Math Books

[doogly]

Alpha Omicron - the man you need to correct six month old diction choices.

[Alpha Omicron]

Alpha Omicron - the man you need to correct six month old diction choices.

I know. But that one was an even worse version of a misuse that really, really drives me nuts.

[lu6cifer]

Alpha Omicron - the man you need to correct six month old diction choices.

I know. But that one was an even worse version of a misuse that really, really drives me nuts.

"Derive" in place of "differentiate" is even worse.

[doogly]

I'm partial to "throw some d's on that bitch."

[lu6cifer]

I'm partial to "throw some d's on that bitch."

This has been sigged.

[Bravemuta]

Hi! I'm looking for a book about the relation between maths and music. I've been searching my college's library, but the only books I've managed to find already assume I have a rather high level of music knowledge. Is there a book that starts from a lower level, perhaps even explaining the basics of music theory from a Maths' and Physics' perspective (like the pitch of notes in term of frequency and why certain notes sound well together, why the golden ratio is so pleasing to the ear etc)

[Mungo0]

I've noticed that my collection of books is sorely lacking in books related to Maths, and I've always been interested in the history, and practical applications of Maths. I don't suppose anyone here has recommendations for the history of Maths, or even practical applications? I realise that this is bit general, but still, help would be appreciated.

[Aleifr]

I'm in my last year of high-school, and I'm looking for a math book to use in addition to the one we use in class.

I also have a suggestion: Arthur Benjamin's "Secrets of Mental Math"
http://www.amazon.co.uk/Secrets-Mental- ... 984&sr=8-1
It improves your mental addition, subtraction, multiplication and division skills to amazing levels, and also teaches you other things like memorizing numbers better.

[doogly]

I'm in my last year of high-school, and I'm looking for a math book to use in addition to the one we use in class.

Any sort of topic you're after?

[Aleifr]

I'm in my last year of high-school, and I'm looking for a math book to use in addition to the one we use in class.

Any sort of topic you're after?

EDIT: The topics I'm most interested in are: calculus, algebra and vectors.

[Other_Calvin]

How much calculus do you know? If you haven't taken it yet, or don't otherwise know it, I would recommend finding a basic calculus textbook. If your school offer calculus, you might be able to borrow one, but it will almost certainly not be very proof-oriented (which, if it's your first introduction to calculus, is probably a good thing).

If you are comfortable with the concepts of basic calculus (differentiation and integration of one variable- you probably won't need series that much, but they can't hurt),you might want to try a multivariable calculus book- see this thread for recommendations.

If you want to find out what real algebra is, you could try group theory (or more advanced abstract algebra), but you've most likely not seen those concepts before if you haven't read outside of the standard math curriculum, and there's a good chance that your math teacher won't have either.

Finally, linear algebra is useful and greatly generalizes the idea of "vector" outside of how you learn them in precalculus/physics. This is a textbook that I've recently worked through, and it seems to be well-written. It also has the advantages of including answers, being free, and being proof-based. Most of the proofs are pretty simple, so they're a good introduction to mathematical rigor and induction.

[MathChief]

"Ricci flow and the Poincaré conjecture" By John W. Morgan, G. Tian
-this is a very nice and thorough book on Perelman's proof of the conjecture, if you know more or less differential manifold

"Lectures on the Ricci Flow (London Mathematical Society Lecture Note Series)" by Peter Topping
-this is for geometry noobs(like me 1 year ago) who wanted to know the proof for the conjecture

[MathChief]

BTW Claymath institute very kindly put the first book I mentioned online sometime ago
you can find it here(together with other awesome math books)
http://www.claymath.org/library/

the pdf link is
http://www.claymath.org/library/monographs/cmim03c.pdf

[qinwamascot]

I'm looking for a good book on the algebraic developments that occured within the last half-century or so, particularly the classification of finite simple groups, mosterous moonshine, and all the rest of the stuff that is used in string theory. I know the basics--undergraduate level classes, introductory graduate classes in algebra (Dummit & Foote), analysis (Royden), topology (Munkres), Number theory (professor's own lecture notes), and convexity theory (I don't remember the book offhand). I also know a decent amount about representation theory and differential geometry from my class in Quantum Field Theory, but certainly not as much as I'd like.

