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Great books about mathematics

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Re: Great books about mathematics

#31

Post by jfarmer »

alok-g wrote: post #29
jfarmer wrote: post #28

Earlier quoted context omitted.

What are you looking for that isn't simply a math textbook? Honestly, that's how mathematicians learn "actual" mathematical subjects, too. If you're comfortable with calculus as a subject, for example, and want a "pure mathematics" approach, I recommend Michael Spivak's Calculus . If you've never worked through a pure math textbook from start to finish, that's a good start. There's not that much interesting in the th…

Good points. I have already gone through several levels of engineering mathematics, so do understand differential and integral calculus well. I am assuming that is that Spivak's book is about -- please correct me if I am wrong; Amazon is not showing a preview of the book. You are right in bringing countability arguments into the picture; I understand them only to some level. I would love to read a book that gives it…

Spivak's Calculus starts with a set of 13 axioms which characterize the real numbers and then derives all the results you're familiar with in calculus. It's rigorous in the mathematical sense, so if you've never worked through a rigorous math textbook before then this might be a good start since you're familiar with the underlying material.

Here are some exercises to give you a sense of the flavor. If you find these exercises trivial then the textbook might not be for you. If you find them hard, well, welcome to math! :)

These are all before we get to any "calculus." Here "function" means a function of the real numbers.

1. Let f be a function that satisfies the conclusions of the Intermediate Value Theorem. Prove that if f takes on each value only once then f is continuous. Generalize this to the case where f takes on each value only finitely many times.

2. Prove that if n is even, then there is no continuous function f which takes on every value exactly n times.

3. A set A of real numbers is said to be sense if every open interval contains a point of A. Prove that if f is continuous and f(x) = 0 for all numbers x in a dense set A then f(x) = 0 for all x.

4. Find a function which is continuous at every irrational point and discontinuous at every rational point (and prove it as such)

Spivak's Calculus is used as a first-year calculus textbook at lots of schools, so if you find the above even a little challenging or strange-seeming then I'd recommend going through the book.

The last chapter of the textbook is a rigorous construction of the real numbers from the rationals using Dedekind cuts (referenced in the first link).

Re: Great books about mathematics

#32

Post by alok-g »

jfarmer wrote: post #31
alok-g wrote: post #29

Earlier quoted context omitted.

Good points. I have already gone through several levels of engineering mathematics, so do understand differential and integral calculus well. I am assuming that is that Spivak's book is about -- please correct me if I am wrong; Amazon is not showing a preview of the book. You are right in bringing countability arguments into the picture; I understand them only to some level. I would love to read a book that gives it…

Spivak's Calculus starts with a set of 13 axioms which characterize the real numbers and then derives all the results you're familiar with in calculus. It's rigorous in the mathematical sense, so if you've never worked through a rigorous math textbook before then this might be a good start since you're familiar with the underlying material. Here are some exercises to give you a sense of the flavor. If you find these…

Sounds good. Thanks!

Re: Great books about mathematics

#33

Post by swatkat »

Tycho wrote: post #18

Anyone have good reading material about 'how to reason mathematically' on a basic level? I mean, not going too deep into any specific topic, but how to get better at interpreting equations and grokking relationships.

Check out 'The Joy of x' by Steven Strogatz.

http://www.stevenstrogatz.com/the_joy_of_x.html

Re: Great books about mathematics

#34

Post by bumbledraven »

Tycho wrote: post #18

Anyone have good reading material about 'how to reason mathematically' on a basic level? I mean, not going too deep into any specific topic, but how to get better at interpreting equations and grokking relationships.

Any artofproblemsolving.com book will be outstanding for this purpose. I suggest starting with Prealgebra by Richard Rusczyk.

Re: Great books about mathematics

#35

Post by calvins »

tmoertel wrote: post #9

Another great book, starting with the basics and then exploring a wide range of mathematical subjects, is The Princeton Companion to Mathematics : http://press.princeton.edu/titles/8350.html It's fun to just flip it open and start reading.

This is a wonderful book. A word of caution though: don't get the Kindle version from Amazon. I thought I'd get it for the Kindle, because the print version is huge and I sometimes like to read in bed and don't want to lug the dead tree version I own around, but I had to return it because of the standard problems with garbled notation (some symbols appearing as a square box or some other incorrect symbol) that affects every math book I've ever purchased on Amazon for the Kindle.

