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Reddit mentions of A Book of Abstract Algebra: Second Edition (Dover Books on Mathematics)
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Reddit mentions: 52
We found 52 Reddit mentions of A Book of Abstract Algebra: Second Edition (Dover Books on Mathematics). Here are the top ones.
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- Dover Publications
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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
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.
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If any undergrads are struggling with those two courses I would highly recommend you check out those two textbooks. They are by far the easiest introduction to those two fields I have found. I also like that you can find solutions to all the exercises so it makes them very valuable for self study also. Both books also have a reasonable amount of excises so that you can in theory do nearly every problem in the book which is also nice compared to standard texts with way too many exercises to realistically go through.
You 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.
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
I actually bought http://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178
On chapter three now :D
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.
Try Pinter. If you think it is too simple for you go for Aluffi.
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.
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.
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.
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.
Machine learning is largely based on the following chain of mathematical topics
Calculus (through Vector, could perhaps leave out a subsequent integration techniques course)
Linear Algebra (You are going to be using this all, a lot)
Abstract Algebra (This isn't always directly applicable but it is good to know for computer science and the terms of groups, rings, algebras etc will show up quite a bit)
General Topology (Any time we are going to deal with construction of a probability space on some non trivial manifold, we will need this. While most situations are based on just Borel sets in R^n or C^n things like computer vision, genomics, etc are going to care about Random Elements rather than Random Variables and those are constructed in topological spaces rather than metric ones. This is also helpful for understanding definitions in well known algorithms like Manifold Training)
Real Analysis (This is where you learn proper constructive formulations and a bit of measure theory as well as bounding theorems etc)
Complex Analysis (This is where you will get a proper treatment of Hilbert Spaces, Holomorphic functions etc, honestly unless you care about QM / QFT, P-chem stuff in general like molecular dynamics, you are likely not going to need a full course in this for most ML work, but I typically just tell people to read the full Rudin: Real and Complex Analysis. You'll get the full treatment fairly briefly that way)
Probability Theory (Now that you have your Measure theory out of the way from Real Analysis, you can take up a proper course on Measure Theoretic Probability Theory. Random Variables should be defined here as measurable functions etc, if they aren't then your book isn't rigorous enough imho.)
Ah, Statistics. Statistics sits atop all of that foundational mathematics, it is divided into two main philosophical camps. The Frequentists, and the Bayesians. Any self respecting statistician learns both.
After that, there are lots, and lots, and lots, of subfields and disciplines when it comes to statistical learning.
A sample of what is on my reference shelf includes:
Real and Complex Analysis by Rudin
Functional Analysis by Rudin
A Book of Abstract Algebra by Pinter
General Topology by Willard
Machine Learning: A Probabilistic Perspective by Murphy
Bayesian Data Analysis Gelman
Probabilistic Graphical Models by Koller
Convex Optimization by Boyd
Combinatorial Optimization by Papadimitriou
An Introduction to Statistical Learning by James, Hastie, et al.
The Elements of Statistical Learning by Hastie, et al.
Statistical Decision Theory by Liese, et al.
Statistical Decision Theory and Bayesian Analysis by Berger
I will avoid listing off the entirety of my shelf, much of it is applications and algorithms for fast computation rather than theory anyway. Most of those books, though, are fairly well known and should provide a good background and reference for a good deal of the mathematics you should come across. Having a solid understanding of the measure theoretic underpinnings of probability and statistics will do you a great deal--as will a solid facility with linear algebra and matrix / tensor calculus. Oh, right, a book on that isn't a bad idea either... This one is short and extends from your vector classes
Tensor Calculus by Synge
Anyway, hope that helps.
Yet another lonely data scientist,
Tim.
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)
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'm not sure what to read into before the Galois class begins.
"A Book of Abstract Algebra" by Charles Pinter
I have no experience with it but perhaps Pinter's text may be of interest as I have heard from a professor that it is quite geometric and that may appeal to you in the sense of "These geometric properties are interesting, let's abstract them and go with it!"
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.
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.
If you'd like an alternative to calculus, try learning linear and/or abstract algebra. Shilov's Linear Algebra is a good book on linear algebra. Linear algebra comes up everywhere, so it's definitely worth learning. The abstractions involved such as fields should also be a good introduction to higher mathematics. For even more abstraction, try A Book of Abstract Algebra by Charles Pinter which is one of my favorite books.
While calculus is also fundamental, personally I find linear and abstract algebra to be much more enjoyable subjects.
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.
Not at Amazon. Dover lets you look at the table of contents and one other page here.
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
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.
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.
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.
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.
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.
I like this abstract algebra book: A Book of 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'm working through the exercises in Pinter's 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.
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.
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
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.
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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.
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I'd recommend these books, which are for the most part short and easy to read, on mathematical thinking:
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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.
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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.
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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:
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A Book of Abstract Algebra, Pinter ( https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178 )
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Understanding Analysis, Abbott ( https://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/1493927116 ).
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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.
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
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.
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.
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).
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.
Sure. https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178
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
I highly recommend Pinter's "A Book of Abstract Algebra" for a quick course and handy refresher book.
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
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!
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.
Pinter's A book of abstract algebra is fantastic and a friendly introduction to the topic. It's also really inexpensive.
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.
Yeah either of those are easier. I don't like Fraleigh cause I think it lacks motivation (also the chapters on splitting/separable fields really suck) but I love Herstein. If you're set on cheap, this guy ain't too bad. If I were self studying though I would try to find a cheap older edition of Artin, as he's very example motivated, and it can sometimes be hard to wrap your head around all the abstraction without a class.
EDIT: Also you might want to find a cheap number theory text, since elementary number theory is probably the most accessible way to see groups and rings in action. And for "how do I prove xxx" questions I always recommend starting with this.
I 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
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
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.