I'm pretty confident I'd be able to read any book recommended, given the time. I have tried reading the original papers, but these aren't in a convenient format for reading and doing independent study. I'm not so much interested in a full proof of the enormous theorem; more like a book that covers things like the Leech lattice or the monsterous moonshine conjecture in somewhat seperate sections, for some light reading. I decided to stay as an undergraduate for an extra year next year, and my university doesn't have anyone whose research is in this particular area. I'm looking for something to keep myself busy in an independent study for a semester or a year before I go to grad school.

[mprime]

Does anyone have experience self-teaching from Shilov's Linear Algebra book? (This one: http://www.amazon.com/Linear-Algebra-Georgi-E-Shilov/dp/048663518X/).

I really want to learn Linear Algebra, but my school sucks ass, so I think I'll just learn it on my own instead of taking their course.

[doogly]

The classification of finite groups is not a string theory topic, and the monstrous group shows up but not in an essential way. If you want some stringy stuff, I think the main thing you will want is complex manifolds. There are a bunch of fun books on this, I don't really have a favorite one. And the way things work in our department, complex manifold stuff mostly is mentioned as a special case in an algebraic topology course.
Vertex Alebras for Beginners might be a good one to look at to get you into some moonshine action.
Introduction to Vertex Operator Algebras and Their Representations also.

[Natty]

I just finished reading the book
------------
John Derbyshire's Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics
------------
and really liked it.

It's set up so that anyone could get a good grasp on the Riemann Hypothesis, even people without any knowledge of calculus. Every even chapter is a brief history of a famous mathematician, and every odd chapter is about the math. I'd strongly recommend it to anyone interested in math or math history, at basically any level above high school.

[Marbas]

Okay, so the book Complex Variables by Robert B. Ash is giving me a super hard time. I think I've only managed to complete four problems out of the 13 I've attempted and have resorted to looking up solutions, it may be that I'm giving up too easily. As a fourth semester undergrad this book shouldn't be too tough for me. But for some reason it's giving me trouble. Does anyone have any other recommendations for books on complex analysis that aren't going to beat me up and make me sad? Should I just give up on Complex Analysis?

Edit: Nevermind, after some persistence, there is success!

[Styhn]

I'm preparing for an upcoming differential geometry course, which I'll take for the second time. The book that the course is using is not very enlightening and I'm having a lot of trouble getting my head around the concepts. I have to learn about manifolds, equivalent definitions of manifolds, transformations between manifolds, one forms, k-forms, etc. etc. Everything is discussed in the context of R^n.

Can someone recommend me some books that will help me make sense of the subject? Thanks in advance!

[doogly]

Spivak's Comprehensive Introduction to Differential Geometry vol 1 is a thing of great and powerful beauty.
I also digged Tu's Introduction to Manifolds.

I don't know what you mean about equivalent definitions of manifolds though, from what I know there is only one (Hausdorff, second countable, locally R^n).

[GyRo567]

Hi! I'm looking for a book about the relation bewteen maths and music. I've been searching my college's library, but the only books I've managed to find already assume I have a rather high level of music knowledge. Is there a book that starts from a lower level, perhaps even explaining the basics of music theory from a Maths' and Physics' perspective (like the pitch of notes in term of frequency and why certain notes sound well together, why the golden ratio is so pleasing to the ear etc)

The Pythagoreans might be of historical interest to you. Their pseudo-religion of number found its strongest evidence in the mathematical ratios of music. One of my professors gave me a book that had been sitting in his closet, The Magic of Numbers by Eric Bell, which he said was all about the Pythagoreans. I still haven't read it (for shame!), but I imagine it has more depth than the average pop math book on patterns found in nature.

i'm searching a good book about "Discrete maths", for semi-begginers (i had that assignature in the college, but i want to know more about Graphs, Groups, and the talky...word thing Theory!).

What could you recommend me?

My apologies for being two years late in replying, but for anyone else still looking, Discrete Mathematics with Applications by Susanna Epp is an excellent introduction that can be fully self-taught if necessary.

