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A classic originally published more than fifty years ago: Mathematics for the Millions: How to Master the Magic of Numbers. Although ostensibly written for the layman, it is not a light work. Its treatment of geometry is particularly good Courant, Richard, Herbert Robins. Revised by Ian Stewart. A sweet book that is similar in spirit to Stillwell's and that should be of interest to students of analysis is Pontrjagin, Lev S. The late Morris Kline wrote several good books for the layman as well as for the professional. My personal favorite is strong on history and art and I think deserves more attention than it has ever had. I think it is more important now then when it was first published in the 's: Mathematics in Western Culture. The following is a book I think every undergraduate math major who is at all serious should have: An Intuitive Journey in Higher Mathematics. There does not seem to be any other single volume that compares. The Princeton Companion to Mathematics. Still anyone who goes into grad school knowing all of this does not need my help. The Skeleton Key of Mathematics: Back to Top General Computer Science The books here tend cover algorithms and computability but don't forget to go the sections Algorithms and Logic and Computability. Dewdney wrote a book of 66 chapters to briefly and succinctly cover the interesting topics of computer science. The emphasis here is theory. This is a book every computer science major should have, and probably every math major and certainly anyone with a serious interest in computer science. The New Turing Omnibus. The Spirit of Computing, 2nd ed. A Foundation for Computer Science. In particular you should take courses in number theory and probability. Abstract algebra, linear algebra, linear programming-these and other areas can be useful. There are two books that are extremely good one-volume introductions at the undergraduate level. Tehy are very well written. I said in printed review that book by Mazur is the best book ever published on combinatorics, or something like that. The second book compares quite favorably. They are both junior-senior level. An Introduction to Combinatorics, 2nd ed. It is based on the notes from a course. As nice an introduction as you will ever see junior-senior level is this: Notes on Introductory Combinatorics. These books are all readable and are selective in their topics. By this I mean they avoid the too common approach of throwing in everything including the kitchen sink. A First Course in Discrete Mathematics. A Path to Combinatorics for Undergraduates. Proofs that Really Count: The Art of Combinatorial Proof. Great for self study. A lot on Fibonacci numbers. Another brief introduction at the sophomore level with some emphasis on logic and Boolean algebra. Does not touch either probability or number theory. Discrete Mathematics for Computing. A pedagogically solid book at the senior-graduate level devoted to counting is Martin, George E. The Art if Enumerative Combinatorics. This is in Springer's Undergraduate Text series but the first hundred pages out of cover generating functions and get well into Polya's counting theory. Another book that is quite formalistic and dry and reflects pre-computer science and yet I come back to again and again and is simply a favorite is: A book that is quite comprehensive and that is well written is: Its level is roughly senior to graduate school. It is divided into undergraduate and graduate halves. If I have to pick a single junior-senior text that is fairly conprehensive and seems designed for the classroom with like most such texts enough material for at least two semesters I would choose: Discrete Mathemtics, 2nd ed. They are both good reads and they overlap a number of disciplines, but arguably belong most to combinatorics. Note they do not belong in Foundations like the book by Ebbinghaus, H. The book by Bunch is excellent for the serious freshman-sophomore. The second book is more advanced and includes a nice treatment of Conway's own surreal numbers. The Kindom of Infinite Number: The Book of Numbers. Hartsfield, Nora, Gerhard Ringel. Pearls in Graph Theory: A Comprehensive Introduction, Revised ed. In truth it is not comprehensive. Secondly, although it covers algorithms it is not computer oriented. Algorithms are very much secondary. For finite geometries go to Batten. Graph theory has become important precisely because of algorithms. Let me mention two excellent books in order of my preference. Stanton, Dennis, Dennis White. The following two are exceptionally clear and well written. See also the book above by Benjamin and Quinn. The Fabulous Fibonacci Numbers. Numerical Analysis Most books on numerical analysis are written to turn off the reader and to encourage him or her to go into a different, preferably unrelated, field. Secondly, almost all of the books in the area are written by academics or researchers at national labs, i. The kind of industry I use to work in was a little different than that. The problem is partly textbook evolution. I've seen books long out of print that would work nicely in the classroom. However, textbook competition requires that newer books contain more and more material until the book can become rather unwieldy in several senses for the classroom. The truth is that the average book has far too much material for a course. Numerical analysis touches upon so many other topics this makes it a more demanding course than others. A marvelous exception to the above is the book by G. It avoids the problem just mentioned because it is based upon notes from a course. It is concise and superbly written. It is the one I am now teaching out of. Afternotes on Numerical Analysis. If you liked the first text you want this: Afternotes goes to Graduate school: Lectures on Advanced Numerical Analysis. The first is absolutely superb. Both books are great to read, but I don't like either as a text. Real Computing Made Real: Preventing Errors in Scientific and Engineering Calculations. Numerical Methods That Work. An interesting book that seems in the spirit of the first book by Acton above is: Breuer, Shlomo, Gideon Zwas. I would like to know however how it has done as a text. A book by a great applied mathematician that is worth having is: Good popular books

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8 Comments

  1. The truth is that the average book has far too much material for a course. Does not touch either probability or number theory. Secondly, almost all of the books in the area are written by academics or researchers at national labs, i.

  2. Number Theory with Applications. Abstract algebra, 2nd ed. Let's just say that Jessica isn't ready for the secrets and discoveries nor the true love that is sure to come.

  3. If I have to pick a single junior-senior text that is fairly conprehensive and seems designed for the classroom with like most such texts enough material for at least two semesters I would choose: Elementary Number Theory and its Applications, 5th ed. Elements of Number Theory.

  4. Both books are great to read, but I don't like either as a text. Afternotes on Numerical Analysis.

  5. A lot on Fibonacci numbers. A Comprehensive Introduction, Revised ed. One book that I assume must be great is the following.

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