(Part 2) Best algebra books according to redditors
We found 1,130 Reddit comments discussing the best algebra books. We ranked the 347 resulting products by number of redditors who mentioned them. Here are the products ranked 21-40. You can also go back to the previous section.
Download this book illegally: https://www.amazon.com/Calculus-James-Stewart/dp/1285740629
This is the book currently used in top-tier high schools to learn calculus. It is highly accessible. It creates a spark, at least in me, that made me take that book to bed and learn so much all night.
You'll get lots of practice if you do their practice problems (especially the more complex and involved ones later). Calc 3 is also covered in this book.
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.
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"
> 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.
Preach it, brother!
Let me highly recommend Conceptual Mathematics: A First Introduction to Categories and Topoi: The Categorial Analysis of Logic as introductions to the topic requiring no more than a completely rudimentary grasp of set theory to get started--and really, they even motivate the rudimentary set theory. These are basically "Category Theory for Dummies," or at least the closest things that I've found so far.
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.
I had to deal with the no internet thing for some time.
Find some place with free wi-fi(you are using phone?).
Download ebook/pdf reader, FBreader + PDF plugin is good (Assuming that you are using Android phone).
Install Firefox and this add-on Save Page WE, it also work for phones (tested with Android).
Then you can save pages from some of these web sites or Wikipedia:
Poor man's library:
http://gen.lib.rus.ec/
http://b-ok.org/
https://archive.org/
Some books:
[Harold Jacobs - Elementary Algebra] (https://www.amazon.com/Elementary-Algebra-Harold-R-Jacobs/dp/0716710471) (that's the most friendly and still not bullshit introduction I was able to find)
[Basic Mathematics by Serge Lang] (https://www.amazon.com/Basic-Mathematics-Serge-Lang/dp/0387967877) - Make sure that you understand the first 100-150 pages from the first book if you want to start this one.
Precalculus by David Cohen - Lang's Basic Mathematics is harder (and much more interesting) than this...
Use the three web sites written above to download them. PDF for better performance (and some times quality) on cheap phone, epub/djvu for the smaller size.
I think that free wi-fi will be easy to find. Reading books wouldn't be like watching videos but you will actually learn more. Compass, straightedge and protractor would be really useful.
"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.
If your goal is mainly to 'understand the adults in the room' then the above is major overkill in my opinion. PCA basically boils down to an application of the Singular Value Decomposition, which is itself a generalization of matrix diagonalization. The book 'Linear Algebra and Its Applications' by David Lay - which is a standard advanced undergraduate text, loaded with examples and great for 'getting the gist' - wraps up with the SVD and a couple of applied examples of using it for PCA.
I'd hazard that you can pretty easily achieve your goals by grasping the SVD and the basic linear algebra concepts that underpin it (multiplication, eigen values, diagonalization and a couple more).
I'll leave you with a site I've had great success with with others for getting to grips with some of the intuition http://www.uwlax.edu/faculty/will/svd/svd/index.html
I have a B.S. in mathematics, statistics emphasis - and am currently in the second semester of Linear Models in a M.S. Statistics program.
Contrary to popular opinion, I don't think Linear Algebra Done Right is suitable for learning linear algebra. Statistics - as far as I've gathered - is more focused on what is called "numerical linear algebra," rather than the more algebraic (and more abstract) approach that Axler takes.
It took a lot of research on my part to find better books. I personally believe that these resources are much better for covering the linear algebra needed for linear models (I recommend these after a first-course treatment in linear algebra):
I have also heard that Matrix Algebra Useful for Statistics by Searle is good, but I haven't read it yet.
If you feel like your linear algebra is particularly strong (i.e., you're comfortable with vector spaces, matrix operations, eigenvalues), you could try diving right into linear models. My personal favorite is Plane Answers to Complex Questions by Christensen. I reviewed this book on Amazon:
>It's a decent text. If you want to understand any part of this text, you need to have at least a first course in linear algebra covering matrices and vector spaces, some probability, and some "mathematical maturity."
>READ THE APPENDICES before you read any part of this text. READ THE APPENDICES. Take good notes on them and learn the appendices well. Then proceed to Chapter 1.
>Definitely one of the most readable books I've read, but it does take a long time to digest everything. If you don't have a teacher to take you through this material and you're completely new to it, you will find that some details are omitted, but these details aren't complicated enough that someone with an undergraduate degree in math wouldn't be able to figure them out.
>Highly recommended. The only thing I don't like about this text is some of its notation. It uses Cov(A) to mean the variance-covariance matrix of a random vector A, and Cov(A, B) to mean E[(A-E[A])(B-E[B])^transpose ]. I prefer using Var(A) for the former case. Furthermore, it uses ' instead of T to denote the transpose of a matrix.
No linear models text will cover all of the linear algebra used, however. If you get a linear models text, you should get your hands on one of the above linear algebra texts as well.
If you need a first course's treatment in Linear Algebra, I prefer [Linear Algebra and Its Applications](http://www.amazon.com/Linear-Algebra-Its-Applications-Edition/dp/0201709708) by Lay. The 3rd edition will suffice, although I think it's in the 5th edition now. Larson's [Elementary Linear Algebra*](http://www.amazon.com/Elementary-Linear-Algebra-Ron-Larson/dp/1133110878/ref=sr_1_1?s=books&ie=UTF8&qid=1458047961&sr=1-1&keywords=larson+linear+algebra) is also a decent text; older editions are likely cheaper, but will likely give you a similar treatment as well, so you may want to look into these too. I learned from the 6th edition in my undergrad.
Please, simply disregard everything below if the info is old news to you.
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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.
Learning Calculus prior would be helpful, but probably not necessary. I'd recommend getting a Linear Algebra book and start working through it. I used this one in my university Linear Algebra course and it is fairly well written, and approachable. You can probably find this book online as a pdf somewhere.
There are essentially "two types" of math: that for mathematicians and everyone else. When you see the sequence Calculus(1, 2, 3) -> Linear Algebra -> DiffEq (in that order) thrown around, you can be sure they are talking about non-rigorous, non-proof based kind that's good for nothing, imo of course. Calculus in this sequence is Analysis with all its important bits chopped off, so that everyone not into math can get that outta way quick and concentrate on where their passion lies. The same goes for Linear Algebra. LA in the sequence above is absolutely butchered so that non-math majors can pass and move on. Besides, you don't take LA or Calculus or other math subjects just once as a math major and move on: you take a rigorous/proof-based intro as an undergrad, then more advanced kind as a grad student etc.
To illustrate my point:
Linear Algebra:
Linear Algebra Through Geometry by Banchoff and Wermer
3. Here's more rigorous/abstract Linear Algebra for undergrads:
Linear Algebra Done Right by Axler
4. Here's more advanced grad level Linear Algebra:
Advanced Linear Algebra by Steven Roman
-----------------------------------------------------------
Calculus:
Calulus by Spivak
3. Full-blown undergrad level Analysis(proof-based):
Analysis by Rudin
4. More advanced Calculus for advance undergrads and grad students:
Advanced Calculus by Sternberg and Loomis
The same holds true for just about any subject in math. Btw, I am not saying you should study these books. The point and truth is you can start learning math right now, right this moment instead of reading lame and useless books designed to extract money out of students. Besides, there are so many more math subjects that are so much more interesting than the tired old Calculus: combinatorics, number theory, probability etc. Each of those have intros you can get started with right this moment.
Here's how you start studying real math NOW:
Learning to Reason: An Introduction to Logic, Sets, and Relations by Rodgers. Essentially, this book is about the language that you need to be able to understand mathematicians, read and write proofs. It's not terribly comprehensive, but the amount of info it packs beats the usual first two years of math undergrad 1000x over. Books like this should be taught in high school. For alternatives, look into
Discrete Math by Susanna Epp
How To prove It by Velleman
Intro To Category Theory by Lawvere and Schnauel
There are TONS great, quality books out there, you just need to get yourself a liitle familiar with what real math looks like, so that you can explore further on your own instead of reading garbage and never getting even one step closer to mathematics.
If you want to consolidate your knowledge you get from books like those of Rodgers and Velleman and take it many, many steps further:
Basic Language of Math by Schaffer. It's a much more advanced book than those listed above, but contains all the basic tools of math you'll need.
