I don't like the typical definition-theorem-proof approach of most textbook in mathematics, including these. It's great for a classroom, no good for self-study. As an alternative, I highly recommend A Book of Abstract Algebra by Pinter. If you work through that first, you may actually enjoy these two later.
Can't agree more. Definition-theorem-proof type of textbooks is way too clean. They don't tell you how ideas came to be or why they mattered. In other words, it's hard for students to learn the intuitions behind the ideas. I wish there are list of "XXX from Ground-Up" type of books that show readers a list of problems, struggles of people trying to solve them, and how ideas emerge from the numerous attempts. Leslie's paper Paxos Made Simple was written in that way. A few chapters of Kleinberg's Algorithm Design were written in that way too.
I think math classes should be paired with history more. My probability professor often offered historical context (for example, the Poisson distribution first being used to model deaths due to horse kicks in the Prussian army) to the ideas we discussed, and the stories were often both interesting and insightful.
I've been refreshing on Calculus and I found that Kline's book was good at application as well as a bit of history, at least I never got the history part at University and I found it very interesting.
I'd love to know all people seriously learning it to tell what they loved. Who like to be stuck on an abstract definition and figure it out on its own (ideal to real), who likes to have gradual build up (real to ideal).
My first AA book was the "European kind", all symbols and definitions, a few proofs every ten pages. It was too dry for me. I never thought other people would think that way.
I love brain teasing but I also need a minuscule amount of inspiration to power my neurons.
When you see Lemma/Proposition/Theorem think of it as an API that you can interface with.
Skip the proofs on the first read (this is the implementation, and may or may not be enlightening.)
But, number one rule with learning maths is: you got to do it yourself. Play with it somehow. It's similar to learning a new (or first) programming language (or API): have a project in mind and try to do it using that language.
Seriously, you absolutely cannot learn this stuff just by reading. Or, at best you may learn a very small fraction of it.
IMO, this text is far from "typical definition-theorem-proof". There is plenty of other prose and examples there aswell.
this is always a tension in writing mathematics textbooks. at one end of the extreme you have e.g. Bourbaki which are very dry, but prove a great deal very efficiently and in the utmost generality. on the other hand you have textbooks which may be not as comprehensive and will intersperse the text with illuminating examples which historically would have been the original motivation for the subject. which is best really depends on your point of view and level of sophistication in the subject. usually I try to have both types of book at hand.
what would be great is if typesetting tools improved sufficiently so that one could choose 'beginner' or 'advanced' mode when reading a maths textbook. perhaps that is too fanciful!
You can say that again. The other day I was looking at proving the Pythagorean theorem in R_n. Merely starting the problem formally is non-trivial. :-(
You can prove it through mathematical induction. Show that if it's valid for n dimensions then it's valid for n+1 dimensions. So then if it's proven for R_2 it's proven in general.
The induction in question here is of course on the dimension of the vector space, which is a natural number -- not the members of the vector space itself.
No need to start with R^2; start with R^1, where it's easy. Starting the proof with R^2 essentially means that you must do the induction step twice.
For an amusing instance of induction on dimension, you might enjoy the proof of the AMGM inequality, which proceeds by upwards induction that doubles the dimension, followed by downward induction: https://proofwiki.org/wiki/Cauchy's_Mean_Theorem#Theorem .
The best undergrad algebra textbook I've studied is Algebra by Mac Lane and Birkhoff (3rd edition! the previous editions aren't quite as good and substantially different; i haven't seen the 4th edition and it is out of print so /shrug). I've used multiple books both in self-study and class and this book is, to me, in a league of its own. Not only does Algebra teach modern algebra, it teaches one to think like a modern algebraist, and not like just any modern algebraist, but like Saunders Mac Lane who was pretty great at algebra.
As an example of Algebra's approach, take the isomorphism theorems [1]. Now many undergraduate textbooks (like Dummit and Foote) will prove these theorems by manipulating cosets and deal with gross "implementation details" at the level of sets. Mac Lane insists otherwise: The only time you have to manipulate cosets is in order to construct the quotient G/N of a group G by one of its normal subgroups N. Once you have constructed this group and proved its universal property, the isomorphism theorems can be proved without ever mentioning cosets again. What is that universal property? It has two parts: First is that there is a morphism p from G to G/N which sends all of N to the identity in G/N. Second is that any morphism f from G to any group L that sends all of N to the identity in L necessarily factors uniquely up to isomorphism as a composition of morphisms g ∘ p. This is the essence of a quotient group.
Mac Lane's approach is to apprehend the essence of what is studied while discarding as much of the set theoretic husk as is possible. It is algebra in its purest form, accessible to and transformative of the mind of an undergraduate. Reading this book is a recurring joy to me.
I second this. This a phenomenal book. I didnt like it at first sight because I thought it looked kind of ugly, but the authors have a knack for clarifying some of the more arcane concepts in math in a very few words. It's not a cutesy bestseller that will end up teaching you jackshit, nor it's an intimidating monster like Lang's Algebra that's meant to put hair on your chest. This book is just right.
Eh, is this supposed to be a good book because I have no idea why the definitions aren't clearly marked and indexed. I only checked the chapter about Group Theory but I was not impressed. Maybe for a quick review of the topic it might be enough but for a beginner it seems that it is not rigorous. The definitions could be much more clear explicit. And there is no reason why they should not be indexed.
Book titles like these are just more evidence that some mathematicians don't understand (or willfully misconstrue) the meaning of words like "basic" or "introduction".
"Basic Algebra" means "material typically covered in late middle or early high school".
Yeah, and watch out for the word "Elementary". Mathematicians use this word to mean "can be understood by a single person". This in contrast to the kind of maths that leans on other difficult maths so much that no one person can figure it all out. That mathematics is the non-elementary kind.
No, the problem here is that the word ALGEBRA has two meanings, in high school vs college+ contexts. They share the same name and in both you can state that ax=bx => a=b ... but the similarities end soon after.
The material here is precisely what you'd expect in your first (i.e., basic) college algebra class.
He states on the website that these texts are intended for first-year graduate students in mathematics. This is an "Introduction" in the same way that a graduate-level course in algorithms is an introduction for a Computer Science Ph.D but probably wouldn't make sense to a first-year undergraduate.
I disagree. The word "introduction" immediately introduces ambiguity: Introduction to whom? An introduction to the aspiring physicist will certainly look drastically different from one to a high school student or an undergraduate mathematics student, but they all have reasonable need of an introduction to abstract algebra.
On an amusing note, I noticed the title, and decided to show my daughter, who is in high school. She knew that I was joking. I had already asked her, after her first day of algebra class, if she had learned about groups.
Somewhat unrelated, but I enjoy the logical dependence chart. It's nice to see the ordering of the topics. Rather than trying to figure that out for oneself
I'm by no means a typography nerd, but the font used in both PDFs is really bad. Not only is it visually unpleasant; most importantly, it is genuinely hard to read.
