporary-Abstract-Algebra-Joseph-Gallian/dp/0547165099), though keep in mind this is geared for undergraduates. Also, look into older editions; new one is expensive
Reading through Algebra: Chapter 0
The process by which one acquires procedural knowledge is trial-and-error, but when that same process is deployed to learn declarative knowledge, it is often derided as "teaching to the test," as if that were somehow inferior to learning through other methods. Yet, when learning how to surf, the goal of learning how to balance on a surf board is not just to do so for its own sake, but rather to move on to more difficult things that rely on being able to balance.
Similarly, in designing good tests, (and here I focus on mathematics in particular), there is a lot of room for pointing out patterns, building up methods for solving problems, and invoking the common sense that is often put to the wayside when students are confronted with a mass of meaningless symbols that are pushed around in seemingly arbitrary ways.
Furthermore, once the upfront cost of creating questions is paid, the yield is more learning and indeed, learning that is personalized to where each student is at on the mastery curve. Lectures can only cover so much in a class period and are only genuinely useful (if that) to those in the middle of the distribution who have the necessary background knowledge and don't already know the material.
One text that serves as an example of this approach is an Abstract Algebra text designed for independent study.
When I was in undergrad we used Abstract Algebra by Beachy and Blair.
If you are asking for classics, in Algebra, for example, there are(different levels of difficulty):
Basic Algebra by Jacobson
Algebra by Lang
Algebra by MacLane/Birkhoff
Algebra by Herstein
Algebra by Artin
etc
But there are other books that are "essential" to modern readers:
Chapter 0 by Aluffi
Basic Algebra by Knapp
Algebra by Dummit/Foot
While /u/PlasticPrison gave an exhaustive list, I would only add one more, which at least in the US is considered a standard on this subject: Lie Algebras in Particle Physics by Georgi.
Sure. If you're looking for something with absolutely no handholding whatsoever, here's what I use as a reference (not that I do much anymore): Howard Georgi's superb book. Though I wouldn't buy it without spending some time with a library copy first.
8 to 12 hours is really not that much, but it should be enough to learn something interesting! I would start with category theory if you can. I liked Emily Riehl's categories in context for an intro, but it will go a little slow for how little time you have to learn the basics. Maybe the first chapter of Algebra: Chapter 0 by Aleffi? [EDIT: you might want to find a "reasonably priced" pdf version of this book if you do decide to use it -- it's pretty expensive] If you can get through that, and understand a little about how types fit into the picture, you should be able to present the basic idea behind curry-howard-lambek. IIRC you do not need functors or natural transformations ("higher level" categorical concepts), as important as they usually are, to get through this topic; Aleffi doesn't go over them in his very first intro to categories which is why I'm recommending him. /u/VFB1210 has some very good recommendations above as well.
I am trying to think of a better introduction to type theory than HoTT -- if you can learn about types without getting infinity categories and homotopy equivalence mixed up in them, I would. Type theory is actually pretty cool and sleek.
Here's a selection of intro-to-type theory resources I found:
Programming in Martin-Löf's Type Theory is
pretty long, but you can probably put together a mini-course as follows: read chapters 1 & 2 quickly, skim 3, and then read 19 and 20.
The lecture notes from Paul Levy's mini-course on the typed lambda calculus form a pretty compact resource, but I'm not sure this will be super useful to you right now -- keep it in mind but don't start off with it. Since it is in lecture-note style it is also pretty hard to keep up with if you don't already kind of know what he's talking about.
Constable's Naïve Computational Type Theory seems to be different from the usual intro to types -- it's done in the style of the old Naive Set Theory text, which means you're supposed to be sort of guided intuitively into knowing how types work. It looks like the intuition all comes from programming, and if you know something functional and hopefully strongly typed (OCaml, SML, Haskell, or Lisp come to mind) you will probably get the most out of it. I think that's true about type theory in general, actually.
PFPL by Bob Harper is probably a stretch -- you won't find it useful right at the moment, but if you want to spend 2 semesters really getting to know how type theory encapsulates pretty much any modern programming paradigm (typed languages, "untyped" languages, parallel execution, concurrency, etc.) this book is top-tier. The preview edition doesn't have everything from the whole book but is a pretty big portion of it.
You can try Hungerford's Introduction to Abstract Algebra or Schaum's Outline of Modern Abstract Algebra. The former is very clearly written and great for self-study, but provides a thorough introductory course that may be much more intensive than you are interested in. The latter covers (in a less rigorous fashion) the principles of linear algebra, number systems and modern algebra. It is exceptionally easy to read and understand, but is not much of a textbook, and lacks most of the depth of the first text. To be honest, it is more of a guide to the foundations of algebra.