I'd like to say soooooooooo much more, but I am sue you're bored by now, so I'll stop here.
Good Luck, buddyroo.
Basic Algebra by Nathan Jacobson
Advanced Calculus by Shlomo Zvi Sternberg, Lynn Harold Loomis
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.
This book has good Google reviews. I haven't read it.
If you know topology and algebra then I think the most fruitful way of approaching categories is by picking up a book on algebraic topology. Hatcher is a canonical reference and although it doesn't really introduce the formal language of category theory, it is shown through most of the book. If you're not that patient and have more mathematical maturity then just pick up May's concise course in algebraic topology which is a wild ride but will get you there.
The most canonical textbook is Mac Lane's categories for the working mathematician but it's kind of dry so you'll need to provide your own class of examples every time.
If you just want to take a look into the topic and see how it is like then I would recommend you to read the first three chapters of this book. The main topic of the book is a little bit advanced so you can ignore any mention of topoi but those first three capters are a very brief introduction to category theory through a couple of examples and as far as I remember it doesn't expect you to know the stuff beforehand. It is however very basic and it won't cover a lot of the useful constructions insipired in algebraic topology/geometry but I still believe it's a pretty nice summary for the language.
An Introduction to Modern Astrophysics is an excellent and easy to read book:
https://www.amazon.com/dp/1108422160/ref=cm_sw_r_cp_apa_omrWBbDYB9MN3
It's commonly used for introductory Astrophysics courses. If you don't have a basic understanding of Calculus it won't make much sense so, if you really want to properly understand the subject, first study basic Calculus. A good introductory Calculus book would be this one:
https://www.amazon.com/dp/1285740629/ref=cm_sw_r_cp_apa_JdsWBbH1KXPAN.
You're also going to want a basic understanding of Physics so one more for that:
University Physics with Modern Physics (14th Edition)
https://www.amazon.com/dp/0321973615/ref=cm_sw_r_cp_apa_LfsWBbHJ83MT6
Those three books together should give you a basic understanding of Astrophysics and put your feet solidly on the road to further understanding. Read the Calculus book first (at least the first half of it or so) and then the Physics book. Then you'll be ready to dive into Carroll and Ostlie's book!
If you don't want to go quite that deep and you just want a really basic overview of the subject, you might consider finding Hawking's "A Briefer History of Time" or watching the PBS SpaceTime series in YouTube.
Edit: If the Calculus book is still a little unclear, your issue probably lies in Algebra. In that case, read this book before any of the others:
College Algebra (10th Edition)
https://www.amazon.com/dp/0321979478/ref=cm_sw_r_cp_apa_MqsWBbR985C30
Good luck on your journey! Give yourself at least a year or two to get through all of them and don't forget to work the problems!
Oh - download Kerbal Space Program and play it for a while. Trust me on this; you'll develop a second sense of basic orbital mechanics ;)
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!
There is no guaranteed was to factor any arbitrary polynomial, sadly. You could look into synthetic division, which is probably the fastest general way to test possible roots of an arbitrary polynomial.
You're never going to escape the "if x is not zero" stuff because 0 really is different than every other number. 0x = y has no solutions for y =/= 0, whereas ax = y does for every a =/= 0. This sort of "we need to exclude this one silly case" thing shows up all over math. For any non-zero number... for any non-empty set... for any non-trivial solution...
In math you always look at the cases where n=1, n=2 first (whatever n is... in your case linear and quadratic functions). Starting with the easy case isn't bad... but losing sight of the harder cases or not learning how to deal with them at all is.
I learned elementary algebra from this book and this one, in that order, and I think they were both excellent at providing the sort of perspective on problems that you were looking for. I don't think that "buy an extra set of textbooks to read in your free time" was the solution you were looking for... but maybe.
The really fundamental rule of algebra is "do to one side what you do to the other", although in practice I usually picture moving terms from one side to the other (flipping the sign) when the thing that I'm doing to both sides is addition or subtraction. The other fundamental rule (haha there are two) is that you can replace any term with something that is equal to it. The third fundamental rule is to check your answer at the end to avoid spurious solutions you got because you took the wrong root or divided by zero or something. If what you have solves the equation, it's a correct answer, even if everything leading up to it was wrong. (It might not get full credit of course...)
Really, with elementary algebra, you will eventually find it to be completely intuitive if you stay in math/science... just by using it so often. If your class isn't giving you a good perspective on the subject, you can at least rest assured that that perspective will come with time. I totally understand your frustration, though. (For what it's worth, I really struggled in high school algebra I (not II for some reason). I eventually just got past that with lots of practice.)
I haven't used it myself, but you might appreciate Gilbert Strang's Linear Algebra and Its Applications.
Category theory is not easy to get into, and you have to learn quite a bit and use it for stuff in order to retain a decent understanding.
The best book for an introduction I have read is:
Algebra (http://www.amazon.com/Algebra-Chelsea-Publishing-Saunders-Lane/dp/0821816462/ref=sr_1_1?ie=UTF8&qid=1453926037&sr=8-1&keywords=algebra+maclane)
For more advanced stuff, and to secure the understanding better I recommend this book:
Topoi - The Categorical Analysis of Logic (http://www.amazon.com/Topoi-Categorial-Analysis-Logic-Mathematics/dp/0486450260/ref=sr_1_1?ie=UTF8&qid=1453926180&sr=8-1&keywords=topoi)
Both of these books build up from the basics, but a basic understanding of set theory, category theory, and logic is recommended for the second book.
For type theory and lambda calculus I have found the following book to be the best:
Type Theory and Formal Proof - An Introduction (http://www.amazon.com/Type-Theory-Formal-Proof-Introduction/dp/110703650X/ref=sr_1_2?ie=UTF8&qid=1453926270&sr=8-2&keywords=type+theory)
The first half of the book goes over lambda calculus, the fundamentals of type theory and the lambda cube. This is a great introduction because it doesn't go deep into proofs or implementation details.
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.
tl;dr: you need to learn proofs to read most math books, but if nothing else there's a book at the bottom of this post that you can probably dive into with nothing beyond basic calculus skills.
Are you proficient in reading and writing proofs?
If you aren't, this is the single biggest skill that you need to learn (and, strangely, a skill that gets almost no attention in school unless you seek it out as an undergraduate). There are books devoted to developing this skill—How to Prove It is one.
After you've learned about proof (or while you're still learning about it), you can cut your teeth on some basic real analysis. Basic Elements of Real Analysis by Protter is a book that I'm familiar with, but there are tons of others. Ask around.
You don't have to start with analysis; you could start with algebra (Algebra and Geometry by Beardon is a nice little book I stumbled upon) or discrete (sorry, don't know any books to recommend), or something else. Topology probably requires at least a little familiarity with analysis, though.
The other thing to realize is that math books at upper-level undergraduate and beyond are usually terse and leave a lot to the reader (Rudin is famous for this). You should expect to have to sit down with pencil and paper and fill in gaps in explanations and proofs in order to keep up. This is in contrast to high-school/freshman/sophomore-style books like Stewart's Calculus where everything is spelled out on glossy pages with color pictures (and where proofs are mostly absent).
And just because: Visual Complex Analysis is a really great book. Complex numbers, functions and calculus with complex numbers, connections to geometry, non-Euclidean geometry, and more. Lots of explanation, and you don't really need to know how to do proofs.
I think it is silly that this requires much of an explanation. I recall this question being asked in a fucking linear algebra book! http://www.amazon.com/Linear-Algebra-Applications-Gilbert-Strang/dp/0030105676/ref=dp_ob_title_bk
I’m not sure Khan Academy is the most useful source; most of the assigned exercises I looked at a few years ago seemed pretty much trivial. You just watch someone solve a problem on a video, and then do exactly the same steps but with slightly different details. It’s an exercise in memory and copying, not in thinking for yourself. Basically the same curriculum as standard high school courses, just at your own pace. See Lockhart, “A Mathematician’s Lament” and Toom, “Word Problems in Russia and America”.
If you are self-studying the Gelfand and Kisilev books /u/TheBloodyNine1 mentioned are nice Russian books with some good problems in them, but also some text. If the text exposition is too fast or high level you could try reading the algebra and geometry books by Harold Jacobs. These have easier (standard American style) exercises but gentler exposition. If you are looking for medium to hard (by typical American standards) problems but also a good amount of step-by-step help with solving them, you might enjoy the Art of Problem Solving books, including those about algebra, geometry, basic number theory, “precalculus”.
