Math Books

[doogly]

I'm not really set on one field yet, just trying to gain the necessary general knowledge that most professors assume you know in higher classes. I'm also looking for a good book that covers the Axiom of Choice. There's one by Horst Herrlich that looks good, is anyone familiar with it, or have another recommendation?

It seems like they don't assume you know anything, just that you are bright. (Which works for me, because I can take graduate math courses without having done an undergraduate major in math.) Getting set on a field (or two) is probably a good thing to do before you arrive. I got nothin for logic though.

[BraveSirRobert]

Any recommendations for excellent, well-written, elegant books for an undergraduate preparing for graduate school? I'm looking for books that are preferably at the graduate level, but can be read by someone who has finished algebra and analysis (at the level of Dummitt/Foote and Rudin), along with undergraduate topology and number theory. So far I have Ireland/Rosen for number theory, and Categories for the Working Mathematician. My professor also wants me to try Atiyah/MacDonald and Ahlfors. Anyone else have some suggestions?

Edit: Oh and I have Hatcher for Algebraic Topology. Yeah, I guess I do have a lot to keep me busy.... but suggestions would still be nice.

It sounds like you're more interested in algebra than analysis, but if you've gotten through Rudin and are looking for general grad-school preparation, you could always move on to so-called Big Rudin (Real and Complex Analysis). This is essentially a continuation of Principles of Mathematical Analysis which covers graduate-level analysis topics.

[kwood4800]

I'm going to school for math education and need to find middle-high school geometry trade books, either fiction or nonfiction. Nothing thats too advanced for a regular school student. If anybody has an suggestions that'd be great!

[dbh2ppa]

I need an introductory book on proof theory, anyone knows any good ones?

[fephisto]

Each section listed by my current liking. * if I own it, | if I'd rather not have it (but of course if it were free, or I already own, I'll take it)

Real Analysis:

Real Analysis and Foundations by Steven Krantz
Principles of Mathematical Analysis by Walter Rudin *
Undergraduate Analysis by Serge Lang
Real and Complex Analysis by Walter Rudin
| Real Analysis by Royden *
| A Friendly Introduction to Real Analysis by Kosmala *

Complex analysis:
Functions of One Complex Variable by Conway
Complex Analysis for Engineers by Stern *
Complex Analysis by Ahlfors
Complex Analysis by Stein *
Real and Complex Analysis by Rudin
| Function Theory of One Complex Variable by Steven Krantz

Algebra:
Contemporary Abstract Algebra by Gallian *
Algebra by Hungerford *
Abstract Algebra by Dummitt and Foote *
Abstract Algebra by Herstein

Linear Algebra:

Linear Algebra by Friedberg Insel and Spence (graduate)
Linear Algebra and its Applications by Lay * (undergraduate)
Matrix Analysis by Horn and Johnson * (useful for QM)
| Elementary Linear Algebra by Andrilli Hecker*

Proof:

Mathematical Reasoning by Sundstrom * (I'd really like to find a better book than this)

POST-SCRIPT:

The "International Series in Mathematics" is a good series. I think this is a good think to build up from if your undergrad or equivalent.

The "Undergraduate Texts in Mathematics" series published by Springer is a slightly more advanced collection.

And then when you've matured pick out the "Graduate Texts in Mathematics" series by Springer in what you're interested in. Or by this point you can probably finally pick out the 'big boy' names. Halmos, Rudin, Royden, Koblitz, Householder, Krantz, Herstein, Rosser, Church, {insert other big-name mathematician}, etc..

[Odin1415]

I'm looking for a good introductory set theory book. I'm an undergraduate math major, and everything seems like it's built on set theory! I want to learn more about it. Any suggestions would be appreciated

[doogly]

I'm looking for a good introductory set theory book. I'm an undergraduate math major, and everything seems like it's built on set theory!

That's what the set theorists want you to believe!

[fephisto]

I'm looking for a good introductory set theory book. I'm an undergraduate math major, and everything seems like it's built on set theory! I want to learn more about it. Any suggestions would be appreciated

Paul Halmos' "Naive Set Theory" maybe? I haven't looked at that one thoroughly though.

[Odin1415]

All right, I'll check into it. Thanks

[epigram]

I'm looking for two books for self study:

I'm using "Learning Bayesian Networks" by R. E. Neapolitan, but want something a bit more rigorous on Bayesian Networks. The proofs in this book are more like explanations than explicit statements and it really irks me.