If you are intent on studying abstract algebra intensively, I would recommend the first book. However, it appears to me that the second text would definitely be more up your alley, as a stronger foundation in algebra will be invaluable when your student is ready to pursue this subject in a more comprehensive way.
His book on Category Theory is also really good if anyone is interested in learning more. I'm reading it now myself.
If you've already read and done all the exercises in Hungerford, why would you be reading Dummit and Foote?
EDIT: Ah, never mind, I thought you meant Algebra by Hungerford. To wit, if you want a supplementary text at the next level up from D&F, you could try Hungerford, Lang (the big daddy of all algebra books), or my personal favorite, Isaacs.
I'm not sure if you mean abstract algebra or linear algebra, but if it's the former, I liked Herstein's Topics in Algebra. There's also Abstract Algebra by Herstein as well, which I think is a cheaper slimmed down version. I used these books for self study and found Herstein's exposition, particularly at the beginning of chapters, very helpful. He isn't as verbose as your typical 7th edition mass market textbook author though.
For linear algebra, I hear Axler's Linear Algebra Done Right is good. I haven't read it, but I read his paper "Down with Determinants" which is, I think, written in the same style and enjoyed the alternative perspective a lot.
I'm soon starting my trek through every problem in the algebra text that Harvard's PhD prelim recommends for study:
Abstract Algebra by Dummit and Foote
I've started the first section of the first chapter, but that was only in a few hours of spare time. I'll be posting solutions by chapter soon and post my stories/insights on Hacker News. Here's section 1.1 (except the last problem, 36):
http://therobert.org/alg/1.1.pdf
Comments are appreciated. Better now than when I start the real journey. :)
Well, Hardy & Wright is the classic book for elementary stuff. It has almost everything there is to know. There is also a nice book by Melvyn Nathanson called Elementary Methods in Number Theory which I really like and would probably be my first recommendation. Beyond that, you need to decide which flavour you like. Algebraic and analytic are the big branches.
For algebraic number theory you'll need a solid grounding in commutative algebra and Galois theory - say at the level of Dummit and Foote. Lang's book is pretty classic, but maybe a tough first read. I might try Number Fields by Marcus.
For analytic number theory, I think Davenport is the best option, although Montgomery and Vaughan is also popular.
Finally, Serre (who is often deemed the best math author ever) has the classic Course in Arithmetic which contains a bit of everything.
I know that in the long run competition math won't be relevant to graduate school, but I don't think it would hurt to acquire a broader background.
That said, are there any particular texts you would recommend? For Algebra, I've heard that Dummit and Foote and Artin are standard texts. For analysis, I've heard that Baby Rudin and also apparently the Stein-Shakarchi Princeton Lectures in Analysis series are standard texts.
I agree with all the suggestions to start with How to Prove It by Velleman. It's a great start for going deeper into mathematics, for which rigor is a sine qua non.
As you seem to enjoy calculus, might I also suggest doing some introductory real analysis? For the level you seem to be at, I recommend Understanding Analysis by Abbott. It helped me bridge the gap between my calculus courses and my first analysis course, together with Velleman. (Abbott here has the advantage of being more advanced and concise than Spivak, but more gentle and detailed than baby Rudin -- two eminent texts.)
Alternatively, you can start exploring some other fascinating areas of mathematics. The suggestion to study Topology by Munkres is sound. You can also get a friendly introduction to abstract algebra by way of A Book of Abstract Algebra by Pinter.
If you're more interested in going into a field of science or engineering than math, another popular approach for advanced high schoolers to start multivariable calculus (as you are), linear algebra, and ordinary differential equations.
A Book of Abstract Algebra by Charles C. Pinter
I really enjoyed reading the book, almost reads like a novel. There is a great first chapter laying out the history of the subject and it just builds from there.
Math is essential the art pf careful reasoning and abstraction.
Do yes, definitely.
But it may be difficult at first, like training anything that’s not been worked.
Note: there are many varieties of math. I definitely recommend trying different ones.
A couple good books:
An Illustrated Theory of Numbers
Foolproof (first chapter is math history, but you can skip it to get to math)
A Book of Abstract Algebra
Also, formal logic is really fun, imk. And excellent st teaching solid thinking. I don’t know a good intro book, but I’m sure others do.
> I'm not sure what to read into before the Galois class begins.
"A Book of Abstract Algebra" by Charles Pinter
It depends on your interests. I thought the machine learning course on coursera was great. Antirez sometimes blogs about the internals of Redis on his blog, and he is a great writer. If you like math, this is the best math book I've read. Finally, you can always start contributing code to an open source project -- learn by doing!
I found Elements of Abstract Algebra to be very accessible. Some people seem to hate it though. For $7, it's not much of a risk. With that said, I have some doubt about the potential for a non-mathematician to learn abstract algebra without developing some "mathematical maturity" with easier material first.