Or for something a bit more poetic, check out Lockhart’s book Measurement.
The best way to learn the “why” of things in a real way is by doing the work for yourself. If someone just tries to tell you it won’t really sink in – you have to struggle with something for yourself before the explanation even has any relevance. Sometimes a book of nothing but problems can be just as useful as a book full of text.
See if you can work your way through problems such as those in Mathematical Circles (Russian Experience) (designed for ambitious Russian middle school students). Or you can look at the problems used at Exeter (famous private high school): Math 1, Math 2, Math 3–4, Math 4–5, Math 6, Discrete Math.
Or see if you can solve some past contest math problems. E.g. pick up a copy of a past AMC 12 (or AMC 10 or AMC 8 if those are too hard), and see how many problems you can do if you let yourself try to solve each one for 20 minutes without looking up the answer.
If you get through some of those and want less typical fare there are some fun topics in A Decade of the Berkeley Math Circle.
For some more general advice about problem solving methods (alongside problems), the book Thinking Mathematically is nice.
To be honest, the fastest way to improve is to find an expert tutor/mentor/coach to meet with face to face. Self-studying from books or websites or learning from class lectures and completely independent work is much more difficult / less efficient. There might be free tutoring resources available in your area if you hunt around (e.g. sometimes colleges will do free tutoring for nearby high school students).
Finally, if you get stuck on anything (problem, topic, ...) in particular, try /r/learnmath.
Elementary Linear Algebra by Larson, Edwards and Falvo.
Amazon link here.
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
For anyone else, I assume the specific book is Goldblatt's "Topoi: The Categorical Analysis of Logic"
Oh boy…
Q1: I have a similar experience in that I did not do well in high school. In fact, I flunked/dropped out of high school and joined the military in an attempt to learn a skill because I was stuck working in a doughnut shop. I ended up in aviation, which is the industry I still work in today as an engineering technician for a major manufacturer.
Toward the end of my military career I was an aircrew member for medevac helicopters. This meant that I had a lot of down time while waiting on medevac request. I was on an alert status which was similar to a living at a firehouse and waiting on calls. To pass the time I started reading pop-science physics books like Neil DeGrass Tyson and Brian Cox on a personal quest to understand how the universe works. One can only get so far in these books before the author says something like "but if you really want to understand how things work you need to know math". I thought that this was something that I was not capable of this so I gave up.
After I was out of the military, I found Khan Academy and started from the lowest level math and worked on all of the exercises. I worked my way up through algebra and started to look at calculus. At this point I went to my local community college and enrolled in precalc just to see if I could do it. I felt like I was kind of writing a wrong I made where I failed so miserably as a teenager. I am now a math major at a state university where I am in my final semester before graduating and I have recently decided to stick around for a masters.
Q2: I can't help you here. Sorry.
Q3: These walls exists for absolutely everybody in different places. The good news is that you can get to the other side of every one of them, and doing so is rewarding enough to make it worth it every single time. You've likely already experienced this at some level and it only gets better. What you need now is an introduction to mathematical proofs, especially if your interested in math for the sake of math. This is the thing that lies between computational topics like calculus and higher math. I will elaborate more on proofs further down.
Q4: I haven’t had experience with the probability book in question, but I worked with A First Course in Probability by Ross. I think this is a pretty standard textbook but my only other exposure to probability was in a statics course for business majors that I hated. I used this book in class for linear algebra and I watched Gilbert Strang’s videos for fun. There are probably better approaches out there but this worked for me.
Q’s 5 and 6: Back to proofs. This book absolutely bridged the gap for me when I made the leap from calculus to higher math which seems to be about where you are now. At the bottom of the page, in the comments, are links to some bite sized videos that go along with the book. These are fantastic as well. Understanding the structures of formal proofs is really what you need in knowing how to read and write mathematics. Dive right into this before something like abstract algebra or real analysis.
Here’s some more good news, your programing experience will likely help you in your next step when trying to understand proofs. The logic parts will sound familiar when learning about things like ‘if-then’ statements.
Here’s the best link I’ve found since becoming a math major. You’re welcome!
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.
Here is a link to John Baez's overview of what Topos theory is.
As /u/ziggurism has already said, you don't need to understand algebraic geometry to understand the theory of elementary toposes but some of the early motivating examples, the category of sheaves on a grothendieck site, are heavily steeped in the language of modern algebraic geometry.
A good, non algebro-geometric introduction to topos theory for those seeking to understand its place in logic is the book 'Topoi: The Categorial Analysis of Logic by Goldblatt. It also serves as a great introduction to the ideas of category theory.
Another fantastic book is Lawvere/Roserugh's Sets for Mathematics. This book seeks to explain the axioms of set theory using the language of category theory. It's not a book on arbitrary topos but it does serve to give you an idea of how topos theory axioms serve to build the logical system that every mathematician is familiar with, the logic in the category of sets. It's a good idea to have this 'concrete' application of topos axioms in Set under your belt before you tackling a book that seeks to explain how an arbitrary topos gives you a more abstract and unfamiliar logical system.
Edit: Also worth looking into is how topos theory can be used in the foundations of physics
Depends on your background. Mac Lane is the standard text and he is a phenomenal author in general, but it builds off knowledge of concepts such as modules, tensor products and homotopy (I still don't have a sufficient background in AT to be honest though). For a more modest background, I would recommend the book "Sets for Mathematics" by Lawvere and Rosebrugh. The book is entirely on category theory, the title is because there is a focus on the category of sets. The first chapter or so is deceptively simple, it gets very difficult as it goes on, but still doesn't require much specific background.
I'll also note that I first got into the subject through a whim purchase in a local Borders of a cheap dover book Topoi by Robert Goldblatt when I was very into mathematical logic. It's 500 pages and requires pretty much no background (I'd know what a topological space is, but I can't think of anything else). It gets very challenging though, and I never got more than 250 pages in before getting overwhelmed, but the first hundred pages really sparked my interest in category theory. Functors (and especially adjoint functors) are postponed much later than you will see in many other sources though. You can find a link to an online version free from the author's webpage too.
David Lay's Linear Algebra and Its Applications
Lecture Notes I used (Go to middle of page)
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
We only need Pythagorean theorem to understand special relativity. Consider two dudes X and Y. Suppose X is on a flying carpet holding up two mirrors distance of h apart. Also assume there's a light particle bouncing between these mirrors vertically like this here. So we see that h = ct_x where c is the speed of light and t_x is the amount of time it takes for the light to go from one mirror to the other. Now have Y stand on the ground and observe the behavior of the light particle as the carpet flies horizontally. From the perspective of Y, the particle flies in a sawtooth pattern like this. The distance the particle travels diagonally depends on the speed of light c and so it is ct_y where t_y is time taken by light to bounce from one mirror to the other as seen by Y. The distance the particle travels horizontally depends on the speed s of carpet and so it is st_y. By Pythagorean theorem, we have h^2 + (st_y)^2 = (ct_y)^2 which implies (t_y)^2 = h^2 / (c^2 (1 - s^2 / c^2 )) which further implies t_y = t_x / (sqrt(1 - s^2 / c^2 )). Thus if s = 0, then t_x = t_y and so time is universal. But as s approaches the speed of light c, the clocks desynchronize.
@ OP, if you want to get into high-falutin physics, you want to know the basics of real, functional (covers linear algebra), complex analyses; some probability and statistics; a bit of group theory.
For analysis the books by Lara Alcock, Amol Sasane, Paul Zorn, Robert Strichartz, Jonathan Kane, Steven Lay, Stephen Abbot, K.G Binmore, Charles Pugh, Mary Hart and many others are very user-friendly. And taking into account your background, Linear Algebra: Step by Step by Kuldeep Singh is perfect for you.
Reading a bunch about Turbulence and Topoi, mostly Tsinober and Goldblatt. Working on a proof about the relation of two manifolds also, basicly a lot of Jacobians.
Two things:
http://www.youtube.com/user/khanacademy#p/c/19E79A0638C8D449
(I dunno why more people here don't seem to know about it).