[davidhbrown]

I'm really enjoying _The Princeton Companion to Mathematics_, edited by Timothy Gowers (ISBN 978-0-691-11880-2). It's a bit less than the price of a textbook (US$90 list; more typically around $60) and in its thousand pages it covers a heck of a lot. I got it because working on a MS in CS, I keep finding references to concepts I haven't encountered before in the papers I'm reading. Usually, these articles give me enough more than Wikipedia to keep going.

It's also just a lot of fun to open at random and read something!

[qinwamascot]

The classification of finite groups is not a string theory topic, and the monstrous group shows up but not in an essential way. If you want some stringy stuff, I think the main thing you will want is complex manifolds. There are a bunch of fun books on this, I don't really have a favorite one. And the way things work in our department, complex manifold stuff mostly is mentioned as a special case in an algebraic topology course.
Vertex Alebras for Beginners might be a good one to look at to get you into some moonshine action.
Introduction to Vertex Operator Algebras and Their Representations also.

Thanks. I've picked up a few books on Vertex Operator Algebras, which our library has a surprisingly good selection for despite no one researching them here. I wasn't so much interested as string theory in particular as any kind of modern algebra that is currently being researched. Vertex Operator Algebras should keep me occupied for a while though; at least over the summer, and hopefully into next year .

equivalent definitions of manifolds

What, exactly, do you mean here? Like how the circle S¹ is diffeomorphic to the space R / Z? Or do you actually mean some kind of alternate definition of the concept of a manifold that is somehow equivalent? If you mean the first, any decent book will have that kind of thing in it. If you mean the second, no book I've read defined manifolds significantly differently.

Regarding your question, I learned differential geometry from "Tensor Analysis on Manifolds" by Bishop and the Spivak volumes 1 and 2. Spivak is nothing short of amazing, but it doesn't seem to match everyone's tastes. Similarly, the Bishop book is sort of abstract at times and can be dense compared to some other undergraduate texts. If you're really having trouble, try looking up some "Differential Geometry for Physicists" books--it's hit and miss whether they'll help, but the concepts they do cover tend to be done in a way that isn't prohibitively abstract. You may need to know some basic physics to use them, though.

[zpconn]

The following two books are, in my opinion, the definitive introductions to the theory of manifolds:

Introduction to Topological Manifolds by John Lee (http://www.amazon.com/Introduction-Topo ... 0387950265)
Introduction to Smooth Manifolds by John Lee (http://www.amazon.com/Introduction-Smoo ... pd_sim_b_1)

I've read a large portion of Tensor Analysis on Manifolds but it just is not on the same level as these books in terms of quality.

A book I've recently stumbled upon and have found to be just fantastic is

A Classical Introduction to Modern Number Theory by Ireland and Rosen (http://www.amazon.com/Classical-Introdu ... 790&sr=1-1)

It should be accessible to anyone with a familiarity with elementary abstract algebra. This book is fast becoming my favorite introduction to the subject.

[jstnice]

i'm searching a good book about "Discrete maths", for semi-begginers (i had that assignature in the college, but i want to know more about Graphs, Groups, and the Language Theory!).

What could you recommend me?

(In e-bay, or something similar, i'm an outsider! =P )

you can come here .http://www.ams.org/mathweb/mi-books.html
I hope that can help you.

[Harg]

For a couple of summers now I've worked through a book on a topic that interested me, but I didn't have the time to focus on during the year. This year I'm looking at set theory. So now I need a book. It should be fairly extensive. It shouldn't be too elementary, as I'm not completely new to set theory (for example Halmos' Naive Set Theory isn't what I'm looking for). If there are exercises included, all the better. And it should serve as a good reference sort of thing, to quickly find some fact in, in the future. As a concrete example, if I were looking for something in category theory, MacLane's CTFTWM would fit perfectly, while Awodey's Category Theory would be a bit on the simplistic/long-winded side.

A quick check on amazon found this and this. Based on the preview pages both of these seem promising. I've looked at Just's and Weese's book before and found it very enjoyable, but I didn't have the time needed to put into it.

So what do you think? Do you recommend either of these two? Or something else?

[Quaternia]

Well, I have Basic Set Theory by Azriel Levy sitting at my desk right now.
It looks quite good; it definitely doesn't waste time (it's crammed with proofs and exercises).
I've put it to limited use for a logic course I had, and it served its purpose well, for whatever that's worth.
It's probably terrible for someone starting out with set theory, but as your post said, that's not the case for you.
I'm unfamiliar with both of the books you linked to, so I can't offer my opinion there.

