No comments about Building a Foundation in Mathematics.

5 comments about The Square Root of Two.

1. Not exactly what I had expected, but still good nonetheless. Would make a good adjunct to an advanced high school math course.



2. An outstanding book by a master teacher, but with serious editing deficiencies.
   
   This is a relatively short and interesting book covering some substantial topics in mathematics. For the most part, it can be read without the need for pencil and paper. It is only five chapters long, and although easy to read, it does require concentration.
   
   This book is written as a discussion between a master teacher and an interested student. The teacher's words are shown in a boldface font, and the student's in an indented normal font. The selection of topics, the book's organization, and the dialog between teacher and student, help guide the reader in the appropriate direction. The book is clearly the work of an excellent teacher.
   
   The first chapter quickly displays the strengths and weaknesses of this book. A major strength is the author's ability to grab and hold your attention, making this book like a well-written mystery story, setting the stage to draw you in and to stay until a solution is revealed. It also displays the book's major weakness, poor editing.
   
   Unfortunately, there are numerous grammatical and substantive errors, some particularly serious for the reader new to the mathematics presented here. These appear as early as the first chapter. Some examples are: On page 11 the term "perfect number" is used incorrectly when "perfect square" is meant. On page 24 the word "is" is left out of a sentence. The illustration on page 23 represents the location of the "square root of two" differently, and incorrectly, as compared to the text discussion. In the sequence of square roots on page 30, the square root of six is inappropriately missing.
   
   Chapter 1. "Asking the Right Questions" shows how the square root of two can arise in the simplest of contexts, as the diagonal of a unit square (i.e., a square one unit on each side). It goes on to show how it is possible to get closer and closer to this square root's true value using integer fractions, but its does not yet prove that this value cannot be exactly represented this way.
   
   Chapter 2 introduces us to the proof of the irrationality of the square root of two and its consequences. After first presenting the proof in English, Dr Flannery shows how it can be concisely presented in mathematical notation. This Chapter explains the connection between the square root of two and the European A-Series paper sizes. It touches on Pell numbers as well as decimal expansions. The term "mixed decimal" as described in this Chapter is incorrect.
   
   Chapter 3, using more algebra than earlier, extends the previous material. Considering that the author assumes minimal mathematical sophistication from the reader, even explaining the term inverse, the material on pages 83-94 seems inappropriately demanding. That material would clearly benefit from a gentler presentation.
   
   The final two Chapters, 4 and 5, present some additional mathematical odds and ends, including the continued fraction expansion to approximate the square root of two, and some concepts connected to Gauss and Ramanujan.
   
   In summary, if the seriously deficient editing, the occasionally inappropriate definitions, and the slightly roller coaster requirements for mathematical maturity were corrected, this book could serve as an exemplar of the best teaching methods, i.e., focused questions that direct the student to find and confirm the right answers.



3. A well written introductory mathematics text that introduces the reader to the concept of irrational numbers, as well as explaining the apparent contradiction of being able to mark out a distance equal to root 2, yet at the same time the impossibility of measuring this distance exactly. The dialog style of writing makes for a very interesting approach to the teaching of mathematics. This book is a good read for all those with a general interest in matters mathematica.



4. This book is somewhat curiously organized, in the form of a dialog between a Master, who is a mathematician well acquainted with the material, and a student, who is apparently someone who has had elementary algebra but is a little uncomfortable with it. By the device of dialog, a lot of mathematics is brought out, all concerned in some way with the square root of 2.
   
   I found it pleasant to read, and recommend this book to anyone who is not so totally afraid of math that an equation scares them. You might learn some interesting math!



5. This is an excellent book for anyone interested in mathematical reasoning using the value of square root of 2 as a test case. The format is akin to a tutor-student (Socratic?) dialogue which makes the text quite interesting. Anyone with basic math skills can grasp it's contents. The text does require concentration to get the most out of it.
   
   There are quite a few editing errors in this book which is a shame because it does take away from it's quality and disrupts one's train of thought. These errors were quite annoying after a while. Some of the errors are howlers (like the one on pi). All in all a good read.

5 comments about Ant Colony Optimization (Bradford Books).

1. This book is a fine compilation of what have been done with the Ant Colony paradigm so far. Highly readable, even for people without previous experience in the field of optimization.