Two great introductions are:
both are short and very well written. Some books particularly close to my heart are:
Tacking onto what /u/ReneXvv said, if you would like a more historical (and in some ways, intuitive approach) Hungerfod's Algebra: An Introduction. Again, this format is slightly different (he begins with rings and then moves onto groups, which is opposite the usual approach), but it's well-designed to get your feet wet into the topic. From there you could graduate onto the more "standard" texts, like Hungerford, Dummit & Foote, or my new favorite, Rotman (who is also slightly nonstandard in his approach and builds nicely into category theory).
I love Pinter's abstract algebra. I haven't read Stewart's book, so I don't know how it compares, but I highly recommend Pinter.
Chartrand's Introductory Graph Theory is the best intro I know of, good for self-study. (BTW, if you're interested in modern algebra, I recommend Pinter's intro. If it's not affordable or findable through a library, you'll probably be able to find it online.)
To answer your broader question, The Elements of Computing Systems: Building a Modern Computer from First Principles should give you a lot of confidence on how computers work.
If you're reading Structure and Interpretation of Computer Programs, feel free to skip any math-y examples. It's written for MIT students whose prereqs are still fresh in their minds. Here's an interesting recent interview with one of the authors, where he discusses the foundations of computer science and what a programmer might want to know. (The notion of abstraction, etc.)
There's no single book that's right for everyone: a suitable book will depend upon (1) your current background, (2) the material you want to study, (3) the level at which you want to study it (e.g., undergraduate- versus graduate-level), and (4) the "flavor" of book you prefer, so to speak. (E.g., do you want lots of worked-out examples? Plenty of exercises? Something which will be useful as a reference book later on?)
That said, here's a preliminary list of titles, many of which inevitably get recommended for requests like yours:
Good luck finding something useful!
I've got a few recommendations:
A First Course in Abstract Algebra. The importance of this subject in mathematics cannot be overstated, even if it seems very counterintuitive. Most number theory problems are solved through advanced algebra. This book examines most aspects of groups, rings, and fields, and many major applications of them. Anyone can read the first chapter, but you're going to have a very bad time if you don't get each chapter DOWN before the next one. This subject matter took me two of the hardest classes ever to get through, so don't be discouraged.
Like I said elsewhere, Rudin's Principles of Mathematical Analysis. Starting from basic set theory, it provides a thorough construction of the concept of real numbers, followed by sequences, series, single-variable calculus, multi-variable calculus, touches on standard and partial differential eqs, and VERY basic functional analysis. Again, a short but extremely dense book, anyone can do it, but not easily. Don't take shortcuts, and it will massively expand your mathematical literacy.
Neither of these requires much set theory, but if you're having problems there is this book. It is what it looks like, but the first few chapters are logic so you can probably skip them. It's an easy read and it seems to me that set theory is very similar in operation to logic.
Fraleigh is a little bit easier to wrap your head around. Get an old edition (or find it at the library), obviously.
Also, I highly recommend Herstein's Topics in Algebra. Again, try to get it from a university library.
Ok this is not going to be very original, but I'd start getting a foundation in algebra, linear algebra and analysis. My suggestions for those topics are Fraleigh, Gilbert Strang's Video Lectures (I'd suggest Heuser for learning analysis but that's german and won't help you).
I guess the most important thing to remember is that you don't have to understand everything when you read it for the first time. Try to get a feel for functions and matrices, sets and maps, etc, because you'll need those all the time.
Good Luck!
Friendly info:
"College Algebra" = Elementary Algebra.
College Level Algebra = Abstract Algebra.
Example: Undergrad Algebra book.
Example: Graduate Algebra book.
Algebra by Serge Lang.
It has a good introduction to category theory, as well as being one of the better all around resources for algebra. He can be quite terse sometimes, but it is well-referenced.
I hear that you can also obtain digital copies of this book.
If you are a newcomer to abstract algebra, you might consider using a text other than Dummit and Foote. I used baby Herstein (as opposed to big Herstein) in an undergraduate class and found it to be a good introduction.
If you can read through gallian's book, I consider dummit and foote's book (http://www.amazon.com/Abstract-Algebra-3rd-David-Dummit/dp/0471433349) as the best math textbook i've ever read. tons of examples, thorough treatment of material, and tons of exercises.
I am a master's student with interests in algebraic geometry and number theory. And I have a good collection of textbooks on various topics in these two fields. Also, as part of my undergraduate curriculum, I learnt abstract algebra from the books by Dummit-Foote, Hoffman-Kunze, Atiyah-MacDonald and James-Liebeck; analysis from the books by Bartle-Sherbert, Simmons, Conway, Bollobás and Stein-Shakarchi; topology from the books by Munkres and Hatcher; and discrete mathematics from the books by Brualdi and Clark-Holton. I also had basic courses in differential geometry and multivariable calculus but no particular textbook was followed. (Please note that none of the above-mentioned textbooks was read from cover to cover).