[doogly]

oh god keep me away from them.

Ah, you've come to the wrong place!

[Mister_Penguin]

I've looked through the topic and I'm still a touch unsure - I'm looking for a text in abstract algebra, and I don't know what to choose at all. I don't have any real problem with formal rigor - I've taken a few proof-intensive courses. I'm just coming at it from kind of an odd angle - It's been a few years since I had an elementary linear algebra course. I'm not a math major (I'm CS), so I've mainly been taking things like theory of computation and combinatorics, rather than the standard "proof-intensive advanced calculus/analysis and linear algebra" sequence that most universities seem to use as a bridge to upper-level courses.

I was thinking about Dummit and Foote, but one professor I spoke with recommended Gallian, Herstein, or Isaacs. I really have no idea what to go with. Isaacs is admittedly a graduate text, but he said that it provided a rather complete coverage, and didn't really presuppose any prior knowledge of the material.
I really have no idea what to do - it seems like everyone loves at least one text from the set of undergraduate texts and hates at least one other. I really just want to learn abstract algebra to broaden my mathematical horizons and brush up on my proof writing. Should I just jump in on one and see how it goes? I want to try the graduate text, but I'm slightly worried about getting in over my head. What makes a text a graduate text, when it doesn't presuppose prior knowledge? Does it just tend to mean the author presumes extensive experience in proof reading/writing, and therefore adjusts the style accordingly (e.g. if the audience has lots of experience with reading proofs, the author might be a bit more laconic)?

[romulox]

As an undergraduate we used Herstein's Abstract Algebra (not to be confused with his Algebra). The book begins with basic ideas about sets and functions and gives a good undergrad introduction to group theory, ring theory and field theory. A few special topics are introduced at the end. One nice thing is the varied difficulty of the exercises.

In graduate school, my first two algebra courses used Hungerford's Algebra. The book gives a good introduction to the many areas in algebra. One of the main differences between an undergraduate and graduate level book is the level of detail provided for the reader. While some major theorems are proved in a given chapter, many are left as exercises. The book is excellent but I would recommend something else entirely for a newcomer to algebra.

I graded for a class which used Gallian's Contemporary Abstract Algebra. The book has a different layout than Herstein's or Hungerford's though includes much of the same material. There are many exercises and at the end of some chapters are computer exercises, applications, suggested readings, etc.

Whether or not to use a graduate text is somewhat dependant on your ability to read between the lines, fill in details, and assess the correctness of your own proofs. I would say most people would have a hard time getting into algebra for the first time from a graduate level book. Gallian and Herstein's are both suitable undergrad level books with enough introduction to satisfy a newcomer but also provide enough room to grow.

hope this helps

-romulox

[doogly]

I also want to give Artin's Algebra a shout out. I really liked this one, definitely worth giving a peek to if you have a chance. He takes a more geometric view, spends time with more some special problems, and gets into some topics often not done in a first course, like representation. I liked it much better than Gallian as an undergrad book, though Artin is harder. (They are both undergrad in that they assume the same level of preparation, but Gallian didn't require as much from the reader). In a more satisfying way, I thought. The only other one I have used is D&F, which is probably best left to a second romp through algebra also.

[quickly]

I completed up to vector calculus my first year at university, but haven't taken math since then. Next year, I am going to be concentrating on formal epistemology, and would like a good primer on model theory for studying formal languages. Any suggestions? The book would preferably be geared towards non-mathematics majors, and focus on logic applications. If the request is basically impossible, some course of books moving from algebra to model theory would be better, I suppose.

[mmmcannibalism]

I will be starting college in fall 2010 and want to use the summer to prepare for a linear algebra course. I have a pretty good background in AB and BC Calc(calc 1 and 2) but their was a prerequesite course for linear algebra I was allowed to skip. Apparently, it was a bit of a proof course that taught things such as proof by induction and introductory topics before linear algebra.

So, can anyone suggest me a good book to bridge the gap between calculus 1+2 and linear algebra?

[ThomasS]

So, can anyone suggest me a good book to bridge the gap between calculus 1+2 and linear algebra?