A book on probability somewhere between, "Probability" by Karr and "A first course in Probability" by Ross, and ideally contains some introductory measure theory.

I'd greatly appreciate any suggestions,

Thanks,

[qinwamascot]

I'm looking for a good strong textbook on introductory level topology. The book I'm using, by Kahn (a tiny 384 page paperback), has nonstandard notation, some typos, and is sometimes hard to read. Ideally, it should be written in a nice reference style and focus just on first semester topology, as I will need a new book anyway for second semester material.

[doogly]

I really liked Armstrong's "Basic Topology."

[Harg]

I've heard good things about Dugundji or Munkres. I haven't gotten around to checking them out myself though.

[auteur52]

Definitely go with Munkres. I'm currently taking a Topology course, in which we rushed through all of the essential Point-Set Topology in 4 weeks, and have now started on Algebraic Topology. Munkres is admittedly weak on Algebraic, but the Point-Set section is top notch, and probably covers everything you could conceivably need to know as a professional mathematician (unless, of course, you become a (non-Algebraic) Topologist or Geometer).

[stockford]

I'm looking for books that cover all the important subjects in maths prior to calculus. So far, that I can think of:

Algebra, Geometry, Trigonometry, Functions and Graphs, Sequences and Sums.

So far I'm reading "Basic Mathematics", by Serge Lang. It's a horrible book because of the way he goes about things, and some of the answers are wrong, but I enjoy the application free, "proofiness" of it, and am I looking for more books with that kind of tone. Sullivan has a comprehensive "Algebra and Trigonometry" book but I'm worried it's going to be chock full of word problems and applied problems. I want to get good at pure maths, applications don't interest me. Another set of books is the Gelfand books (Algebra, Trigonometry, Functions and Graphs and Method of Coordinates). The Trig book looks good, but Functions and Graphs didn't look so good.

So yeah, any treatments of pre-calculus topics would be much appreciated, I want to be very comfortable with all kinds of functions, sequences, series, etc. For instance, I have no idea how to solve something like \frac{1}{x-a} + \frac{2}{2x-a} + \frac{5}{x+3a} = \frac{4}{x+a}. That's how good I want my algebra to be. Or how to work with sequences and sums- I know nothing about them, really.

[aro3n]

What is the best calculus book that is clear and concise and also easy to self-read/ teach yourself since I heard my school's AP BC Calculus book is miserable.

[Exeter]

What is the best calculus book that is clear and concise and also easy to self-read/ teach yourself since I heard my school's AP BC Calculus book is miserable.

I don't know about "best," but the Schaum's outline series is very cheap ($10 on Amazon) and easy to read. Gilbert Strang's text is decent and available for free via MIT's OpenCourseWare here. Finally, for a more geometric perspective, try Calculus Made Easy by Silvanus P. Thompson and Martin Gardner. This book is an update by Martin Gardner (an excellent mathematics writer, if I do say so myself) of a 1908 book of the same title. Don't let the fact that it's old bother you. After all, we're just talking about elementary real variable calculus, and all that stuff's been known for 3-400 years.

[Bluggo]

For Set Theory, I heard extremely good comments about "The Joy of Sets", by Devlin.

A question: I want to get a good statistics/probability theory book, suitable for both self-study and reference.

Any advice?

[doogly]

http://www.dartmouth.edu/~chance/teachi ... /book.html

This one has the bonus of being free! Just probability though.

[jaimeastorga2000]

I can think of a few free books which I don't think have been linked to. I'm pretty sure these have all expired in copyright, or have been released under creative commons, or have been uploaded by their creators. They are arranged from the "easy" subjects to the "hard." I hope this list helps somebody!

Geometry: Euclid's Elements
Algebra: Algebra
Trigonometry: Trigonometry
Calculus: Calculus
Calculus: Elementary Calculus
Calculus: Calculus Made Easy
Calculus: Multivariate Calculus In 25 Easy Lectures
Differential Equations: A Short Course On Differential Equations
Differential Equations: Higher Mathematics For Students Of Chemistry And Physics

As far as commercial books go, I found Stewart's Calculus - Early Transcendentals and Thomas' Calculus to be the best calculus textbooks. How To Prove It: A Structured Approach is a good proof and logic book.