As you can see, I didn't learn much geometry during my past 4 years of undergraduate mathematics. In high school, I learnt a good amount of Euclidean geometry but after coming to university geometry appears very mystical to me. I keep hearing terms like hyperbolic/spherical geometry, projective geometry, differential geometry, Riemannian manifold etc. and have read general maths books on them, like the books by Hartshorne, Ueno-Shiga-Morita-Sunada and Thorpe.
I will be grateful if you could suggest a series of books on geometry (like Stein-Shakarchi's Princeton Lectures in Analysis) or a book discussing various flavours of geometry (like Dummit-Foote for algbera). I am aware that Coxeter has written a series of textbooks in geometry, and I have read Geometry Revisited in high school (which I enjoyed). If these are the ideal textbooks, then where to start? Also, what about the geometry books by Hilbert?
If you are enjoying your Calc 3 book, I highly recommend reading Topology, which provides the foundations of analysis and calculus. Two other books I would highly recommend to you would be Abstract Algebra and Introduction to Algorithms, though I suspect you're well aware of the latter.
Dummit and Foote's Abstract Algebra is an excellent book for the algebra side of things. It can be a little dense, but it's chock full of examples and is very thorough.
To help get through the first ten or so chapters, Charles Pinter's A Book of Abstract Algebra is an incredible resource. It does wonders for building up an intuition behind algebra.
I'm working through the exercises in Pinter's Abstract Algebra.
If you're not afraid of math there are some cheap introductory textbooks on topics that might be accessible:
For abstract algebra: http://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178/ref=sr_1_1?ie=UTF8&qid=1459224709&sr=8-1&keywords=book+of+abstract+algebra+edition+2nd
For Number Theory: http://www.amazon.com/Number-Theory-Dover-Books-Mathematics/dp/0486682528/ref=sr_1_1?ie=UTF8&qid=1459224741&sr=8-1&keywords=number+theory
These books have complimentary material and are accessible introductions to abstract proof based mathematics. The algebra book has all the material you need to understand why quintic equations can't be solved in general with a "quintic" formula the way quadratic equations are all solved with the quadratic formula.
The number theory book proves many classic results without hard algebra, like which numbers are the sum of two squares, etc, and has some of the identities ramanujan discovered.
For an introduction to analytic number theory, a hybrid pop/historical/textbook is : http://www.amazon.com/Gamma-Exploring-Constant-Princeton-Science/dp/0691141339/ref=sr_1_1?ie=UTF8&qid=1459225065&sr=8-1&keywords=havil+gamma
This book guides you through some deep territory in number theory and has many proofs accessible to people who remember calculus 2.
I like this abstract algebra book: A Book of Abstract Algebra
This is one of the best books of abstract algebra I've seen, very well explained, favoring clear explanations over rigor, highly recommended (take your time to read the reviews, the awesomeness of this book is real :P): http://www.amazon.com/Book-Abstract-Algebra-Edition-Mathematics/dp/0486474178/ref=sr_1_6?ie=UTF8&qid=1345229432&sr=8-6&keywords=introduction+to+abstract+algebra
On a side note, trust me, Dummit or Fraileigh are not what you want.
If you're interested in this sort of thing, I'd recommend you look at abstract algebra later. There's lots of proofs that certain constructions are impossible with it. This book is a pretty accessible introduction.
Apart from the mentioned books, some people also like Pinter's book, which is also a gentle introduction to algebra.
Specifically for group theory I can suggest these notes.
You might want to pick up a copy of Pinter and skim the first few chapters.
I would recommend getting familiar with how the cyclic groups work (they are basically clock arithmetic), how dihedral groups work (flipping and rotating polygons), and how the symmetric group works (ways you can shuffle things).
Work out the multiplication table for a handful, including the cyclic group of order 12, the symmetric group on 3 symbols, and the dihedral groups for the triangle and square.
WARNING: Don't think you need to get quick at mental multiplication in these groups. It's better you get a "feel" for how they work. Just like matrix multiplication, multiplying group elements is (in general) very tedious for people to do.
Try to think about groups in other areas of math or in everyday life. They appear anywhere you think of symmetry. Rotations and other rigid motions in space are a common example. But even something as simple as tic-tac-toe.... if you rotate (or invert) the board in a game of tic-tac-toe, a player with the advantage still has the advantage. If you know some physics, you should immediately look up Noether's theorem.
Having a head start on those, you can spare yourself some mental strain early on and focus on the harder first-year ideas: subgroups, homomorphisms, Lagrange's theorem, and quotient groups.
"A Book of Abstract Algebra" by Charles C. Pinter is nice, from what I've seen of it--which is about the first third. I'm going through it in an attempt to relearn the abstract algebra I've forgotten.