2. Fifteen years after the elegant double-bridge experiments by Deneubourg et al. that formed the basis of the Ant Colony Optimization algorithm, Marco Dorigo, the inventor of ACO, and Thomas Stützle, an expert on stochastic local search methods, have pooled their knowledge to summarize the current state of the art.
   This book gives a well paced introduction to ACO, describes its use in various optimization problems and gives interesting examples of its applications in industry. Explanations are clear and concise and, with the exception of a few well defined technical terms, free of scientific jargon. It is a pleasure to read for everyone with an interest in optimization theory. However, if you are looking for a book that celebrates the beauty of nature's problem solving capabilities, you are better of with "Swarm Intelligence" or Flake's "Computational Beauty of Nature". The initial idea of ACO may be bio-inspired, but this book has a crystal clear focus of the computational considerations in optimization theory.



3. Being an ant isn't very complex, but it's a daily fight for life. The losers in that fight don't count, but the winners get to vote.
   
   That is the basis of ant colony optimization. There are many parts to the idea, all of them very simple. First, there are many routes to the goal (food, if you're an ant) - some are better, some worse, you don't know which are which in advance, and the answer may change over time. Second, it's a random search. If you find any answer at all, no matter how convoluted, you get to vote on your route. Third, there are many other ants, all voting. Any leg of a trip that is heavily followed must be part of a good route, and gets many votes. There are details, but that's about it.
   
   Chapters 1-3 are the most readable, and convey the basic spirit of the family of algorithms. Ch. 4-6 will drag a bit, for the general reader, but go into significant detail about the ant algorithm and specific applications.
   
   Ch. 7 ends the book with a warm, informal discussion of the algorithm's history and some delightful variations. Dorigo, the principal author and founder of the ant school, uses this chapter to express his pure joy at having found such a wonderful thing, and at the similar approaches that others have also found.
   
   The approach has some real limits. For example, it can solve only problems that look like finding the shortest route. The good news is that a wide range of unlikely problems can all be cast in these terms. The better news is that, given the many variations available, some form of the 'stigmergic' approach will probably solve any problem in that range. Best of all, though, is the sheer cleverness and the sincere appreciation expressed by the authors.
   
   Nature is economical, but a brilliant problem solver. This is written by someone who as able to listen in on one of the lessons.
   
   //wiredweird



4. The central idea in the book is to analyse what evolution has provided us. In the form of ants being able to find the shortest path over terrain. This ability has inspired the research described herein.
   
   The book can be read as a fascinating deconstructionist approach to observing and manipulating ant colonies. By trying to look under the observations to discern the fundamental algorithms at work. And then to apply these to such longstanding contexts as the Travelling Salesman Problem.



5. Ant Colony Optimization focuses on the fact that ants foraging for food will quickly form a trail that is the shortest possible ditance betwen the food and home. Rach ant follows the scent trail laid on a path by previous travelers and adds its own pheromone to the scent, both going and coming. With a choice, ants tend to follow the strongest scent. Of a pioneer pair, the one choosing the shortest path will make the round trip before the other. Each pheromone trace evaporates in time, but an ant's antenna can detetct the slightest trace. That is a simplification of the introductory chapters of the book. The "pheromone trail" scheme is used to devise "artificial ant" which then takes part in the comnstruction of powerful ant algorithms for solving intractable problems such as the classical "Traveling Salesman" and other routing problems. The book is a complete text for a college course, with a large bibliography and many internal references to sources on the Internet. It is well written, with pseudo code showing how each algorithm can form computer programs. I can't evaluate the difficulty, but for me the math in later chapters is above my reach, but gratifying, nevertheless.

4 comments about The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes.

1. From over 300 references, David Darling has compiled what I, a non-mathematician, consider to be an excellent encyclopedia of mathematics. There are over 1800 entries. There are simple definitions, more in-depth explanations, graphs and many photos. He illustrates well the application and appearance of many abstract mathematical concepts in the real world of art, architecture, etc.
   
   In addition to the hundreds of 'pure' mathematical references, he also includes many entries that are fun for everyone. These entries include puzzles, games and tricks.
   
   I enjoyed the background and historical information included in the biographies of the many mathematicians covered. Historical information about concepts and values, e.g. pi, is also included, such as the time Indiana almost voted to round pi off to 3.2!
   
   This book would be an excellent library builder. It is hard to read straight through - I tried it and failed - but as reference and reading here and there it is great. It is good enough that I want to find his other references and check them out as well.