[mmx49]

Someone posted this on GameFAQs of all places. I checked, but I didn't see it earlier in this thread or in the websites thread, so here it is:

http://www.e-booksdirectory.com/mathematics.php

A whole lotta free, online math books.

[dbh2ppa]

has anyone worked through spivak's "calculus"?

1) is it possible to work through it without prior mathematical training (other that high-school math)?
2) if the answer to the prior question is "no way, never, you're crazy", what's a good book to study to prepare for this one?
3) how long should it take to work though the exercises on the chapters? a day? two days? a week? a month? indefinite? go on until you solve them, regardless of the time? work on them for x hours, then look at the answer book or skip them?
4) is one supposed to solve the exercises based strictly on what's stated in the chapter (using only the definitions, axioms and theorems in the chapter and the previous ones)? or is one supposed to have/use prior knowledge?

[Alpha Omicron]

Here is Byrne's beautiful version of Euclid's Elements.

[ThomasS]

has anyone worked through spivak's "calculus"?

1) is it possible to work through it without prior mathematical training (other that high-school math)?
2) if the answer to the prior question is "no way, never, you're crazy", what's a good book to study to prepare for this one?
3) how long should it take to work though the exercises on the chapters? a day? two days? a week? a month? indefinite? go on until you solve them, regardless of the time? work on them for x hours, then look at the answer book or skip them?
4) is one supposed to solve the exercises based strictly on what's stated in the chapter (using only the definitions, axioms and theorems in the chapter and the previous ones)? or is one supposed to have/use prior knowledge?

I mostly know it by reputation, though I did just now look at a few pages via Amazon preview.

1) Beyond high school algebra, it appears to require only "mathematical maturity". Depending on your high school, logic puzzle experience, and so on you may already have some of this. If you don't, it may well be a decent book to work through in order to develop such maturity. Just understand that things might be slow going until you develop it. Also note that more typical calculus books tend to require and exercise far less mathematical maturity.

Roughly speaking, if you want to learn math for the sake of learning math - if you love puzzles and think you might be a mathematician - then it is probably a decent book to try. If you just want to get ahead of a normal college calc class there are probably easier books with more pretty pictures.

2) To develop mathematical maturity, I recommend searching technical or university bookstores for a book which has proofs and which can be read with work. One size doesn't fit all, but Dover has a range of inexpensive paperbacks which might be a decent place to start.

3) Spivak appears to have stars in front of some problems. At a glance I'd estimate that the star free problems should normally take 5-10 minutes. Starred problems appear to involve more brainstorming, I'd probably be ready to give them 30 minutes or more before asking for help.

4)It does appear to be fairly axiomatic in that sense.

[simmons]

Does anyone know of a book on multivariable calculus that is more of a list of definitions, theorems, etc.

I've already taken a full undergraduate level of mathematics, but I'm starting to do a lot of work in physics and I want a book that I can quickly skim for theorems and definitions from multivariable calculus that is small, light, and cheap.

[doogly]

Div Grad Curl and All That purports to be precisely what you are looking for. I haven't actually used it myself though.

[Fafnir43]

Div, Grad, Curl and All That is for vector calculus rather than general multivariable calculus. I remember it as being very, very good at explaining concepts, but not making any effort at all to do more than that - no exercises, no rigor, very few proofs. It also explains things in terms of electromagnetism, which could be good or bad depending on your physics background. It's definitely worth reading if you're having trouble understanding, say, surface integrals or Stokes' theorem on an intuitive level (for which I don't blame you), but not if you want a reference or if you're having trouble actually solving problems. I haven't really run across the sort of thing you're after - Riley, Hobson and Bence is very good for skimming definitions and theorems, but since it has a far wider scope than multivariable calculus it's not small or light and it's only cheap if you get an old edition secondhand.

[Penitent87]

For set theory and logic, "computability and logic" by boolos, burgess and jeffrey is a fairly good read. It contains problems at the end of each chapter, and is fairly easy to understand if read carefully.

"Combinatorics" by Peter J. Cameron is another very good read. It contains a wide range of subjects from counting to Ramsey theory to error correcting codes. The latter half of the book is, however, intended for graduate readers, so it proceeds at a brisk pace.