Personally I'd recommend these two:
-http://www.amazon.com/gp/product/0470073330/ref=pd_lpo_k2_dp_sr_2?pf_rd_p=486539851&pf_rd_s=lpo-top-stripe-1&pf_rd_t=201&pf_rd_i=0471316598&pf_rd_m=ATVPDKIKX0DER&pf_rd_r=1S7Z2QKSMQM70SWRZQFT
-http://www.amazon.com/Calculus-Stewarts-James-Stewart/dp/0495011606/ref=sr_1_1?ie=UTF8&s=books&qid=1279124596&sr=1-1
You're probably gonna wanna go with earlier editions cuz they're way cheaper and say the same thing.
Linear Algebra and [Linear Algebra and Its Applications] (http://www.amazon.com/dp/0321385179).
Here's mine
To understand life, I'd highly recommend this textbook that we used at university http://www.amazon.com/Campbell-Biology-Edition-Jane-Reece/dp/0321558235/ That covers cell biology and basic biology, you'll understand how the cells in your body work, how nutrition works, how medicine works, how viruses work, where biotech is today, and every page will confront you with what we "don't yet" understand too with neat little excerpts of current science every chapter. It'll give you the foundation to start seeing how life is nothing special and just machinery (maybe you should do some basic chemistry/biology stuff on KhanAcademy first though to fully appreciate what you'll read).
For math I'd recommend doing KhanAcademy aswell https://www.khanacademy.org/ and maybe a good Algebra workbook like http://www.amazon.com/The-Humongous-Book-Algebra-Problems/dp/1592577229/ and after you're comfortable with Algebra/Trig then go for calc, I like this book http://www.amazon.com/Calculus-Ron-Larson/dp/0547167024/ Don't forget the 2 workbooks so you can dig yourself out when you get stuck http://www.amazon.com/Student-Solutions-Chapters-Edwards-Calculus/dp/0547213093/ http://www.amazon.com/Student-Solutions-Chapters-Edwards-Calculus/dp/0547213107/ That covers calc1 calc2 and calc3.
Once you're getting into calc Physics is a must of course, Math can describe an infinite amount of universes but when you use it to describe our universe now you have Physics, http://www.amazon.com/University-Physics-Modern-12th/dp/0321501217/ has workbooks too that you'll definitely need since you're learning on your own.
At this point you'll have your answers and a foundation to go into advanced topics in all technical fields, this is why every university student who does a technical degree must take courses in all those 3 disciplines.
If anything at least read that biology textbook, you really won't ever have a true appreciation for the living world and you can't believe how often you'll start noticing people around you spouting terrible science. If you could actually get through all the work I mentioned above, college would be a breeze for you.
I recommend Stewart's book for calculus. There are many computational problems and examples.
http://www.amazon.com/Calculus-6th-Edition-Stewarts-Series/dp/0495011606/ref=sr_1_2?ie=UTF8&qid=1371868204&sr=8-2&keywords=calculus+stewart
It's not too expensive if you buy it from an Amazon user. And the 6th edition will be fine. I actually like it more than the 7th edition.
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.
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.
Perhaps you might find Shilov's Linear Algebra or Roman's Advanced Linear Algebra to be useful. Both of them treat bilinear and quadratic forms.
I think Shilov does actually discuss Gram-Schmidt orthonormalization, but he doesn't call it that, and it seems to be spread over several sections in chapters 7 and 8. Roman might be better for that. Anyway, you can peruse both of these at libgen.
My favorite linear algebra text is Paul Halmos' Finite-Dimensional Vector Spaces. As far as textbooks go, it's cheap, and it's written very well. It does expect a certain amount of mathematical maturity (a familiarity with proof techniques).
Gilbert Strang's book, Linear Algebra and Its Applications might be better for someone looking into applied mathematics than Halmos'. He makes frequent references to applications and uses geometric arguments fairly liberally. It is 3 times the price of Halmos' text as well, but I'm sure your university library has a copy or two.
I agree with urish, that learning linear algebra fairly well, especially considering the fields that you're interested in.
Hope this helps.
You should check out an Infinitely Large Napkin. It’s a good book for building up intuition. Unfortunately the probability section isn’t complete yet, but I think you would enjoy the measure theory section.
If you want some more formal books, there’s Kolmogorov’s Introductory Real Analysis and Folland’s book on Real Analysis
I think Linear Algebra by Kuldeep Singh is the best fit for newcomers to LA. It's unpretentious and meant to be actually read by students (can you imagine?). This book will take you from someone who just discovered there exists such a thing as LA to someone who solves problems in Linear Algebra Done Right By Axler cold. After Kuldeep Singh you can pick up Advanced Linear Algebra by Steven Roman which is an extreme overkill even for mathematicians.
Basically, once you get the basics of LA down, you can simply read up on the newest matrix algos for machine learning on ArXiv or something. BTW, if your goal is working with data you need to learn some probability.
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.
Holy crap OP, where do you go to school? All the topics you mentioned are (in my experience) usually split up into two separate courses (Calc II and III), and part of a third as well (the diff eq. stuff). As far as textbooks go I can only recommend what I used:
http://www.amazon.com/Calculus-Early-Transcendentals-James-Stewart/dp/0538497904/ref=sr_1_1?ie=UTF8&qid=1418617662&sr=8-1&keywords=calculus+early+transcendentals+7th+edition
http://www.amazon.com/Calculus-Early-Transcendentals-Howard-Anton/dp/0470647698/ref=sr_1_2?ie=UTF8&qid=1418617682&sr=8-2&keywords=anton+calculus
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.
My undergrad class used David Lay’s Linear Algebra and it’s Applications. The most recent version isn’t too difficult to illicitly stumble upon.
https://www.amazon.com/Linear-Algebra-Its-Applications-5th/dp/032198238X/ref=nodl_
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
Linear Algebra: Step by Step https://www.amazon.com/dp/0199654441/ref=cm_sw_r_cp_api_6l2QzbNAXRANT
This is what I used when I was learning it on my own. It explains things very well and has lots of practice with the correct and worked out answers online.
I've been kinda sorta watching it but I think he was holding up Artin's book.. not sure..
edit: Okay, the beginning of the second video confirms my guess..
From my experience, Calculus in America is taught in 2 different ways: rigorous/mathematical in nature like Calculus by Spivak and applied/simplified like the one by Larson.
Looking at the link, I dont think you need to know sets and math induction unless you are about to start learning Rigorous Calculus or Real Analysis. Also, real numbers are usually introduced in Real Analysis that comes after one's exposure to Applied/Non-Rigorous Calculus. Complex numbers are, I assume, needed in Complex Analysis that follows Real Analysis, so I wouldn't worry about sets, real/complex numbers beyond the very basics. Math induction is not needed in non-proof based/regular/non-rigorous Calculus.
From the link:
For Calc 1(applied)- again, in my experience, this is the bulk of what's usually tested in Calculus placement exams:
Solving inequalities and equations
Properties of functions
Composite functions
Polynomial functions
Rational functions
Trigonometry
Trigonometric functions and their inverses
Trigonometric identities
Conic sections
Exponential functions
Logarithmic functions
For Calc 2(applied) - I think some Calc placement exams dont even contain problems related to the concepts below, but to be sure, you, probably, should know something about them:
Sequences and series
Binomial theorem
In Calc 2(leading up to multivariate Calculus (Calc 3)). You can pick these topics up while studying pre-calc, but they are typically re-introduced in Calc 2 again:
Vectors
Parametric equations
Polar coordinates
Matrices and determinants
As for limits, I dont think they are terribly important in pre-calc. I think, some pre-calc books include them just for good measure.
If you're trying to learn calculus on your own you're better off buying a used version of either of these books for cheap (or going to a library)
http://www.amazon.com/Thomas-Calculus-11th-George-B/dp/0321185587
or Stewart: http://www.amazon.com/Calculus-Stewarts-James-Stewart/dp/0495011606/ref=sr_1_1?ie=UTF8&s=books&qid=1268447623&sr=1-1
Schaums provides basic insight, and several practice problems. If you want to understand the theory, go for Stewart or Thomas.
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.
Lay's Linear Algebra is a great introduction to linear algebra book.
