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Advanced Algebra textbooks

math.stonybrook.edu

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Re: Advanced Algebra textbooks

#21

Post by tnecniv »

g9yuayon wrote: post #8
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.

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.

Re: Advanced Algebra textbooks

#22

Post by gaur »

andrepd wrote: post #20
gaur wrote: post #13

Earlier quoted context omitted.

Looks like Times to me. It's a standard mathematical font, and anyway is more pleasant and easier to read than the abominable Computer Modern fonts.

Can you elaborate on that? I find computer modern much more pleasant to read than Times.

It's an inferior clone of Monotype Modern, just like Arial is an inferior clone of Helvetica.

In terms of being easier to read, I find the extreme contrast between thick and thin strokes does not work well on a screen.

Re: Advanced Algebra textbooks

#23

Post by trentmb »

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.

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...

Re: Advanced Algebra textbooks

#24

Post by mturmon »

trentmb wrote: post #23
andrepd wrote: post #19

Earlier quoted context omitted.

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.

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.

Re: Advanced Algebra textbooks

#25

Post by agumonkey »

g9yuayon wrote: post #8
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.

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'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.

Re: Advanced Algebra textbooks

#26

Post by eastWestMath »

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".

I feel like mathematicians are the ones who get to decide what constitutes basic algebra.

Re: Advanced Algebra textbooks

#27

Post by eastWestMath »

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".

I feel like mathematicians are the ones who get to decide what constitutes basic algebra.

Re: Advanced Algebra textbooks

#28

Post by kzhahou »

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".

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.

Re: Advanced Algebra textbooks

#29

Post by mathgenius »

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".

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.

Re: Advanced Algebra textbooks

#30

Post by mathgenius »

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.

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.

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