2. I enjoy reading mathematical dictionaries. Whether I read it from cover to cover or scattershot style, I always learn something new. In this dictionary, I learned about the "Seventeen or Bust" distributed computing project, where the goal of the project is to check the remaining seventeen possibilities to be the smallest Sierpinski number. I also was reminded of many other mathematical facts that I have encountered sometime in the past.
   The manuscripts that I receive as co-editor of Journal of Recreational Mathematics contain a wide variety of mathematical ideas. To handle them all, it is necessary to keep a mathematical dictionary handy. Since this book is well written and has over 1,800 entries, I have placed it on my essential reference shelf.
   
   Published in Journal of Recreational Mathematics, reprinted with permission.



3. I have had an interest in Recreational Mathematics ever since my High School days; at least 55 years ago. I can still remember constructing a 15 Puzzle using a block of wood and my mother's wooden yardstick,with numbers pasted on from a calendar.Then there was the cardboard set of Tangrams I made after reading about them in a book.The next thing I can recall was Magic Squares and learning how to solve any odd-number.I was then nearly driven to distraction trying to find the "secret" to solving even- numbered Magic Squares. In High School,after encountering Logrithms for the first time,I couldn't get enough of them.I was talking to my Math teacher one day after school,telling him how fascinating I found them. He reached into his desk drawer and pulled out a Slide Rule.He asked me if I knew what it was. I had never seen or heard of one. He showed me how it was based on Logrithims and could be used to multiply,divide ,etc. Then he handed me a little booklet and sent me packing with it and the slide rule. It was the most fascinating thing I'd ever seen and almost as much fun as girls and shooting pool.I guess my love for puzzles and such was what led me into pursuing college and eventually becoming an Engineer.
   All through the years,I've retained this interest in Puzzles and Recreational Mathematics. I started to acquire books on the subject,and once reading about Sam Loyd and the "Cyclopedia of Puzzles" ,I couldn't contain myself until I could at least see it and maybe even acquire a copy.It was published in 1914,and included over 5,000 puzzles,tricks,conundrums,riddles,etc.,of which about 1'000 are illustrated.Solutions are printed in the last pages.However,a number of puzzles were selected as "Prize Puzzles" and the solutions were withheld.A prize of $100 would be awarded to the person who sent in the best set of correct answers before January,1915.Because of errors,multiple solutions,impossible solutions,etc., it was impossible to determine a winner. It was quite a story!.After much effort,I finally got myself a copy and without doubt it is my prized puzzle book possession. I have never seen another copy,but it is sometimes shown in other puzzle books. I believe it has been reprinted ,and a condensed version was also published.
   Over the years,I have added many Math and Puzzle books to my library and now have about 600.
   I go through all this ,just to show you how much I think of this new book by David Darling.
   I don't know whether it should be called a Dictionary,an Encyclopedia, (it's not a "Cyclopedia" as Sam Loyd called his),a Compendium, or what. It is all those things and much more.But not to worry,Darling had to call it something and I guess his title is as good as any.
   It is a basic reference book dealing with all kinds of things,people ,definitions,theories,puzzles,terms, and Recreational mathematics. If you are reading about anything or anybody in these areas and want to know more;this will be the book to turn to to get started.
   The book is organized alphabetically,like an encyclopedia,but also has an Index by category. It has an extensive and excellent Reference as well.
   There is no doubt that I will turn to this book often in the future.It's only a shame I didn't have it ;or something similar to it many years ago.
   Please don't take a few things I'm going to say as nitpicking. That is not my intent,So,here goes;
   