Other books I'd recommend are "Introduction to Topology" by Mendelson and "The Mathematical Theory of Communication" by Shannon. With both of these, you would benefit from a good mathematical background.

Lastly, "Mathematical Methods for Science Students" by Stephenson is a good introduction to mathematical methods at university. It was a useful reference for me during my studies.

[Alpha Omicron]

Can anyone offer an opinion on Multivariable Calculus, Linear Algebra, & Differential Equations by Grossman?

[asciifin]

I'm currently taking elementary linear algebra, and i <3 it!! I have to wait to take Algebraic Structures I, and am looking for good books on algebraic theory to read in the mean time.

[doogly]

My favorite intro algebra book is the one by Artin, just called Algebra. You may also be well served by Linear Algebra Done Right, it is a more abstract look at the same structures you are studying now (depending on what you do in your course). I've heard good things about it; haven't read it myself though.

[Qoppa]

To keep my math mind sharp over the summer, I want to teach myself as much about some field of math as possible, preferably not something I'll be taking courses in next year. So, I've narrowed my choices down to either graph theory or number theory. What's a good text book for either? Something for an undergrad pure math major who just finished his second year... For number theory, I'm looking at Introduction to the Theory of Numbers by Hardy, and for graph theory, Modern Graph Theory by Bollobas. Any other suggestions for texts, or for subjects I could study?

[doogly]

This raises the question, what will you be taking next year? And what have you had?

[Qoppa]

I guess all first two year curricula aren't created equal.

I have taken:
Calc I-III - Mostly non-rigorously, though I do know some basics of analysis as well.
Linear Algebra - Stuff up to canonical forms, inner product spaces, spectral theory...
Algebra - One semester of group and ring theory.
Probability - Just pretend I didn't learn this, because I hated this class. I'm only including it for completeness.

Next year will be (a guess, since I haven't picked next years classes yet):
Real analysis (redo calc I-III rigorously), complex analysis, more group theory, more field theory, modules and rings, ODE's.

So basically, I want to stay away from algebra and analysis since chances are whatever I can teach myself over the summer, I'll end up learning next year.

[doogly]

Your two suggestions sound like they'd work just fine, though those aren't areas I know much about. Someone else might chime in though.
You could pick up some geometry too, if you liked. I did Do Carmo's Differential Geometry in my second year, it was one of my favorite courses. Such a treat!
Also there are lots of topics in algebra you could go into after your first course, that your next two (or three, or four...) will avoid. So, if you really enjoyed the algebra you took, you might enjoy something like representation theory. Maybe you feel you've OD'ed on groups though.

[Durin]

Anyone have an opinion on Calculus with Analytic Geometry by George Simmons? I plan on beginning to teach Calculus to myself this Summer. I'm going to be in AP Calculus BC next year and I want to get a head start.

[Qoppa]

Also there are lots of topics in algebra you could go into after your first course, that your next two (or three, or four...) will avoid. So, if you really enjoyed the algebra you took, you might enjoy something like representation theory. Maybe you feel you've OD'ed on groups though.

Representation theory looks interesting... Any good introductory books?

Also, anyone have any experience with Fundamentals of Number Theory by William LeVeque?

[doogly]

Not too sure, but http://arxiv.org/abs/0901.0827 is easily downloadable, maybe a good place to take a look? And Fulton and Harris* I have heard is good, and full of explicit examples. Representation theory is a very examply topic so this kind of approach is much better than something too general and detail obscuring. This is a topic where I've never done a course, just picked up what was needed as needed, so I don't know particular books as well - just that it winds up frequently needed!

*Also easily downloadable depending on your scruples wrt Eknigu.

[Haeche]

If you're interested in more advanced mathematics, I have a list some where. I'll try to dig it up one day. Out of the top of my head, I was very impressed by Trefethen's Numerical Linear Algebra. If anyone is interested in Stochastic Calculus, I have a review of a couple books here.

[Haeche]

Don't mean to post twice, but just wanted to tell everyone that Dover's series on various mathematics is pretty awesome. I'm using Gelfand and Fomin's Calculus of Variations right now to brush up on the topics, and I'm definitely impressed. A couple other graduate students I know had nothing but good things to say about the same serie's books on Topology and Real analysis. A major factor is how cheap they are. 10 bucks for a pretty complete introductory treatment on Variations!??! Topology?!?! WOW!