Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Advanced Algebra textbooks

math.stonybrook.edu

Post reply on HN
Print view View original

31–40 of 41 posts

Re: Advanced Algebra textbooks

#31

Post by vitriol83 »

eliteraspberrie wrote: post #4

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.

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!

Re: Advanced Algebra textbooks

#32

Post by greydius »

danharaj wrote: post #10

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 moder…

I also found this book to be excellent.

Re: Advanced Algebra textbooks

#33

Post by jacinda »

gaur wrote: post #11

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".

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.

Re: Advanced Algebra textbooks

#34

Post by jostylr »

tnecniv wrote: post #21
g9yuayon wrote: post #8

Earlier quoted context omitted.

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…

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.

Nature and Growth of Mathematics by Edna Kramer is a fantastic book interweaving history and math.

Re: Advanced Algebra textbooks

#35

Post by craigching »

tnecniv wrote: post #21
g9yuayon wrote: post #8

Earlier quoted context omitted.

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…

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.

http://store.doverpublications.com/0486404536.html

Re: Advanced Algebra textbooks

#36

Post by trentmb »

mturmon wrote: post #24
trentmb wrote: post #23

Earlier quoted context omitted.

If anyone else got curious about induction on the reals, my first interesting result was this Math.StackExchange post: http://math.stackexchange.com/questions/4202/induction-on-re...

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.

Yes- it just kick-started my curiosity is all!

Re: Advanced Algebra textbooks

#37

Post by navi54 »

danharaj wrote: post #10

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 moder…

Do you have any suggestions on other topics and books in math? This books is amazing.

Re: Advanced Algebra textbooks

#38

Post by tzs »

codeofthedamned wrote: post #7
eliteraspberrie wrote: post #4

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.

I agree. Pinter's book is very easy to read and great for self-study.

And it's a Dover book, so is quite reasonably priced (around $12).

Re: Advanced Algebra textbooks

#39

Post by JadeNB »

andrepd wrote: post #19
sound_of_basker wrote: post #9

Earlier quoted context omitted.

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.

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 .

Re: Advanced Algebra textbooks

#40

Post by danharaj »

navi54 wrote: post #37
danharaj wrote: post #10

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 moder…

Do you have any suggestions on other topics and books in math? This books is amazing.

Math is too big. Got a particular desire?
Post reply on HN