   On page 278 it talks about the Rubik's Cube solution record being around 20 seconds. In 2003,the World Championship was held in Toronto.The winner was able to set a new record at around 16 seconds. This year,2007 a Canadian Championship was held. The winner did it with an average of 14.21 seconds.including one solving under 10 seconds.One solver managed to solve it in 4 minutes and 54 seconds...blindfolded.
   On page 117,the 15 Puzzle is said to have been invented by Sam Loyd who could not obtain a patent. In 2006,Jerry Slocum published a magnificient book on this puzzle.In it you will see who the real inventer was,and how Sam Loyd fooled everyone about it for 115 years. It was actually invented by Matthias J. Rice in 1879,and was originally called the Gem Puzzle.This book does a marvelous job of researching the history of this puzzle.(See my review,June 6,2006.) Mr Slocum has also recently published another excellent book on Tangrams which I also reviewed on Jan 6,2004.He has written several other books on Puzzles,has the world's largest puzzle collection,and heads up the Slocum Puzzle Foundation and Museum in Beverly Hills,California.
   The author has done a very good job of discussing Polyhedra,and has shown several in his book. I would like to point out the work done by Magnus Wenninger,who is the world's expert on constructing Polyhedron Models. He has written an excellent book on the subject as well. He has a web site showing many dozens of his models. Just search the net under Mangus Wenninger .If you enjoy these models,it will blow you away. You can even purchase them at extremely reasonable prices. I first became aware of him through someone who constructs similar models.He takes a different approach,and builds his using balls and sticks. They are simply delightful and remind me of the illustration of the molecule at the top of the cover of this book.
   Just to keep things interesting,there are many puzzles included throughout the book. If you have been interested in this sort of thing ,you'll have encountered many of them over the years. If Mathematical Recreations is new to you;this book is a wonderful introduction to it all.
   What more can I say? If you're into Mathemtical Recreations,Mathematics in general and Puzzles of all types ;you are going to love this book and want a copy.
   I have been searching for the name to describe someone who is interested in solving puzzles. The best I have been able to find so far are; METAGROBOLOGIST and OMNIHEURIST. if you know of any others ,I would be pleased to hear from you.
   
   Thank you David,it's a great book!



4. This encyclopedia of mathematics was quite worthwhile to read nevertheless.I expected an in-depth review of various famous math formulas and procedures for calculating numbers and short stories about the people who created them.Most of the items listed were interesting to read about.Yet,some were just trivial in nature.Based upon the glowing reviews,i felt compelled to investigate this book further.For the serious math student,this book is more crust than bread.For a book entitled,'Universal',it's fairly enclusive to the British and German mathematicians only. Now,puzzlers and ratzlers may be entertained by the charming entries within and even given the impetus to advance their research into weightier mathematic descriptions.However,this book only appears on the 'heavyside',not really enriching enough to nurture a sprouting engineer into fructation.A better title would simply be,"A Short History of Popular Mathematics and Puzzles".

5 comments about Proofs that Really Count: The Art of Combinatorial Proof (Dolciani Mathematical Expositions).

1. I haven't read this book yet, but I have a signed copy after seeing Jenny Quinn speak at the 2005 meeting of the Northwest chapter of the Mathmatics Association of America. If her written work is anything like her speaking, then this should be a great book. Her combinatorial proofs are an interesting approach to old equations, and she presents them in a very clear manner. A most enthusiastic lady.



2. I was introduced to this book by a talk that one of the authors (Arthur Benjamin) gave at the MAA Mathfest in Albuquerque in August of 2005. The talk was one of the very best mathematics talks that I've ever attended. Everyone in the audience could follow what was going on, and we all left with an understanding of the basic approach to combinatorial identities used in this book. The authors' approach is to prove combinatorial identities by defining a quantity and then obtaining different formulas for that quantity. One formula becomes the left hand side of an identity while another formula becomes the right hand side.
   
   When I read the book I found that it was just as clearly written, with lots of beautiful examples.



3. "Thoroughly engaging... Accessible to a very broad audience... While the theorems covered may not be new to research mathematicians, I would wager that very few of us have seen them proven in quite this way." -- American Mathematical Monthly [http://www.maa.org/reviews/reallycount.html]
   
   I am not a mathematician and I learn something cool and useful from this book every few paragraphs. Highly recommended.



4. The proofs in this book are easy enough for a bright high schooler or even an exceptional middle schooler to understand, while still making use of insightful tricks that keep the solutions far from being obvious.



5. This is just a fantastic book on combinatorics. Although not as long as most combinatorics texts it packs in a great deal of information. Very well-written with a lively style. Probably, the best thing about it for me is how clear the exposition is. I really learned a lot from this book.

4 comments about Mathematical Models of Financial Derivatives (Springer Finance).

1. The book is only a undergraduate textbook with no surprise. The author just tried to collect every method in linear pde and applied to finance area. Content is more or less same as Paul Wilmott's "option pricing : mathematical models and computations".



2. This is a well-written textbook for beginners in financial derivatives. It is very comprehensive as it covers various financial products. The main attraction of this book is its exercises. Many problems come from past academic papers. I benefit a lot from doing those drills.



3. This is a really lucid and detailed introduction to derivative pricing theory from the pde way of doing things. The author is an applied mathematician, of the fluid mechanics variety, and this should tell you right away what the drift of the presentation is like.

