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The Enjoyment of Math Reprint edition by Rademacher, Hans, Toeplitz, Otto (1966) Paperback
Requiring no more background than plane geometry and elementary algebra, this book leads the reader into some of the most fundamental ideas of mathematics, the ideas that make the subject exciting and interesting.
- GenresMathematics
Paperback
First published March 1, 1990
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Displaying 1 - 3 of 3 reviews
February 2, 2014
"The Enjoyment Of Mathematics" is one of those books that's either really easy to recommend, or really easy to not recommend. Not really interested in math? This book will definitely not change your mind. Simply curious about mathematics? Even so, stay away. "The Enjoyment of Mathematics" will make your eyes glaze over unless you either majored in mathematics, or kinda wish you did. In which case - this book is pretty dope.
This is essentially a collection of proofs. The chapters vary as far as the subject matter is concerned, but the format of each chapter is pretty consistent: 1) An easily digestible, page-long introduction to a problem. This might include historical background, of simply some toy examples that exhibit a property we're interested in exploring. 2) The statement of a theorem. 3) A full, uncompromising proof of said theorem, along with all its supporting lemmas. 4) A brief discussion of why this result motivates related questions.
Your reaction to the above paragraph is a good indicator as to whether or not you'll enjoy this book. If you're still interested, there are three further considerations:
1) how interesting are the problems?
2) how good are the proofs?
3) how difficult is the subject matter?
For the most part, the problems are pretty awesome. Most of the material falls under the umbrella of geometry and number theory, though the authors also touch on combinatorics, set theory, and topology. Notably, this means that none of the material presupposes any mathematical knowledge beyond the high school level. The star of the show is the reasoning, not the results.
Here are some representative chapters, along with the theorems they're concerned with proving:
The Sequence of Prime Numbers: there is an infinite number of primes; the set of numbers that have a reminder of 2 when divided by 3 include an infinite number of primes; for any given N, it's possible to find N consecutive composite numbers
Some Maximum Problems: the square is larger than all rectangles of equivalent perimeter; the geometric mean is always smaller than the arithmetic mean; the equilateral triangle has a larger area than any other triangle inscribed in the same circle
Pythagorean Numbers and Fermat's Theorem: the derivation of a formula for all Pythagorean triplets; proof by infinite descent that "a^4 + b^4 = c^4" has no integer solutions
"Periodic Decimal Fractions: the period of a/b can be no longer than the number of remainders that are prime to b; the length of the period when a/b is written in decimal form is the smallest n for which 10^n - 1 is evenly divisible by b; all reduced fractions a/b with the same denominator have periods of the same length; Fermat’s little theorem
The proofs themselves are, for the most part, quite good. The authors never make large leaps in logic, so with enough effort, it's always possible to follow along. My main complaint here is the some of the proofs are totally unmotivated. More than once I found myself thinking "sure, but how does this get us any closer to the result we're after?" It all makes sense in hindsight, of course, but it's often nice to outline a difficult proof before taking the plunge.
Finally - this is a difficult book, though never an impossible one. You're unlikely to enjoy this book unless you've previously encountered at least a few of its results, so some of the material makes for easy reading (think: the infinitude of the primes, perfect numbers, and diagonalization in set theory). For the most part, however, you'll have to pay close attention, and a few of the chapters are downright difficult (ex "The Indispensability of the Compass for the Constructions of Elementary Geometry"). Fortunately, the difficulty is a function of the material, not its presentation, so it never feels unfair. If you take the time to make sure you agree with the correctness of every last bit of reasoning, expect to make it through between 10 to 20 pages an hour.
In sum, "The Enjoyment of Mathematics" is a demanding but enjoyable foray into proof-based math. Again, this is not your everyday pop-math book aiming to convince non-believers that math can, in fact, be fun. "The Enjoyment of Mathematics" reads more like a textbook than anything else - and that isn't necessarily a bad thing. Presupposing only a minimal foundation of background knowledge and using nothing but deductive reasoning, this book presents varied and ingenious proofs of some fascinating results. As "best of" collection of proofs, then, this book will delight some - and sedate others. If you truly enjoy mathematics, there's plenty of enjoyment to be had here. If not, this book will only further the conviction that its title is a contradiction in terms.