I was using Herstein (which was what I learned from the first time), and was doing fine, but saw the Pinter book at Barnes & Noble. I've found it is often helpful when relearning a subject to use a different book from the original, just to get a different approach, so gave it a try (it's a Dover, so was only ten bucks).
What is nice about the Pinter book is that it goes at a pretty relaxed pace, with a good variety of examples. A lot of the exercises apply abstract algebra to interesting things, like error correcting codes, and some of these things are developed over the exercises in several chapters.
You don't have to be a prodigy to be able to understand some real mathematics in middle school or early high school. By 9th grade, after a summer of reading calculus books from the local public library, I was able to follow things like Niven's proof that pi is irrational, for instance, and I was nowhere near a prodigy.
From the ground up, I dunno. But I looked through my amazon order history for the past 10 years and I can say that I personally enjoyed reading the following math books:
An Introduction to Graph Theory
Introduction to Topology
Coding the Matrix: Linear Algebra through Applications to Computer Science
A Book of Abstract Algebra
An Introduction to Information Theory
I love Aluffi! It's a fun read, and more "modern" than texts like Dummit and Foote (in that it uses basic category theory freely). I like category theory, so I really enjoy Aluffi's approach.
My understanding is this is the standard
This is a really good intro/study guide. It explains proofs and the math concepts you'll need. It assumes no prior proof experience and would be good for self study for that reason.
https://www.amazon.com/Abstract-Algebra-Student-Friendly-Laura-Reis/dp/1539436071
Also brilliant.org has a cool group theory module but it doesn't really explain the proofs. Its also subscription based.
I don't know the first but I didn't really like the second book. Right now I do seem to make some good progress understanding stuff (not all but most) in Algebra Chapter 0 which is a lot bigger but introduces a lot of the Algebra I was missing (or forgot since school) along with the Category Theory terms.
I think category theory is best learned when taught with a given context. The first time I saw category theory was in my first abstract algebra course (rings, modules, etc.), where the notion of a category seemed like a necessary formalism. Given you already know some algebra, I'd suggest glancing through Paolo Aluffi's Algebra: Chapter 0. It is NOT a book on category theory, but rather an abstract algebra book that works with categories from the ground level. Perhaps it could be a good exercise to prove some statements about modules and rings that you already know, but using the language of category theory. For example, I'd get familiar with the idea of Hom(X,-) as a "functor"from the category of R-modules to the category of abelian groups, which maps Y \to Hom(X,Y). We can similarly define Hom(-,X). How do these act on morphisms (R-module homomorphisms)? Which one is covariant and which one is contravariant? If one of these functors preserves short exact sequences (i.e. is exact), what does that tell you about X?
Dummit (or just D&F), Artin, [Lang] (https://www.amazon.com/Algebra-Graduate-Texts-Mathematics-Serge/dp/038795385X), [Hungerford] (https://www.amazon.com/Algebra-Graduate-Texts-Mathematics-v/dp/0387905189). The first two are undergraduate texts and the next two are graduate texts, those are the ones I've used and seen recommended, although some people suggest [Pinter] (https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178) and Aluffi. Please don't actually buy these books, you won't be able to feed yourself. There are free versions online and in many university libraries. Some of these books can get quite dry at times though. Feel free to stop by /r/learnmath whenever you have specific questions
The canonical text. Sorry for the mobile link.
MTW is the classic text on GR (maybe Weinberg's book used to be except its a little dated). Its also ~1000 pages long and not necessarily a best introduction to GR and is expensive. The nice thing about F&N is that its only ~200 pages and gives a nice intro of the subject. (There are other books like Hartle that I am not familiar with).
I'm never really did a GR outside of a grad course taken in undergrad that used F&N. I'm very familiar with many QM books, and strongly recommend Shankar. I enjoyed Zee more than Peskin & Schroeder, though again you need P&S if you want to really learn QFT. And for particle physics Perkins should be a first introduction and he doesn't emphasize the Group Theory/Lie Algebra, which you may be able to get from a book like Georgi's Lie Algebra in Particle Physics.
Contemporary Abstract Algebra by James Gallian https://www.amazon.com/Contemporary-Abstract-Algebra-Joseph-Gallian/dp/1133599702
Gallian is a good intro:
https://www.amazon.com/Contemporary-Abstract-Algebra-Joseph-Gallian/dp/1133599702
Doesn't require to much background, but is a good intro to the subject. D&F is a great book, but feels more referency
Hungerford's Abstract Algebra is seen at my university as the best book for Abstract Algebra. This is mostly due to the fact that he covers Ring Theory before covering Groups. My university teaches Algebra this way, covering Rings before groups. However, Hungerford conveniently includes a study plan in his book for covering Groups before Rings. In my opinion, Hungerford has a great book with plenty of examples and exercises, with additional sections on applications.