There error rate was much lower than in any other math book I've tried, but still much too high.

Re: Great books about mathematics

#36

Post by anaphor »

Here is a great one: http://www.amazon.com/Introduction-Graph-Theory-Dover-Mathem...

"A stimulating excursion into pure mathematics aimed at "the mathematically traumatized," but great fun for mathematical hobbyists and serious mathematicians as well. Requiring only high school algebra as mathematical background, the book leads the reader from simple graphs through planar graphs, Euler's formula, Platonic graphs, coloring, the genus of a graph, Euler walks, Hamilton walks, and a discussion of The Seven Bridges of Konigsberg."

Re: Great books about mathematics

#37

Post by calvins »

Tycho wrote: post #18

Anyone have good reading material about 'how to reason mathematically' on a basic level? I mean, not going too deep into any specific topic, but how to get better at interpreting equations and grokking relationships.

You might enjoy one of the many books that exist for undergraduates to ease the transition into higher math classes, where there is a shift from the strong calculation-focus and rote-learning of most math teaching through calculus, to the more proof-centric and understanding-based approach that one finds in classes like abstract algebra, real and complex analysis, and other post-calculus math courses.

Here are some examples of the kinds of books I mean, and you can find others by following Amazon recommendations from those:

http://www.amazon.com/Nuts-Bolts-Proofs-Fourth-Edition/dp/01... http://www.amazon.com/Mathematical-Proofs-Transition-Advance...

With math textbooks especially, it pays to look for a previous edition, as the current edition can be ridiculously expensive, and the previous edition might be only 20% of the price, with no significant differences between the two.

Also, don't get them for the Kindle, as Amazon doesn't seem capable of publishing a math book with lots of notation that doesn't also have tons of errors where symbols get incorrectly imported. I've bought at least 20 and yet have to see one that didn't have lots of incorrect symbols.

Re: Great books about mathematics

#38

Post by auctiontheory »

lovemath wrote: post #24
Tycho wrote: post #18

Anyone have good reading material about 'how to reason mathematically' on a basic level? I mean, not going too deep into any specific topic, but how to get better at interpreting equations and grokking relationships.

My wife gave me this great book for Christmas: Love and Math: The Heart of Hidden Reality by Edward Frenkel It begins with the author's struggle to learn the math behind quantum physics in spite of cold-war era soviet educational obstacles and leads bit by bit into the Langlands program, drawing connections between group theory, number theory and harmonic analysis. It's definitely my favorite book of 2013. http://www…

Yup. My thought as I read Love and Math a couple of months ago was "this would be great to give to a high school senior who is wondering whether to continue with mathematics." If you truly love doing math, you'll feel it when you read this book.

Re: Great books about mathematics

#39

Post by Someone »

jfarmer wrote: post #31
alok-g wrote: post #29

Earlier quoted context omitted.

Good points. I have already gone through several levels of engineering mathematics, so do understand differential and integral calculus well. I am assuming that is that Spivak's book is about -- please correct me if I am wrong; Amazon is not showing a preview of the book. You are right in bringing countability arguments into the picture; I understand them only to some level. I would love to read a book that gives it…

Spivak's Calculus starts with a set of 13 axioms which characterize the real numbers and then derives all the results you're familiar with in calculus. It's rigorous in the mathematical sense, so if you've never worked through a rigorous math textbook before then this might be a good start since you're familiar with the underlying material. Here are some exercises to give you a sense of the flavor. If you find these…

"3. A set A of real numbers is said to be sense if every open interval contains a point of A."

For those googling: there's a typo: sense => dense (http://en.wikipedia.org/wiki/Dense_set)

Re: Great books about mathematics

#40

Post by jfarmer »

Someone wrote: post #39
jfarmer wrote: post #31

Earlier quoted context omitted.

Spivak's Calculus starts with a set of 13 axioms which characterize the real numbers and then derives all the results you're familiar with in calculus. It's rigorous in the mathematical sense, so if you've never worked through a rigorous math textbook before then this might be a good start since you're familiar with the underlying material. Here are some exercises to give you a sense of the flavor. If you find these…

"3. A set A of real numbers is said to be sense if every open interval contains a point of A." For those googling: there's a typo: sense => dense ( http://en.wikipedia.org/wiki/Dense_set )

Indeed! Me and my fat fingers. :)
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