For the intuition behind linear algebra, watch Essence of linear algebra.
https://www.amazon.com/Calculus-James-Stewart/dp/1285740629/ref=sr_1_1?ie=UTF8&qid=1484794699&sr=8-1&keywords=calculus+8th+edition
>what is the difference really between 'calculus' and 'real analysis'
At the undergraduate level, "calculus" typically means the what. For example: what is this limit? What is the derivative of a given function? What is the value of this integral?
"Analysis" more typically gets into the why behind calculus. Why does this function have a limit? Justify why the typical rules for differentiation—product rule, chain rule, etc.—are valid. Define what it means for a function to be integrable over a given interval, and justify your computation of a given integral.
There's a lot more going on than just that, but to first approximation, making the distinction between the what of calculus and the why of analysis is a good starting point.
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I don't have a copy of Kolmogorov's text, so I'm at a disadvantage. I assume you mean something like this book in the Dover series? If so, then the table of contents suggests it's a pretty ambitious book, at least for typical undergraduates—and especially if it's one's introduction to the subject matter. That text by Kolmogorov covers some of both metric space topology and point-set topology, as well as linear algebra, measure theory, integration, and differentiation (itself in the context of Lebesgue integration). I'm no expert on the matter, but Kolmogorov's (and Fomin's) text seems more representative of what's often called "functional analysis" rather than just "real analysis". I suspect that pedagogically, you might benefit from a more "concrete" introduction to real analysis before tackling something like this textbook.
As for the inverse and implicit function theorems, there are a handful of ways to approach those results. One way is to show that the two theorems are equivalent: the inverse function theorem is true if and only if the implicit function theorem is true. The way a lot of books proceed is to establish the inverse function theorem by making some suitable simplifications—e.g., that the derivative map is being evaluated at the origin, and that this derivative map is the identity map—then apply the contraction mapping theorem. (Of course, the two theorems are equivalent, so one could instead prove the implicit function theorem first, instead.)
Rudin is emphatically not the only suitable textbook for something like this, but nearly any such "suitable" textbook will inevitably be challenging. It will help you considerably to have already had linear algebra, at least, especially if you turn to a textbook that presupposes linear algebra as a prerequisite. I'm not sure what to recommend to you, but here are a few textbooks I've used over the years (in addition to those already mentioned above):
Stromberg and Browder are challenging, on the general level of Rudin in that respect. Marsden and Hoffman has an unusual structure (at least in my volume), where the statements of the theorems and propositions are separated from the proofs of those assertions. (And, if memory serves, M&H may have suffered from a number of typos.) Edwards may be the most accessible of the books above, and it covers quite a bit from both a "classical" and higher-order "theoretical" perspective. But which of these books—if any—would be the best fit for you would necessarily be a matter of speculation for me.
I hope this was nonetheless helpful. Good luck!
Hungerford has both an undergraduate text and his GTM text. The undergraduate text starts off with ring theory and then moves to group theory. The GTM text starts with group theory and moves onto ring/field theory.
I would say that the best way to start is to pick a single book in Calculus, such as this or this or even this, and work all the way through it.
Then it is up to you; you could go straight towards Real Analysis; I recommend starting with a book that bears Intro in the name.
Or you could pursue a more collegiate curriculum and move onto Differential Equations and Linear Algebra, then Real Analysis.
I assume you are doing this all independently, so you should look at college sequences for math majors and the likes. You can mimic those, and look for online syllabi of the courses to make sure you are covering the appropriate material. This helps because it gives a nice structure to your learning.
Whatever the case, work through a calculus book, then decide what further direction you wish to take.
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.
http://www.khanacademy.org/ is a pretty solid resource up through Linear Algebra. I'd recommend picking up a textbook in each subject so that you have a good list of examples and problems to work through to supplement Khan Academy. A used older edition of Stewart's Calculus would do good, it has everything as the newer ones and it is the standard calc textbook. Remember to keep doing problems, and don't stop, especially on the ones that are giving you the biggest headache! If you have any questions or problems ask /r/cheatatmathhomework or /r/learnmath.
Once you have an understanding of the basics, the MIT Open Courseware is a good source.
I've heard some good things about Michael Artin's book (http://www.amazon.com/Algebra-Michael-Artin/dp/0130047635).
Wow, do you go to some school where mathochism is cool? This is not a junior-level course in my academic worldview. It was not too too long ago that linear algebra was almost exclusively a graduate course. It was pushed down to the undergraduate level because of its extreme usefulness in ODEs and DSP, among other things. Undergraduates did not get that much smarter, instead the curriculum for linear algebra just got that much more streamlined. Your prof is either ignorant of or doesn't care about that evolution. If this is supposed to be a "regular" class, then you might voice a complaint to the chairperson of the department. Junior level courses usually are the introduction to mathematical rigor, not the launchpad for the study of Lie Algebras or other specialized areas. However if you are in an honors class or a hardcore mathematics school, you'll just have to strap in and enjoy the ride.
So here's some rope. All my references are old because I am old(ish). However, you can probably do better with keyword searches in Wikipedia and WolframAlpha based on your lecture notes. Do something like a mind map of the connections. The only thing you are missing online will be problems.
Go to your library and get Linear Algebra and Its Applications. I learned from an earlier edition of the book, but I can't imagine it getting worse. The people who hate the book are the ones who didn't do the exercises. If you stick with it, it is very cool and things start to build and just make sense. Strang is an excellent, excellent expositor, but you have to be a big picture person. He also tells you exactly what the core of the book is, The Fundamental Theorem of Linear Algebra. Grok that and linear algebra is your oyster, e.g., Gram-Schmidt will seem like an obvious thing. (And wouldn't you know, a reference on that Wikipedia page is to a paper by Strang on just that(pdf).)
If you can put up with older notation, you will find a lot in the famous book by Halmos, Finite Dimensional Vector Spaces.
A lot of this carries through to graduate algebra and functional analysis, so find whatever texts your graduate courses require and check their indices. From the above it sounds like your prof is trying to hit all the connections to other areas.
This next book will probably not help you, but it is just crazy enough to make me think you may find some of your professor's thoughts hidden there, Mathematical Physics by Geroch. You don't have time to learn category theory, but his exposition ends up at the spectral theorem, I seem to recall. Seeing another presentation of those powerful theorems might be illuminating. (It's a beautiful book, but I've never heard of it being used in a class.)
If you don't have MATLAB, get a (free?/cheap?) student edition and play with it for "real" examples of what you are doing. Going through the Theorem-Proof process never worked for me with things like linear algebra: Seeing how you can pull things apart and put them back together is what makes the power of linear algebra come alive and gives you some motivation.
The last piece of advice is not a guarantee, but has always worked for me when in a draconian course: Drill yourself on your old tests and quizzes and homework. When everyone is failing and the final comes around, chances are good (for various reasons, including pity and laziness) that the earlier exams are almost exactly recapitulated. Use your prof's office hours to go over the subtleties of the exam problems. If you are engaged with the material, the chances are good that he will extend the scope of discussion and pull in examples from the current lectures. That's a very handy insight to have.
If the notes of your class do make it online, please think of linking it back here. I'm curious as to how deep this course is since it is pretty wide.
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.
Best, and most commonly suggested, resource for independently studying the calculus sequence is Khan Academy.
The videos go into a great deal of detail, outline plenty of examples, and give the intuition behind all the concepts as well.
Don't just rely on the videos though; find a supplementary text book (Stewart is an excellent textbook for calculus), and do a ton of exercises. The best way to learn math is do as much math as possible.
Math texts are among the most egregious examples of unnecessarily updated texts. Almost none of the math you'll learn as an undergrad has changed in my lifetime, but students have to buy the newest version just so they can have the right homework problems. Stewart's text is at version 8, and you can [buy it on Amazon for $183] (https://www.amazon.com/Calculus-James-Stewart/dp/1285740629/ref=pd_sbs_14_1?_encoding=UTF8&pd_rd_i=1285740629&pd_rd_r=JVSPPXJ3JZSTMBTMY7ZA&pd_rd_w=y7iOT&pd_rd_wg=2XT4J&psc=1&refRID=JVSPPXJ3JZSTMBTMY7ZA). I just looked and found a used copy of the 7th edition for $12. It's so cheap because student's cant use the damned thing. I used Thomas and Finney's book thirty years ago, and there haven't been any developments in Calculus since then that are relevant to a Freshman Engineering Major. The teacher of my Functional Analysis class in grad school was fed up with this, so we used Riesz and Sz.-Nagy's Dover Edition. This was a graduate-level math class, and we were able to use a text that was decades old.
