Customer Reviews

100 Great Problems of Elementary Mathematics (Dover Books on Mathematics)

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This is a translation of a book originally published in 1932 under the title Triumph der Mathematik. The original title was better. Most of the problems here are far from elementary. For example, there is a nine-page proof of the Hermite-Lindemann theorem on the transcendence of pi and e, and a 12-page proof of Abel's theorem on the insolvability in closed form of equations higher than fourth degree. These are not what you normally call elementary problems. To understand them, and to understand their solutions, one might do better to consult more specialized texts in the areas under discussion.

On the other hand, the book is a gold mine of fascinating mathematics: How much must a sailboat tack with a north wind in order to get north as quickly as possible? From the altitude of two known stars determine your time and position. Construct the five regular solids. Prove that of all solids of equal surface the sphere possesses the maximum volume. Determine pi experimentally by throwing a needle across parallel lines.

The selection of problems is outstanding and lives up to the book's original title. The proofs are concise, clever, elegant, often extremely difficult and not particularly enlightening. To say that this book requires a background in college math is like saying that playing chess requires a background in how to move the pieces; it also requires a lot of thought and, preferably, a lot of experience.

I would recommend this book to practicing mathematicians, both amature and professional. For the rest of us, the author has surveyed more than 2,000 years of mathematical problems and picked out some real beauties.

100 Great Problems of Elementary Mathematics is such a goldmine of ingenuity that it is hard to comprehend how it could be sold for so low a price. Ten dollars is practically a steal.

This publication, which was translated into English back in 1965, is a concise summary of some of the greatest works of mathematics throughout mankind's history. The problems contained are quite challenging. Many are such that if you understood any one of them, then you would probably know something that even the best math professor nearest you would not. This may sound like an overstatement, but in a day and age where some PhD's in math have either forgotten or never really learned how to determine so little as the square root of a number by just pencil and paper, it is probably not.

It is from analyzing the book's passages of Bernoulli's Power Sum Problem that I was able to achieve a great mathematical triumph after discovering the following challenge found in William Dunham's The Mathematical Universe: determining a precise mathematical formula to figure out how Jakob Bernoulli could take all the positive integers from 1 to 1000, raise each of them to the tenth power, and then add them up to where the sum came up to over 30 digits! I tried to develop algorithms that would work but failed each time, until I, once again, read this volume.

The situations presented are quite difficult to grasp, but once you get to where you know how to apply any one of them in solving mathematical puzzles, you feel elated. I know I did.

For the individual who enjoys looking at mathematics in a historical context and who wants to approach problems that are perhaps not entirely solvable with the use of the calculator and/or the computer, I recommend this book.

Elementary algebra and ingenius ideas are combined to solve some of the most difficult problems in the history of math.This book helped me solve several difficult technical problems .
The concise treatment and cross reference to other solutions is outstanding . This is the finest treatment of advanced mathematical treatments I have ever seen. First published in 1932, it represents the best from the masters and can be used to discover tricks which were helpful to me in algorithm development . The treatment of astronomical problems alone is worth the price .

Perhaps the stress given to geometry gives evidence to the age of the book, but it still represents an example of how a collection of problems should be written. It is too entangled with mathematics to be defined an issue about mathematical games, but also fans of games can find out some enjoying items. Because, if much room has been given to proofs and resolutions, the boundary of elementary curiosity never goes out of sight, even if it can sometimes look like a far horizon. It is surely the best book about elementary problems, mathematical games and jokes I have ever read till now, and I have found its language as clear and straight as a non-English reader (like me) usually finds a non-English writer.

I love this book, and recommend it very highly, if you're the type who would like to understand, say, why the Fundamental Theorem of Algebra (every polynomial equation as a (possibly complex) root, is true. Yes, it takes intellectual effort to follow the proofs, but that can be incredibly rewarding, once you finally understand.

But this review is mostly to clarify the term "Elementary" in the title. This is used in a technical sense. Many (most?) of the theorems have multiple proofs. And sometimes the clearest proofs involve calculus, and often the calculus of complex variables. But if a proof doesn't involve calculus, then mathematicians refer to these as "elementary". It is in this sense that the title uses the term.

What does boat-sailing have to do with mathematics? Perhaps it is not mainstream mathematics, but apparently some think it is Elementary Mathematics, along with other easy things, say the Hermite-Lindemann theorem and the Abel Impossibility theorem. D"orrie apparently shares this view, but with whom?

Not everything in the book is so bizarre or non-elementary, though it does have full sections on Projective Geometry and Astronomy. There are some nicer problems, but the solutions seem to be a hard read --- and sometimes they seem to be incorrect (as in the proof of the fundamental theorem of algebra).

But the worst problem is the translation --- the book still has a German look and feel. While this may be considered a feature by some, I find it very annoying. For example, did you know that putting an overbar over a number denotes factorial? Have you heard of Fermat's Prime Number Theorem? (solution in the book)

So I can recommend it for the German audience --- unless they can find the original, which they will probably prefer...

squeezing 100 problems into less than 400 pages is not an easy task. There seems to be consistent tradeoff for brevity at the expense of readability. But if you are fairly experienced in elementary mathematics, most of the stuff can be followed. The degree of difficulty in following problems covered in this book varies greatly. Some can be appreciated by people with middle/high school math backgrounds. Others are very challenging to me (4th undergrad in engineering with good math background relative to undergrad math students). The big advantage of this book is that it deals with widely applicable, and historically significant, and applied math problems. This is in contrast to the bulk of math problem books out there that just deal with recreational problems (puzzle for example. but math olympic problems are recreational too, since the best high school students have to be able to solve them in around an hour). Look at the table of content and you will see. Highly stimulating.