This is essentially a collection of proofs. The chapters vary as far as the subject matter is concerned, but the format of each chapter is pretty consistent: 1) An easily digestible, page-long introduction to a problem. This might include historical background, of simply some toy examples that exhibit a property we're interested in exploring. 2) The statement of a theorem. 3) A full, uncompromising proof of said theorem, along with all its supporting lemmas. 4) A brief discussion of why this result motivates related questions.
Your reaction to the above paragraph is a good indicator as to whether or not you'll enjoy this book. If you're still interested, there are three further considerations:
1) how interesting are the problems?
2) how good are the proofs?
3) how difficult is the subject matter?
For the most part, the problems are pretty awesome. Most of the material falls under the umbrella of geometry and number theory, though the authors also touch on combinatorics, set theory, and topology. Notably, this means that none of the material presupposes any mathematical knowledge beyond the high school level. The star of the show is the reasoning, not the results.
Here are some representative chapters, along with the theorems they're concerned with proving:
The Sequence of Prime Numbers: there is an infinite number of primes; the set of numbers that have a reminder of 2 when divided by 3 include an infinite number of primes; for any given N, it's possible to find N consecutive composite numbers
Some Maximum Problems: the square is larger than all rectangles of equivalent perimeter; the geometric mean is always smaller than the arithmetic mean; the equilateral triangle has a larger area than any other triangle inscribed in the same circle
Pythagorean Numbers and Fermat's Theorem: the derivation of a formula for all Pythagorean triplets; proof by infinite descent that "a^4 + b^4 = c^4" has no integer solutions
"Periodic Decimal Fractions: the period of a/b can be no longer than the number of remainders that are prime to b; the length of the period when a/b is written in decimal form is the smallest n for which 10^n - 1 is evenly divisible by b; all reduced fractions a/b with the same denominator have periods of the same length; Fermat’s little theorem
The proofs themselves are, for the most part, quite good. The authors never make large leaps in logic, so with enough effort, it's always possible to follow along. My main complaint here is the some of the proofs are totally unmotivated. More than once I found myself thinking "sure, but how does this get us any closer to the result we're after?" It all makes sense in hindsight, of course, but it's often nice to outline a difficult proof before taking the plunge.
Finally - this is a difficult book, though never an impossible one. You're unlikely to enjoy this book unless you've previously encountered at least a few of its results, so some of the material makes for easy reading (think: the infinitude of the primes, perfect numbers, and diagonalization in set theory). For the most part, however, you'll have to pay close attention, and a few of the chapters are downright difficult (ex "The Indispensability of the Compass for the Constructions of Elementary Geometry"). Fortunately, the difficulty is a function of the material, not its presentation, so it never feels unfair. If you take the time to make sure you agree with the correctness of every last bit of reasoning, expect to make it through between 10 to 20 pages an hour.
In sum, "The Enjoyment of Mathematics" is a demanding but enjoyable foray into proof-based math. Again, this is not your everyday pop-math book aiming to convince non-believers that math can, in fact, be fun. "The Enjoyment of Mathematics" reads more like a textbook than anything else - and that isn't necessarily a bad thing. Presupposing only a minimal foundation of background knowledge and using nothing but deductive reasoning, this book presents varied and ingenious proofs of some fascinating results. As "best of" collection of proofs, then, this book will delight some - and sedate others. If you truly enjoy mathematics, there's plenty of enjoyment to be had here. If not, this book will only further the conviction that its title is a contradiction in terms.
November 28, 2009
I feel like there was probably a good amount of good math in this book, but I just couldn't get into it. I don't know if it's because I was trying to read it at the end of sort of busy days, or if it's because most of the things that I'd find interesting in the book are things I basically already know.
April 20, 2007
classic problems.
Displaying 1 - 3 of 3 reviews