Michael Artin's Algebra's first few chapters is probably one of the best explanations of linear algebra that you'll get.
http://www.amazon.com/Algebra-Michael-Artin/dp/0130047635
If you want to learn serious mathematics, start with a theoretical approach to calculus, then go into some analysis. Introductory Real Analysis by Kolmogorov is pretty good.
As far as how to think about these things, group theory is a strong start. "The real numbers are the unique linearly-ordered field with least upper bound property." Once you understand that sentence and can explain it in the context of group theory and the order topology, then you are in a good place to think about infinity, limits, etc.
Edit: For calc, Spivak is one of the textbooks I have heard is more common, but I have never used it so I can't comment on it. I've heard good things, though.
A harder analysis book for self-study would be Principles of Mathematical Analysis by Rudin. He is very terse in his proofs, so they can be hard to get through.
My understanding is this is the standard
For math you're going to need to know calculus, differential equations (partial and ordinary), and linear algebra.
For calculus, you're going to start with learning about differentiating and limits and whatnot. Then you're going to learn about integrating and series. Series is going to seem a little useless at first, but make sure you don't just skim it, because it becomes very important for physics. Once you learn integration, and integration techniques, you're going to want to go learn multi-variable calculus and vector calculus. Personally, this was the hardest thing for me to learn and I still have problems with it.
While you're learning calculus you can do some lower level physics. I personally liked Halliday, Resnik, and Walker, but I've also heard Giancoli is good. These will give you the basic, idealized world physics understandings, and not too much calculus is involved. You will go through mechanics, electromagnetism, thermodynamics, and "modern physics". You're going to go through these subjects again, but don't skip this part of the process, as you will need the grounding for later.
So, now you have the first two years of a physics degree done, it's time for the big boy stuff (that is the thing that separates the physicists from the engineers). You could get a differential equations and linear algebra books, and I highly suggest you do, but you could skip that and learn it from a physics reference book. Boaz will teach you the linear and the diffe q's you will need to know, along with almost every other post-calculus class math concept you will need for physics. I've also heard that Arfken, Weber, and Harris is a good reference book, but I have personally never used it, and I dont' know if it teaches linear and diffe q's. These are pretty much must-haves though, as they go through things like fourier series and calculus of variations (and a lot of other techniques), which are extremely important to know for what is about to come to you in the next paragraph.
Now that you have a solid mathematical basis, you can get deeper into what you learned in Halliday, Resnik, and Walker, or Giancoli, or whatever you used to get you basis down. You're going to do mechanics, E&M, Thermodynamis/Statistical Analysis, and quantum mechanics again! (yippee). These books will go way deeper into theses subjects, and need a lot more rigorous math. They take that you already know the lower-division stuff for granted, so they don't really teach those all that much. They're tough, very tough. Obvioulsy there are other texts you can go to, but these are the one I am most familiar with.
A few notes. These are just the core classes, anybody going through a physics program will also do labs, research, programming, astro, chemistry, biology, engineering, advanced math, and/or a variety of different things to supplement their degree. There a very few physicists that I know who took the exact same route/class.
These books all have practice problems. Do them. You don't learn physics by reading, you learn by doing. You don't have to do every problem, but you should do a fair amount. This means the theory questions and the math heavy questions. Your theory means nothing without the math to back it up.
Lastly, physics is very demanding. In my experience, most physics students have to pretty much dedicate almost all their time to the craft. This is with instructors, ta's, and tutors helping us along the way. When I say all their time, I mean up until at least midnight (often later) studying/doing work. I commend you on wanting to self-teach yourself, but if you want to learn physics, get into a classroom at your local junior college and start there (I think you'll need a half year of calculus though before you can start doing physics). Some of the concepts are hard (very hard) to understand properly, and the internet stops being very useful very quickly. Having an expert to guide you helps a lot.
Good luck on your journey!
Many of the current Dover books were used as textbooks not too long ago when they were still in print from the original publisher. For example, this was the book used in my undergrad intro to PDEs course, and this was used for the undergrad intro to algebraic topology. Those were the hardcover versions, before Dover bought them. Even the Kolmogorov & Fomin book was used in grad school for real analysis.
To me the real Dover gems are the old ones (pre-1960) that are hard to find now except at used book stores or on Amazon. Though they may seem "outdated" many of them actually cover topics that are still useful but not covered nowadays.
Hi there,
It'd be a good idea to know what your level of mathematics is like. I assume you know basic set theory? (I assume this is needed to analyze music from the second viennese school - but how detailed is the set theory you learn? Are there proofs?). What other maths do you have knowledge of? I believe the standard algebra book is Artin's but this may be too hard/dense for you. I can think of some easier books off the top of my head, but they start off with ring/fields instead of groups (such as the Hungerford)...
Just curious, do you have any suggestions on what a random person can read to learn about transformational theory on the net besides wikipedia?
Edit: Random googling found this. Seems like a good start for both mathematicians to learn about 20th century music theory and musicians to learn about set theory/algebra.
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!
My school used this for the first two years of calculus.
One thing I found useful for doing this is Stewart's Calculus (many people will disagree with me here, but it was my old Calc book, so I didn't have to buy a new one, and I thought it was pretty decent). Don't worry about buying the latest version. you can probably find an old one in a used book store, or ebay or something, which will save you some bucks. The thing that kills Calc students is their poor algebra, so make sure you are rock-solid on that. You should be able to solve linear equations, quadratic equations, rational equations, and equations involving square-roots without a problem. You should also be able to graph all of these, and you should have a good understanding of exponents and logs. Don't spend much time reading the book, spend your time practicing, doing problem after problem until you really nail each one. If you can find a study-buddy, this will help a lot, as they will be able to point out where you are going wrong, and you will be able to teach them things (which is one of the best ways to learn).
Anyway, that's just some random advice, but I hope it helps. Good luck!
I did an independent study of category theory from Goldblatt's Topoi: The Categorical Analysis of Logic. Having not studied category theory further than that, I can't offer much comparison, but I found it just barely accessible (which is generally about the best I hope for) and pretty cheap to boot.
Roman is great for what you want.
I'm about 90% sure that your calculus course will use this book.
I would go back and refresh yourself on your college algebra and trigonometry. Knowing things like your identities and how to move things around/re-write a term algebraically is about 50% of mastering the classes. In some of the more complex differentiation and integrals later on, simplifying the equation you start with helps immensely.
Where I teach they use Linear Algebra by Lay for the introductory class. I'm not sure what level you need but Linear Algebra Done Right is also commonly recommended; could be more abstract than what you need?
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
Discrete mathematics with ducks!
Contemporary Abstract Algebra
An Illustrated Theory of Numbers
Visual Complex Analysis
Number-Crunching
The Magic of Math: Solving for x and Figuring Out Why
The Princeton Companion to Mathematics
The Number Devil: A Mathematical Adventure
Professor Stewart's Cabinet of Mathematical Curiosities
Gödel, Escher, Bach: An Eternal Golden Braid
One Two Three . . . Infinity: Facts and Speculations of Science
Recently made a move and these are a few books on my shelf that stand out more.
There are so many beautiful mathematical images that could be chosen though and I do wish more publishers would have more creative book covers also.
For algorithms, I would recommend checking out Skiena's "Algorithm Design Manual." One of the cool features are his "War Stories" which give various examples of how the author used and adapted algorithms to solve real-world problems.
For linear algebra, I haven't read it myself but you might try Lay's "Linear Algebra and Its Applications" which probably will have more of a focus on applications than the titles mentioned by KolmogorovTuring.
https://www.amazon.com/Calculus-James-Stewart/dp/1285740629/ref=sr_1_2?ie=UTF8&qid=1466199140&sr=8-2&keywords=james+stewart+calculus+8th+edition
It covers Calculus 1-3. I have a downloaded version from a friend of the whole book.
I don't have suggestions on a lecture series, but this is a pretty common book.
nice, we're using a book by the same guy.
I'm currently working through Munkres' book on Topology, and I am using the video lectures found here. I know these are in an annoying form factor, but, trust me, these are the only videos that go into any depth you will find on the internet. They use Munkres, too, which is a plus.