   Some will argue that all of Wilmott's books are along exactly the same line, so why do we need another pde book? Given the amazing number of textbooks dedicated to the martingale approach, it is great to have yet another, fresh way of looking at the pde approach.

   The derivations come with all the necessary technical details, the style is very down to earth, and to my mind original. There are many details that I personally haven't seen in any other textbook before, and there are plenty of what seem like very useful exercises.

   I really like this book, and it was a pleasant surprise to see it in a local library.




4. The goal of this book is to disseminate the knowledge of a very technical subject to a very wide range of audience, including finance professionals. The author did a respectable job in that regard. With some improvement in future revisions, this book seems to be one of the best introductionary texts on stochastic calculus.

No comments about MP Basic College Mathematics (softcover).

5 comments about The HarperCollins Dictionary of Mathematics.

1. I'm yet to find a better maths dictionary. Lots of definitions. Well organized/cross-referenced. As an undergraduate, I'm yet to look up something and be disappointed. Handy tables at the back.

   It doesn't get 5 stars because of a few glaring misprints.




2. If you're anything like me, you enjoy reading while in the toilet, doing one of the necessities of human nature, this reference has a great "random open" feature, where you can just slide your thumb at any page and find an interesting term you can think and learn about, I dont know how much the shallow discussion to these complex terms is really useful, but it gives you interesting things to think about, it also contains some mathematics history and some information about famous mathematicians, the paperback cover provides for some nice comfortable format, this dictionary has replaced the old Almanac I used to open randomly when I'm bored.
   A 5 star for the cuteness.
   
   Enjoy.



3. My job requires extensive technical writing skills in the form of developing mathematical algorithms or creating presentations or other documentation that frequently requires that mathematical concepts be clearly defined for the reader. Even when you think you know a good definition for a mathematical term, this dictionary will likely give you a better one. Good reference to keep around.
   
   This is perhaps one of the best, if not the best, mathematics dictionary. But it is just that, a dictionary of mathematical terms and phrases. Do no expect it to teach you any subject in mathematics. It would be great for any student taking a mathematics course. If the student runs across a term that he/she might have forgotten, the student can refer to this dictionary for a quick reminder.



4. Well written and edited. Definitions and explanations are clear and to the point. As I flipped through the book, I ran into many, many entries which I did not understand, but in that regard, an unabridged dictionary of the English language has many, many entries in it about which I know nothing.



5. I teach math in high school, and I have found this dictionary more useful than the textbook. The definitions are more precise and understandable than in a book designed to be understandable. It's compact, lightweight (because it's paperback) yet it's packed with information. It has been one of my better purchases this year.

No comments about Lectures on Discrete Geometry (Graduate Texts in Mathematics).

2 comments about Popular Woodworking Practical Shop Math (Popular Woodworking).

1. This book is one of those rare ones where the title literally says it all: It's a practical shop math guide; however, it's also a little more than that.
   
   The book starts with a review of basic math, focusing on operations involving fractions, followed by a review of basic geometry. The material about converting decimals to standard fractions is particularly good. The author does a very good job of offering a comprehensive review without being condescending or insulting. There are lots of examples and even exercises designed to allow you to test your understanding.
   
   The heart of the book is the section called "Using Math in the Workshop." This focuses on those things that should be simple, but which can drive you nuts, such as enlarging grid patterns, determining miter angles for polygons, dividing angles and arcs, calculating board feet, and so on.
   
   There's a sort of bonus feature I liked a lot, a very clear explanation of how to convert a picture into a dimensional drawing. This allows you to do something that seems impossible: You can make an actual plan out of a picture in a catalog or magazine. (It's not really easy to do so, but at least it's possible.)
   
   The book is very high quality and easy to use. The paper is thick and glossy, so it should be durable in the shop. It's spiral-bound, so the pages lay flat.
   
   After having it for just a few days, I can see that this book will become one of my most useful woodworking tools. I'm VERY happy I bought it, and I recommend it most highly.



2. Remember your Geometary books at school.. that is what this book reminds me of, as it acts as If you are just starting out. (these is a complete explanation of "Zero as a Placeholder", working with fractions, How to long devide, etc)
   Needs less explanations, and more diagrams... A bit too long winded. And while I am at it, ONLY US measure & Metric. I know it was prited for the US market, but a short explanation or noted in Imperial measure would make it international. (The US Gallon is aprox 20% shorter than the English Imperial Gallon.. etc..)
   Ring binder styles however is a nice idea, as it will lay flat in the workshop when many others dont.