Edit: Sorry, should have included what textbook I was talking about. It's fixed now!
I really enjoyed going through Abstract Algebra: An Introduction by Hungerford. There is an option to begin with Group Theory or begin with Ring Theory, if that's important to you. I think Hungerford's book is a very good introduction with plenty of exercises.
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.
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.
There's a good explanation of introductory quaternion theory in this book, which sets it in the larger context of group and field theory.
You'll remember and forget formulae as you use them. It's the using them that makes things concrete in your head.
Once you're comfortable with algebra, trig. I'm assuming you've had geometry, since you were taking algebra 2; if not, geometry as well.
Once you're comfortable with those topics, you'll have enough of the basics to start branching out. Calculus is one obvious direction; a lot people have recommended Spivak's book for that. Introductory statistics is another (far too few people are even basically statistically literate.) Discrete math is yet another possibility. You can also start playing with "problem math", like the Green Book or Red Book. Algebraic structures is yet another possibility (I found Herstein's abstract algebra book pretty easy to read when we used it in school).
Edit: added Amazon links.
Whoa, great questions, but I think you want a textbook, not a reddit post response. I used Dummit & Foote but it is probably a bit "heavier" than what you want/need at this point.
Griffiths' Quantum Mechanics has a crash course in most of the linear algebra required to do a first course in quantum mechanics. It's not very complicated - you just need basic understanding of vector spaces, linear transformations and functionals, and inner products, with a little bit of practice using dual notation of vectors (not too much, just enough for the Dirac notation which the book explains). Griffiths' also has a good explanation of simple fourier series/transform.
The key thing is being able to do basic linear algebra without matrices since in most of the cases, the vector space is infinite dimensional. But spin is a good example where almost everything can be done with matrices.
Additionally, solving ordinary differential equations and using separation of variables for partial differential equations in 3-d quantum mechanics would help.
Group theory will be of help in more advanced classes. Dummit and Foote or Arton's books on algebra are decent introduction. They are a bit dense though. If you want a real challenge, try Lang's Algebra book. I don't know of any easier books though. My first algebra book was Dummit and Foote which can be done without any real prerequisites beyond matrix algebra, but isn't really well written.
Links to books: Griffiths, Dummit and Foote.
PS: I have ebooks of these two books in particular.
Yes: Dummit and Foote. I used it in my freshman algebra class. It has excellent proofs and exercises. It will teach you the mathematical maturity faster than analysis and will most likely be more useful to you later on.
No worries for the timeliness!
For Measure and Integration Theory I recommend Elements of Integration and Measure by Bartle.
For Functional Analysis I recommend Introductory Functional Analysis with Applications by Kreyszig.
And for Topology, I think it depends on what flavor you're looking for. For General Topology, I recommend Munkres. For Algebraic Topology, I suggest Hatcher.
Most of these are free pdf's, but expensive ([;\approx \$200;]) to buy a physical copy. There are some good Dover books that work the same. Some good ones are this, this, and this.
Hello! I'm interested in trying to cultivate a better understanding/interest/mastery of mathematics for myself. For some context:
 
To be frank, Math has always been my least favorite subject. I do love learning, and my primary interests are Animation, Literature, History, Philosophy, Politics, Ecology & Biology. (I'm a Digital Media Major with an Evolutionary Biology minor) Throughout highschool I started off in the "honors" section with Algebra I, Geometry, and Algebra II. (Although, it was a small school, most of the really "excelling" students either doubled up with Geometry early on or qualified to skip Algebra I, meaning that most of the students I was around - as per Honors English, Bio, etc - were taking Math courses a grade ahead of me, taking Algebra II while I took Geometry, Pre-Calc while I took Algebra II, and AP/BC Calc/Calc I while I took Pre-Calc)
By my senior year though, I took a level down, and took Pre-Calculus in the "advanced" level. Not the lowest, that would be "College Prep," (man, Honors, Advanced, and College Prep - those are some really condescending names lol - of course in Junior & Senior year the APs open up, so all the kids who were in Honors went on to APs, and Honors became a bit lower in standard from that point on) but since I had never been doing great in Math I decided to take it a bit easier as I focused on other things.
So my point is, throughout High School I never really grappled with Math outside of necessity for completing courses, I never did all that well (I mean, grade-wise I was fine, Cs, Bs and occasional As) and pretty much forgot much of it after I needed to.
Currently I'm a sophmore in University. For my first year I kinda skirted around taking Math, since I had never done that well & hadn't enjoyed it much, so I wound up taking Statistics second semester of freshman year. I did okay, I got a C+ which is one of my worse grades, but considering my skills in the subject was acceptable. My professor was well-meaning and helpful outside of classes, but she had a very thick accent & I was very distracted for much of that semester.
