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A Computational Introduction to Number Theory and Algebra

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Re: A Computational Introduction to Number Theory and Algebra

#11

Post by S4M »

Ebbit wrote: post #2

What background of mathematics does this book assume?

I read a bit the chapter about the distribution of prime numbers. I could follow because I have an Msc in maths (not in number theory) but the progression is fast, and while I don't think one needs an advanced background to read it, one needs to be used to formal reasoning, and I feel like the author is moving fast, so I guess for someone with a low background in maths it would be a hard (but not impossible) read.

Re: A Computational Introduction to Number Theory and Algebra

#12

Post by tomku »

anton-107 wrote: post #8
xyzzyz wrote: post #3

Earlier quoted context omitted.

From the preface: Prerequisites. The mathematical prerequisites are minimal: no particular mathematical concepts beyond what is taught in a typical undergraduate calculus sequence are assumed. The computer science prerequisites are also quite minimal: it is assumed that the reader is proficient in programming, and has had some exposure to the analysis of algorithms, essentially at the level of an undergraduate course…

What would be a good textbook for Math 101, specifically to learn some advanced mathematical formalism without actually diving in applied science behind it?

Keeping with the free theme, Book Of Proof by Richard Hammack is a nice introduction to proofs and formalism. It's available free from the author as a PDF[1], and also as a physical book on Amazon[2].

An alternative if you're willing to spend a little is How to Prove It by Daniel J. Velleman, also available from Amazon[3] and probably many other retailers. Both books cover roughly the same topics.

[1]: http://www.people.vcu.edu/~rhammack/BookOfProof/

[2]: http://www.amazon.com/Book-Proof-Richard-Hammack/dp/09894721...

[3]: http://www.amazon.com/How-Prove-It-Structured-Approach/dp/05...

Re: A Computational Introduction to Number Theory and Algebra

#13

Post by jordigh »

anton-107 wrote: post #8
xyzzyz wrote: post #3

Earlier quoted context omitted.

From the preface: Prerequisites. The mathematical prerequisites are minimal: no particular mathematical concepts beyond what is taught in a typical undergraduate calculus sequence are assumed. The computer science prerequisites are also quite minimal: it is assumed that the reader is proficient in programming, and has had some exposure to the analysis of algorithms, essentially at the level of an undergraduate course…

What would be a good textbook for Math 101, specifically to learn some advanced mathematical formalism without actually diving in applied science behind it?

This stuff is usually rolled into courses called "abstract algebra". If a textbook is called "abstract algebra", it's usually designed for a first year undergraduate. If it's just called "algebra", it's usually aimed at a more mature audience. Herstein and Hungerford are the texts I learned from. This book seems more recent and popular:

http://www.amazon.com/Book-Proof-Richard-Hammack/dp/09894721...

The other big stream of basic undergraduate mathematics is analysis. For that I recommend Spivak's Calculus.

Re: A Computational Introduction to Number Theory and Algebra

#14

Post by jordigh »

Hackernaut wrote: post #5

How relevant is this book to artificial intelligence

Depends what you mean by AI. Back in the day, a computer algerbra system was considered AI, in which case, this book is very relevant. But I can't think of another interpretation of AI that would make this book relevant.

Re: A Computational Introduction to Number Theory and Algebra

#16

Post by williamstein »

mrcactu5 wrote: post #6

Algebraic Number Theory: A Computational Approach http://wstein.org/books/ant/ant.pdf A Course in Computational Algebraic Number Theory http://bit.ly/1heah8l

Author of "Algebraic Number Theory: A Computational Approach" here, in case anybody has any questions. Here's the history of that book. I first taught an undergraduate class at Harvard in maybe 2002 and went over the first 20 pages of Swinnerton-Dyer's brief course on algebraic number theory book -- expanding it into course-length notes. I taught the course next to grad students at UC San Diego, and added more content inspired by the excellent "Algebraic Number Theory" by Cassels-Frohlich. Then I taught it again twice at Univ of Washington, adding more modern computational content, and resulting in a rough draft of this book. Finally, Travis Scholl (a UW grad student) and I spent the last year polishing it and making it look a bit nicer. The book is under contract to be published by the American Mathematical Society soon.

Re: A Computational Introduction to Number Theory and Algebra

#17

Post by pm90 »

williamstein wrote: post #16
mrcactu5 wrote: post #6

Algebraic Number Theory: A Computational Approach http://wstein.org/books/ant/ant.pdf A Course in Computational Algebraic Number Theory http://bit.ly/1heah8l

Author of "Algebraic Number Theory: A Computational Approach" here, in case anybody has any questions. Here's the history of that book. I first taught an undergraduate class at Harvard in maybe 2002 and went over the first 20 pages of Swinnerton-Dyer's brief course on algebraic number theory book -- expanding it into course-length notes. I taught the course next to grad students at UC San Diego, and added more conten…

Thanks for putting this out there for free! I think its an amazing thing to do, especially for the more academic books. Anecdotally, my friend, who is now a mathematics grad student in University of Western Ontario, learned mathematics entirely from ebooks and the low price editions that you find in India.

Re: A Computational Introduction to Number Theory and Algebra

#18

Post by visarga »

Not saying the book isn't good, but I have a general observation to make. Such books would be better if they provided a sequence of carefully difficulty-graded exercises that would build on towards practical mastery. Instead, there is a flood of theorems with a spattering of exercises. With two exercises one can't feel confident about learning a theorem or other deep complex math concept.

It's one thing to read a theorem, another to be confident to apply it. When I was learning math in university, it was the same. Theorems, axioms and definitions by the truckload, but exercises - nada. In reality it all comes down to applying math.

Re: A Computational Introduction to Number Theory and Algebra

#19

Post by semigroupoid »

williamstein wrote: post #16
mrcactu5 wrote: post #6

Algebraic Number Theory: A Computational Approach http://wstein.org/books/ant/ant.pdf A Course in Computational Algebraic Number Theory http://bit.ly/1heah8l

Author of "Algebraic Number Theory: A Computational Approach" here, in case anybody has any questions. Here's the history of that book. I first taught an undergraduate class at Harvard in maybe 2002 and went over the first 20 pages of Swinnerton-Dyer's brief course on algebraic number theory book -- expanding it into course-length notes. I taught the course next to grad students at UC San Diego, and added more conten…

Thank you for such a great work and especially for making it available for free! For anyone interested, I can also recommend trying out Sage! There is also this free book (http://abstract.pugetsound.edu/sage-aata.html) on Abstract Algebra which includes Sage exercises. The most prominent application of number theory in computer science is probably cryptography. I can highly recommend "An Introduction to Mathematical Cryptography" by Hoffstein, Pipher and Silverman in that area.
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