On the same site are video lectures for Algebraic Topology. For this, I definitely recommend buying Artin's "Algebra" (1st edition can be found cheaply, and I don't think there's really any significant difference from 2nd), and watch these video lectures by Harvard. Then, you can finally move on to the Algebraic Topology video lectures which uses the free textbook "Algebraic Topology" by Allen Hatcher.
Hope this helps.
I found Calculus from Larson and Edwards pretty good.
That doesn't really convey the fact that this is actually Prerequisite Algebra. This is math that you should know before entering college. Just saying "basic algebra" gives the impression that this is the first level.
On the other hand, Basic Algebra can mean "set theory, group theory, rings, modules, Galois theory, polynomials, linear algebra, and associative algebra." Context is everything, I suppose.
I used these lectures and skipped 23-28 and all of the review lectures. Though, you may want to review if there is any material in there that would be on the exam. I just ran out of time / got lazy towards the end. It helped me to buy the book and do homework assignments in the relevant chapters as I watched each video. It's not the same book used in the lectures, but for the most part it follows, and if it doesn't it was just out of order. The textbook is okay but is more or less the video lectures with the chalkboard diagrams and examples in print; there's not that much additional information. Doing practice problems is invaluable. Much of Math 415 is algorithmic.
Stewart, Larson, Rogawski
I would suggest combining the linear algebra and abstract algebra text into Artin's text. I have found it to be a very good text on algebra with a heavy emphasis on the theory of linear algebra. I glanced through Hungerford's text and didn't take to it. Too verbose with too many examples.
I second the Rudin and Munkres. I found that reading through Hocking and Young's text helped me get the intuition i needed to plow through Munkres.
Why are you doing the exercises? Is it for a class? Are you self studying?
I've done all the exercises in Hungerford. When we had a section assigned for homework, I would just do everything in that section. Maybe I was doing 1 or 2 sections a week. I can't really remember. After the class ended, I finished the book in the name of completionism (and because I enjoyed the material). It was a really fun project because my professor had done the same thing years before. She still had her hand-made answer key, and we'd compare solutions during office hours.
I've also done all the exercises in an old version of Vakil's notes and all the exercises in the first few chapters of Hartshorne. However, in both of those books the exercises contain key material so you have to do them.
If I could do it again, I'd force myself to type up everything. I did all this before I learned TeX. Now I have pages and pages of exercises wasting away in a box somewhere...
I went ahead and grabbed a copy of this.
Basic Algebra? I hope he uses Jacobson's textbook :D
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!
Two of my all time favorite math subjects: Linear Algebra and Number Theory.
My Number Theory professor actually wrote a textbook that I think you’d find is, generally speaking, pretty easy to follow and has some neat applications. Bear in mind though that he wrote it specifically to teach Number Theory at a University level so sometimes reading through some proofs may not be exactly clear. here is the libgen link, if you’d rather another source this is the amazon link to purchase or rent the book.
You will learn some cool techniques and patterns and things you (possibly) never even knew about numbers if you start studying Number Theory.
Now my linear algebra textbook I never actually used because my professor never required it but from what I hear it’s a good foundation but doesn’t expound much upon certain topics - idk if that’s true but I heard the author is a chill dude so here ya go
You can libgen that too.
I don’t exactly think studying calculus on your own will be too exciting, but if you do study calculus, you can then carry it over to other fields - like physics, which has a foundation in calculus from Isaac Newton. I don’t have a good reading source for introductory calculus.
It's not ecology-centric but David Lay's "Linear Algebra" is a great text at an intro level that's great for developing intuition. I remember one example that uses demography of owl populations as a way to use matrix projection models. Overall a great book to teach yourself.
https://www.amazon.com/Linear-Algebra-Its-Applications-5th/dp/032198238X
I really liked Linear Algebra and it's Applications. I thought it was a good textbook with plenty of problem sets.
That helps a little. I'm not too familiar with that world (I'm a physics major), but I took a look at a sample civil engineering course curriculum. If you like learning but the material in high school is boring, you could try self-teaching yourself basic physics, basic applied mathematics, or some chemistry, that way you could focus more on engineering in college. I don't know much about engineering literature, but this book is good for learning ODE methods (I own it) and this book is good for introductory classical mechanics (I bought and looked over it for a family member). The last one will definitely challenge you. Linear Algebra is also incredibly useful knowledge, in case you want to do virtually anything. Considering you like engineering, a book less focused on proofs and more focused on applications would be better for you. I looked around on Amazon, and I found this book that focuses on applications in computer science, and I found this book focusing on applications in general. I don't own any of those books, but they seem to be fine. You should do your own personal vetting though. Considering you are in high school, most of those books should be relatively affordable. I would personally go for the ODE or classical mechanics book first. They should both be very accessible to you. Reading through them and doing exercises that you find interesting would definitely give you an edge over other people in your class. I don't know if this applies to engineering, but using LaTeX is an essential skill for physicists and mathematicians. I don't feel confident in recommending any engineering texts, since I could easily send you down the wrong road due to my lack of knowledge. If you look at an engineering stack exchange, they could help you with that.
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You may also want to invest some time into learning a computer language. Doing some casual googling, I arrived at the conclusion that programming is useful in civil engineering today. There are a multitude of ways to go about learning programming. You can try to teach yourself, or you can try and find a class outside of school. I learned to program in such a class that my parents thankfully paid for. If you are fortunate enough to be in a similar situation, that might be a fun use of your time as well. To save you the trouble, any of these languages would be suitable: Python, C#, or VB.NET. Learning C# first will give you a more rigorous understanding of programming as compared to learning Python, but Python might be easier. I chose these three candidates based off of quick application potential rather than furthering knowledge in programming. This is its own separate topic, but my personal two cents are you will spend more time deliberating between programming languages rather than programming if you don't choose one quickly.
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What might be the best option is contacting a professor at the college you will be attending and asking for advice. You could email said professor with something along the lines of, "Hi Professor X! I'm a recently accepted student to Y college, and I'm really excited to study engineering. I want to do some rigorous learning about Z subject, but I don't know where to start. Could you help me?" Your message would be more formal than that, but I suspect you get the gist. Being known by your professors in college is especially good, and starting in high school is even better. These are the people who will write you recommendations for a job, write you recommendations for graduate school (if you plan on it), put you in contact with potential employers, help you in office hours, or end up as a friend. At my school at least, we are on a first name basis with professors, and I have had dinner with a few of mine. If your professors like you, that's excellent. Don't stress it though; it's not a game you have to psychopathically play. A lot of these relationships will develop naturally.
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That more or less covers educational things. If your laziness stems from material boredom, everything related to engineering I can advise on should be covered up there. Your laziness may also just originate from general apathy due to high school not having much impact on your life anymore. You've submitted college applications, and provided you don't fail your classes, your second semester will probably not have much bearing on your life. This general line of thought is what develops classic second semester senioritis. The common response is to blow off school, hang out with your friends, go to parties, and in general waste your time. I'm not saying don't go to parties, hang out with friends, etc., but what I am saying is you will feel regret eventually about doing only frivolous and passing things. This could be material to guilt trip yourself back into caring.
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For something more positive, try to think about some of your fun days at school before this semester. What made those days enjoyable? You could try to reproduce those underlying conditions. You could also go to school with the thought "today I'm going to accomplish X goal, and X goal will make me happy because of Y and Z." It always feels good to accomplish goals. If you think about it, second semester senioritis tends to make school boring because there are no more goals to accomplish. As an analogy, think about your favorite video game. If you have already completed the story, acquired the best items, played the interesting types of characters/party combinations, then why play the game? That's a deep question I won't fully unpack, but the simple answer is not playing the game because all of the goals have been completed. In a way, this is a lot like second semester of senior year. In the case of real life, you can think of second semester high school as the waiting period between the release of the first title and its sequel. Just because you are waiting doesn't mean you do nothing. You play another game, and in this case it's up to you to decide exactly what game you play.
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Alternatively, you could just skip the more elegant analysis from the last few paragraphs and tell yourself, "If I am not studying, then someone else is." This type of thinking is very risky, and most likely, it will make you unhappy, but it is a possibility. Fair warning, you will be miserable in college and misuse your 4 years if the only thing you do is study. I guarantee that you will have excellent grades, but I don't think the price you pay is worth it.