Now this semester I'm taking Applied Finite Mathematics, and am doing alright. Much of the content so far has been a retread, but that's fine for me since I forgot most of the stuff & the presentation is far better this time, it's sinking in quite a bit easier. So far we've been going over the basics of Set Theory, Probability, Permutations, and some other stuff - kinda slowly tbh.
 
Well that was quite a bit of a preamble, tl;dr I was never all that good at or interested in math. However, I want to foster a healthier engagement with mathematics and so far have found entrance points of interest in discussions on the history and philosophy of mathematics. I think I could come to a better understanding and maybe even appreciation for math if I studied it on my own in some fashion.
So I've been looking into it, and I see that Dover publishes quite a range of affordable, slightly old math textbooks. Now, considering my background, (I am probably quite rusty but somewhat secure in Elementary Algebra, and to be honest I would not trust anything I could vaguely remember from 2 years ago in "Advanced" Pre-Calculus) what would be a good book to try and read/practice with/work through to make math 1) more approachable to me, 2) get a better and more rewarding understanding by attacking the stuff on my own, and/or 3) broaden my knowledge and ability in various math subjects?
Here are some interesting ones I've found via cursory search, I've so far just been looking at Dover's selections but feel free to recommend other stuff, just keep in mind I'd have to keep a rather small budget, especially since this is really on the side (considering my course of study, I really won't have to take any more math courses):
Prelude to Mathematics
A Book of Set Theory - More relevant to my current course & have heard good things about it
Linear Algebra
Number Theory
A Book of Abstract Algebra
Basic Algebra I
Calculus: An Intuitive and Physical Approach
Probability Theory: A Concise Course
A Course on Group Theory
Elementary Functional Analysis
I bought a copy of Dover's Linear Algebra (Border's Blowout) which I plan to go through after I finish A Book of Abstract Algebra.
I feel like I have a long way to go to get anywhere. :S
If you're teaching algebra at all, I highly reccomend reading the first chapter of Pinter's A Book of Abstract Algebra which gives a very short and entertaining history of the subject.
http://www.amazon.com/Book-Abstract-Algebra-Edition-Mathematics/dp/0486474178
You can probably handle this book, and it's all of ten bucks anyways.
What do you want to do, though? Is your goal to read math textbooks and later, maybe, math papers or is it for science/engineering? If it's the former, I'd simply ditch all that calc business and get started with "actual" math. There are about a million books designed to get you in the game. For one, try Book of Proof by Richard Hammack. It's free and designed to get your feet wet. Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand/Polimeni/Zhang is my favorite when it comes to books of this kind. You'll also pick up a lot of math from Discrete Math by Susanna Epp. These books assume no math background and will give you the coveted "math maturity".
There is also absolutely no shortage of subject books that will nurse you into maturity. For example, check out [The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs by Grinberg](https://www.amazon.com/Real-Analysis-Lifesaver-Understand-Princeton/dp/0691172935/ref=sr_1_1?ie=UTF8&qid=1486754571&sr=8-1&keywords=real+analysis+lifesaver() and Book of Abstract Algebra by Pinter. There's also Linear Algebra by Singh. It's roughly at the level of more famous LADR by Axler, but doesn't require you have done time with lower level LA book first. The reason I recommend this book is because every theorem/lemma/proposition is illustrated with a concrete example. Sort of uncommon in a proof based math book. Its only drawback is its solution manual. Some of its proofs are sloppy, messy. But there's mathstackexchange for that. In short, every subject of math has dozens and dozens of intro books designed to be as gentle as possible. Heck, these days even grad level subjects are ungrad-ized: The Lebesgue Integral for Undergraduates by Johnson. I am sure there are such books even on subjects like differential geometry and algebraic geometry. Basically, you have choice. Good Luck!
All the books listed can be found on libgen.io
If interest is theoretical mathematics:
Become adept at writing proofs.
I recommend
https://www.amazon.com/Discrete-Transition-Advanced-Mathematics-Undergraduate/dp/0821847899
Do some exercises in the first chapter, and go around the book doing whatever is of interest. I suggest learning about proofs/truth tables, functions, infinite sets, and number theory. This book will have chapters approaching all of these.
After this, you have some choice. I would take a beginners book in any of the following fields
Abstract algebra: https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178
Linear algebra: Linear Algebra Done Right by axler
Analysis: foundations of mathematical analysis by rudin (this will be hard but don’t be afraid!)
Approach each of these books slowly. Do not rush. Self-studying math is HARD. You might only get through 3 pages in a week, but I guarantee that you will get the ropes, and a few weeks later, look back and wonder how it was difficult at all.
In making the choice of what to study first, go to the subjects Wikipedia page or google “should I study x or y first” and you’ll likely find good resources
I highly recommend Pinter's "A Book of Abstract Algebra" for a quick course and handy refresher book.