If you are willing to learn to think about proofs (and it sounds like you are), Lax's Linear Algebra is decent. Or are you looking for basic proof practice before linear algebra? I can try to think of something, but at the end of the day I think these things are best learned by simply reading books which require them. More to the point, there isn't really a gap between calc and linear algebra, they are not really connected so tightly at this level.

[martman]

Any one have any recommendations for Coding Theory? Maybe something with some Algebra tossed in too? This seems like a field where there should be a "standard" text or two, but I'm not finding much on my own.

Thanks

[mikhail]

I've had a lot of success applying basic geometry to problems which can be expressed that way over the years. I have a PhD in engineering, but I reckon most of the geometry I know comes from secondary (high) school. I've always compensated with a strong faculty for spatial relations. Recently, I´ve found myself struggling with the limitations of that - I'm working on generalising a very useful 2D result to 4D, and my lack of formal geometry has slowed me down a lot. Even something as simple as co-ordinate transformation of a line in 4D space - I've had to look up how to write the line (I quickly realised that l w + m x + n y + o z + c = 0 defines a volume!), and my intuition struggles with the transformations (it's not easy to picture 4D!)

Can anyone recommend a geometry textbook or books which would suit me? I think something which is simple but rigourous, or which compiles a lot of ideas and trickses for intuitive work would be appropriate (both would be ideal!).

[B.Good]

I took Linear Algebra last semester and I was into the abstract stuff (once I understood it and wrapped my head around it of course) enough to even try to do some work on my work on whatever class is after Linear Algebra. So I was wondering what is after Linear Algebra and what are some good books that cover that subject? Or if some book recommendations for the subject have already been given in this thread you can just name the subject and I can just look through the thread for the books.

[Talith]

If you like the abstract stuff (i.e the stuff to do with vector spaces as a mathematical object), you might like to read about group theory (or in general, abstract algebra). The subject in itself doesn't rely on much prior knowledge, although you'll need to know some basic structures like the different subsets of the real numbers and a little bit of modular arithmetic, other than that you're set to go. There are many many books on group theory and algebraic structures and a few have been mentioned in this thread (and others) so just see what you can find.

[B.Good]

If you like the abstract stuff (i.e the stuff to do with vector spaces as a mathematical object), you might like to read about group theory (or in general, abstract algebra). The subject in itself doesn't rely on much prior knowledge, although you'll need to know some basic structures like the different subsets of the real numbers and a little bit of modular arithmetic, other than that you're set to go. There are many many books on group theory and algebraic structures and a few have been mentioned in this thread (and others) so just see what you can find.

Thank you very much! There are some recommendations for abstract algebra even on this page (or the previous page if this post starts a new page) so that makes it quite easy for me.

[B.Good]

As an undergraduate we used Herstein's Abstract Algebra (not to be confused with his Algebra). The book begins with basic ideas about sets and functions and gives a good undergrad introduction to group theory, ring theory and field theory. A few special topics are introduced at the end. One nice thing is the varied difficulty of the exercises.

In graduate school, my first two algebra courses used Hungerford's Algebra. The book gives a good introduction to the many areas in algebra. One of the main differences between an undergraduate and graduate level book is the level of detail provided for the reader. While some major theorems are proved in a given chapter, many are left as exercises. The book is excellent but I would recommend something else entirely for a newcomer to algebra.

I graded for a class which used Gallian's Contemporary Abstract Algebra. The book has a different layout than Herstein's or Hungerford's though includes much of the same material. There are many exercises and at the end of some chapters are computer exercises, applications, suggested readings, etc.

Whether or not to use a graduate text is somewhat dependant on your ability to read between the lines, fill in details, and assess the correctness of your own proofs. I would say most people would have a hard time getting into algebra for the first time from a graduate level book. Gallian and Herstein's are both suitable undergrad level books with enough introduction to satisfy a newcomer but also provide enough room to grow.

hope this helps

-romulox

What book would you recommend if I wanted to try to learn Abstract Algebra on my own, or would you advise against that idea entirely and just wait until I can take an Abstract Algebra class? I made it through Linear Algebra all right, but I'm no genius. I can usually pick out stuff from a decent undergrad text without a teacher's help though.