A beautiful and relatively elementary account of a part of mathematics where three main fields - algebra, analysis and geometry - meet. The book provides a broad view of these subjects at the level of calculus, without being a calculus book. Its roots are in arithmetic and geometry, the two opposite poles of mathematics, and the source of historic conceptual conflict. The resolution of this conflict, and its role in the development of mathematics, is one of the main stories in the book. Stillwell has chosen an array of exciting and worthwhile topics and elegantly combines mathematical history with mathematics. He covers the main ideas of Euclid, but with 2000 years of extra insights attached. Presupposing only high school algebra, it can be read by any well prepared student entering university. Moreover, this book will be popular with graduate students and researchers in mathematics due to its attractive and unusual treatment of fundamental topics. A set of well-written exercises at the end of each section allows new ideas to be instantly tested and reinforced.
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41 of 41 people found the following review helpful:
5.0 out of 5 stars
A starting point for independent mathematical explorers,
This review is from: Numbers and Geometry (Hardcover)
This is a book on elementary mathematics that treats its readers as curious human beings rather than intellectually spineless fools that must be drilled. There are excellent explanations of a lot of mathematics--some common, some unusual--and there are exercises that are actually interesting. Even more important, I believe, is that it opens so many doors for further reading, not least in areas that many undergraduates never get to see: classical geometry, classical number theory, foundations, non-Euclidean geometry, elliptic curves, algebraic number theory, projective geometry, etc. Also unique is the way in which every topic comes with truly meaningful and interesting applications: complex numbers lead to Gaussian integers and Fermat's two squares theorem, conic sections lead to projective geometry, trigonometry leads to Dehn's solution of Hilbert's third problem, etc.
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Inside This Book
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First Sentence:
The beauty and fascination of numbers can be summed up by one simple fact: anyone can count 1, 2, 3, 4,..., but no one knows all the implications of this simple process. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
prime divisor property, natural number divisors, smallest nontrivial solution, largest unit fraction, unique prime factorization, rational triangles, common prime divisor, two squares theorem, previous exercise set, equidistant set, rational right triangle, ordinary primes, bottom line sum, constructible points, numerical area, quadratic integers, step polygon, outer polygons, vertically opposite angles, rational slope, integer points, reflections theorem, ring properties, irrational lengths, focal property Key Phrases - Capitalized Phrases (CAPs): (learn more)
Euclid's Elements, Exercises There, Proof Suppose, Exercises Another, Use Exercise New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
The beauty and fascination of numbers can be summed up by one simple fact: anyone can count 1, 2, 3, 4,..., but no one knows all the implications of this simple process. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
prime divisor property, natural number divisors, smallest nontrivial solution, largest unit fraction, unique prime factorization, rational triangles, common prime divisor, two squares theorem, previous exercise set, equidistant set, rational right triangle, ordinary primes, bottom line sum, constructible points, numerical area, quadratic integers, step polygon, outer polygons, vertically opposite angles, rational slope, integer points, reflections theorem, ring properties, irrational lengths, focal property Key Phrases - Capitalized Phrases (CAPs): (learn more)
Euclid's Elements, Exercises There, Proof Suppose, Exercises Another, Use Exercise New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 100
books:
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85
books
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- Disquisitiones Arithmeticae by Carl F. Gauss on 4 pages
- The Heritage of Thales (Undergraduate Texts in Mathematics / Readings in Mathematics) by W.S. Anglin in Front Matter, and Back Matter
- Projective Geometry (Undergraduate Texts in Mathematics / Readings in Mathematics) by Pierre Samuel in Front Matter, and Back Matter
- Mathematics: A Concise History and Philosophy (Undergraduate Texts in Mathematics / Readings in Mathematics) by W.S. Anglin in Front Matter, and Back Matter
- Second Year Calculus: From Celestial Mechanics to Special Relativity (Undergraduate Texts in Mathematics / Readings in Mathematics) by David M. Bressoud in Front Matter, and Back Matter
See all 100 books this book cites
- The Heritage of Thales (Undergraduate Texts in Mathematics / Readings in Mathematics) by W.S. Anglin in Front Matter, and Back Matter
- Mathematical Expeditions: Chronicles by the Explorers (Undergraduate Texts in Mathematics) by Reinhard Laubenbacher in Front Matter, and Back Matter
- The Pleasures of Probability (Undergraduate Texts in Mathematics / Readings in Mathematics) by Richard Isaac in Front Matter, and Back Matter
- Glimpses of Algebra and Geometry by Gabor Toth in Front Matter, and Back Matter
- Basic Topology (Undergraduate Texts in Mathematics) by M.A. Armstrong in Back Matter (1), and Back Matter (2)
See all 85 books citing this book
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