Many functions don't take real numbers or integers as their arguments. Consider the multiplication of an MxN matrix and an NxM matrix where M != N. The result of which is an NxN matrix. In this context, matrix addition doesn't even have a relation to matrix multiplication.
If you're interested, these relationships are what group theory tries to explore. My favorite book on the subject is A Book of Abstract Algebra
Sure. https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178
How about some nice, inexpensive classics from Dover Publications?
For number theory, Andrews, Number Theory or Leveque, Elementary theory of numbers or the more advanced Leveque, Fundamentals of Number Theory
For linear algebra, Cullen, Matrices and Linear Transformations.
I bet you haven't read Edwards, Riemann's Zeta Function.
Edit: Oops! Now I see that you wanted to avoid linear algebra. Cullen might still be good as a second source. Maybe Pinter, A book of Abstract Algebra would appeal to you for a taste of field theory. However, vector spaces just naturally go with fields, so you may want to wait until after you have studied linear algebra.
As others have said, learning proofs first is the way to go. Book of Proof is a free online book you could try working through.
If you're set on abstract algebra for some reason, A Book of Abstract Algebra seems perfect for your level. You could try to dive right into this if you wanted, but if so you should do the appendix first (Book of Proof would be a more in depth treatment of the content in the appendix of this book).
I hear D&F is too tough for one's first brush with Abstract Algebra. On the other hand, people swear by "A Book of Abstract Algebra" by Charles Pinter to get one started on the path of AA.
There are these videos and there is also this book. The book is better if you struggled the first time, and it includes a short section on number theory.
This is just my perspective, but . . .
I think there are two separate concerns here: 1) the "process" of mathematics, or mathematical thinking; and 2) specific mathematical systems which are fundamental and help frame much of the world of mathematics.
​
Abstract algebra is one of those specific mathematical systems, and is very important to understand in order to really understand things like analysis (e.g. the real numbers are a field), linear algebra (e.g. vector spaces), topology (e.g. the fundamental group), etc.
​
I'd recommend these books, which are for the most part short and easy to read, on mathematical thinking:
​
How to Solve It, Polya ( https://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069111966X ) covers basic strategies for problem solving in mathematics
Mathematics and Plausible Reasoning Vol 1 & 2, Polya ( https://www.amazon.com/Mathematics-Plausible-Reasoning-Induction-Analogy/dp/0691025096 ) does a great job of teaching you how to find/frame good mathematical conjectures that you can then attempt to prove or disprove.
Mathematical Proof, Chartrand ( https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094 ) does a good job of teaching how to prove mathematical conjectures.
​
As for really understanding the foundations of modern mathematics, I would start with Concepts of Modern Mathematics by Ian Steward ( https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Books/dp/0486284247 ) . It will help conceptually relate the major branches of modern mathematics and build the motivation and intuition of the ideas behind these branches.
​
Abstract algebra and analysis are very fundamental to mathematics. There are books on each that I found gave a good conceptual introduction as well as still provided rigor (sometimes at the expense of full coverage of the topics). They are:
​
A Book of Abstract Algebra, Pinter ( https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178 )
​
Understanding Analysis, Abbott ( https://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/1493927116 ).
​
If you read through these books in the order listed here, it might provide you with that level of understanding of mathematics you talked about.
For classes like number theory and abstract algebra, I would suggest just picking up a book and attempting to read it. It will be hard, but the main prerequisite for courses like this is some mathematical maturity. That only comes with practice.
Realistically there is probably no preparation that you could have which would prepare you in such a way that a book on advanced mathematics would be super easy.
I like this number theory book
http://www.amazon.com/Number-Theory-Dover-Books-Mathematics/dp/0486682528/ref=sr_1_1?ie=UTF8&qid=1348165257&sr=8-1&keywords=number+theory
I like this abstract algebra book
http://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178/ref=sr_1_2?s=books&ie=UTF8&qid=1348165294&sr=1-2&keywords=abstract+algebra
AH HA, one of the few times I will link a dover book in good heart!
http://www.amazon.com/Book-Abstract-Algebra-Edition-Mathematics/dp/0486474178
Pinter offers a fine introduction to abstract algebra.
I haven't yet started practicing for the GRE, but does it include Linear Algebra or Modern/Abstract Algebra? Also is there Calculus on it? I'm taking (or have taken, or will take by the time of the GRE) all of those classes and they're all very interesting. I just bought this book on Abstract Algebra, if you're interested.
I'd recommend Linear Algebra Done Right. I think there are links to the pdf floating around, but I don't know if it's free or not. If not pay the money. I heard you like proofs.
Also good not because of but despite being way cheap, Elements of Abstract Algebra which is like a puzzle book, Socratic method. There are a lot of good books to recommend, I chose this one.