I had a stellar professor and a great book. I thought it was a breeze. I used this book in undergrad: http://www.amazon.com/Elementary-Linear-Algebra-Ron-Larson/dp/0618783768/ref=sr_1_25?ie=UTF8&qid=1421682001&sr=8-25&keywords=linear+algebra
As far as notation: it will change from book to book. Learn as you go. I certainly didn't have a class or a book dedicated to the notation of mathematics. Generally the author will briefly explain their notation as they introduce the topic.
There's a good explanation of introductory quaternion theory in this book, which sets it in the larger context of group and field theory.
I used a book by Larson and Falvo called Elementary Linear Algebra. So far the book was really good at explaining every topic.
My recommendation, get a Calculus book like this: Calculus by Larson. It explains everything from the beginning (analytical geometry) and with your algebra background you should understand it. Study section by section.
In one year you should have covered to Integral Calculus single variable. If you get stuck with some topic, post your question here!
Good luck!
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.
I keep seeing this book recommended in a lot of places. How is it different from the one by Axler and one by Roman?
Apart from the other good advice in this thread, if you're able to invest some money, I've heard wonderful things about this book.
It sounds like you had a really shitty teacher. Bad teachers can fuck with your confidence. But then, you already know that. I know it's easy for me to say, but: don't let that get in the way of learning math. There's nothing wrong with you. I promise you're capable of learning algebra.
There's also a /r/learnmath subreddit where you can ask questions about particular math problems.
Finally, I'm curious: does a problem like "Solve for x: 5 + x = 17" make sense to you (and would you be able to answer it)? If not, I'd be happy to write a few paragraphs about that. Maybe that's what you need to get started.
Cool.
Linear Algebra Don't waste your time with anything other than Lay, pretty much. Sounds like you're 100% new to LinAlg (it's not about polynomial equations) so it may be a bit tough to get off the ground working by yourself, but not impossible. It'd be worth finding a MOOC on the subject, there should be plenty. Otherwise, it's a pretty standard freshman maths course and a lot of people struggle with it (not because it's hard, just because it's different to HS maths), so there's a ton of resources on the internet.
Calculus Kinda just gotta slog away with where you're at tbh. I had Stewart as a freshman, didn't think it was overly great though. Still, that's the kind of level you need, so search for "alternatives to Stewart calculus" and anything that comes up should be appropriate. I wouldn't be able to tell you which to pick though.
Stats Basically, completing both of the above is pretty much a prerequisite for being able to understand linear regression properly, so don't expect to gain much by diving straight into stats. You could probably find a "business analytics" style textbook that would let you do more stats without understanding what's really going on under the hood, but if you want to stick with it in the long term you'll benefit more from getting stuff right at the beginning.
If the "Early Transcendentals" book does not have "Single-Variable" or "Multi-Variable" in its name, then it has all of the content of both books; if it does have "Single-Variable" in its name, then it might share a chapter with "Multi-Variable" but that's it.
The response by /u/jimmy_rigger was made as if you were asking about a textbook explicitly labeled as "Single-Variable", and my reply pointed out that you weren't.
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You can look inside the books to see for yourself:
I even noticed almost the same sequence of chapters in a competing textbook (Thomas), except that it puts infinite series before polar/parametric (chapters 10 and 11).
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
Sweet. I think the best curriculum to approach this with, assuming you're in this for the long haul, would be to start with building a good understanding of calculus, cover basic classical mechanics, then cover electricity and magnetism, and finally quantum mechanics. I'm going to leave math and mechanics mostly for someone else, because no textbooks come to mind at the moment. I'll leave you with three books though:
For Math, unless someone else comes up with something better, the bible is Stewart's Calculus
The other two are by the same author:
Griffith's Introduction to Electrodynamics
Griffith's Introduction to Quantum Mechanics
I think these are entirely reasonable to read cover to cover, work through problems in, and come out with somewhere near an undergraduate level understanding. Be careful not to rush things. One of the biggest barriers I've run into trying to learn physics independently is to try and approach subjects I don't have the background for yet: it can be a massive waste of time. If you really want to learn physics in its true mathematical form, read the books chapter by chapter, make sure you understand things before moving on, and do problems from the books. I'd recommend buying a copy of the solutions manuals for these books as well. It can also be helpful to look up the website for various courses from any university and reference their problem sets/solutions.
Good luck!
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.
Okay.. Someone had to do it.. right?
Statistics for business and economics ~$131.15 Used and ~$188.39 New.
Principles & Practices of Physics v1 hardcover ~$51.55 Used and ~$164.02 New.
Chemistry - The Molecular Nature... ~$124.00 Used and ~$239.87 New.
Principles & Practices of Physics v2 ~$129.74 Used and ~$126.78 New.
Differential Equations and Linear Algebra ~$79.89 Used and ~$151.29 New. I am the least sure about this book in particular. But for a wag, I'm sure the numbers will work.
Calculus - Early Transcendentals ~$86.03 Used and ~$236.81 New.
So by my calculations your current "TV Stand" cost ~$1107.16. I'd recommend you go to amazon and sell the books you probably aren't ever going to crack the cover on again for ~$602.36 and buy yourself an actual TV stand with a little money left in your pocket.
I do all this because most of my friends in college complained about the costs of text books and then never sold them again. Or did the absolutely stupidest thing you could ever do with a book you've paid over $200 for and sold them back to the bookstore for ~$20 a pop. Don't be lazy, use amazon to sell your books back and the sting of your new found education won't be so bad. The idea is to get smarter right?
I probably should have waited for a response but I bought. https://www.amazon.co.uk/Introductory-Analysis-Dover-Books-Mathematics/dp/0486612260 yesterday as it's so much cheaper. So hey let's hope it's worth the £5 I paid
The canonical text. Sorry for the mobile link.
Linear Algebra And Its Applications by Lay. I think I'll download the Kindle apps on my laptop and iPhone and see if the sample from the book renders decently there.
The increased portability would be a godsend, but I'm not sure if the marginal cost of going digital is justified yet.
This is what my school requires of us. https://www.amazon.com/gp/aw/d/1285740629/ref=mp_s_a_1_1?ie=UTF8&qid=1501294155&sr=8-1&pi=AC_SX236_SY340_FMwebp_QL65&keywords=calculus+stewart+8th+edition
Don't forget the $130 Web assign!
I would assume that if you're a music major and "been good at math", you might be referring to the math of high school. In any case, it would help if you spend some time doing/reviewing calculus in parallel while you go through some introductory physics book. So here's what you could do:
Other than that, feel free to google your question. You'll find good info on websites like physicsforums.com, physics.stackexchange.com, as well as past threads on this subreddit where others have asked similar questions.
Once you're past the intro (i.e., solid grasp of calculus, and solid grasp of mechanics, which could take up to a year), you are ready to venture further into math and physics territory. In that regard, I recommend you look at posts by Gerard 't Hooft and John Baez.
Strang's Linear Algebra.
A friend of mine used this book in her undergraduate abstract algebra course. It looks fantasitic: Gallian - Contemporary abstract algebra.
Hawking is a theoretical physicist. His craft is closer to math than it is to classical physics.
You made a lot of erroneous and hot-headed statements, but that's understandable. Since you seem to be very, very ignorant of math, I don't even know where to even begin to show you the differences - I am at a disadvantage here :) How about we talk about levels, then?
Most math an engineer knows is barely a first year material for a math undergrad. Math is so vast that even the grad students of math are at the very base of a huge mountain.
Here's Basic Algebra for a math major(flip through the first pages and checkout the contents).
Here's Algebra for engineers.
Notice how the algebra for engineers is a very small part of general algebra and non-rigorous at that.
Here's Calculus for engineers.
Here's Calculus for math majors.
This is not to say engineers are mentally inferior to mathematicians, it's just these two professions are concerned with fundamentally different things.
The only serious next step.
lol thanks i'm furious.
btw, i have something for you:
http://www.amazon.com/Elementary-Algebra-Harold-R-Jacobs/dp/0716710471
don't thank me now. buy it read it fail to understand it realize you suck at math move on to something you're good at like raping/robbing/killing then come back and thank me for setting you on your proper life's path