Gabor Toth - Elements of Mathematics: A Problem-Centered Approach to History and Foundations
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- Book:Elements of Mathematics: A Problem-Centered Approach to History and Foundations
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Elements of Mathematics: A Problem-Centered Approach to History and Foundations: summary, description and annotation
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This textbook offers a rigorous presentation of mathematics before the advent of calculus. Fundamental concepts in algebra, geometry, and number theory are developed from the foundations of set theory along an elementary, inquiry-driven path. Thought-provoking examples and challenging problems inspired by mathematical contests motivate the theory, while frequent historical asides reveal the story of how the ideas were originally developed.
Beginning with a thorough treatment of the natural numbers via Peanos axioms, the opening chapters focus on establishing the natural, integral, rational, and real number systems. Plane geometry is introduced via Birkhoffs axioms of metric geometry, and chapters on polynomials traverse arithmetical operations, roots, and factoring multivariate expressions. An elementary classification of conics is given, followed by an in-depth study of rational expressions. Exponential, logarithmic, and trigonometric functions complete the picture, driven by inequalities that compare them with polynomial and rational functions. Axioms and limits underpin the treatment throughout, offering not only powerful tools, but insights into non-trivial connections between topics.
Elements of Mathematics is ideal for students seeking a deep and engaging mathematical challenge based on elementary tools. Whether enhancing the early undergraduate curriculum for high achievers, or constructing a reflective senior capstone, instructors will find ample material for enquiring mathematics majors. No formal prerequisites are assumed beyond high school algebra, making the book ideal for mathematics circles and competition preparation. Readers who are more advanced in their mathematical studies will appreciate the interleaving of ideas and illuminating historical details.
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Undergraduate Texts in Mathematics are generally aimed at third- and fourth-year undergraduate mathematics students at North American universities. These texts strive to provide students and teachers with new perspectives and novel approaches. The books include motivation that guides the reader to an appreciation of interrelations among different aspects of the subject. They feature examples that illustrate key concepts as well as exercises that strengthen understanding.
For further volumes: http://www.springer.com/series/666 and http://www.springer.com/series/4672
This Springer imprint is published by the registered company Springer Nature Switzerland AG
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
To my children
Evelyn, Isabel, Gerald,
Gregory, Gabriel, and Gerda.
If youre teaching a class, you can think about elementary things
that you know very well. These things are kind of fun and delightful.
It doesnt do any harm to think them over again. Is there a better way
to present them? Are there any new problems associated with them?
Are there any new thoughts you can make about them?
Richard P. Feynman (19181988)
This textbook aims for a rigorous, precise, and transparent presentation of mathematics before the advent of calculus. In developing nave and axiomatic theories alike, and with geometry and algebra hand in hand, the text takes a new and fresh look at many a mathematical concept, never losing sight of the importance of intuition, and the ultimate quest for mathematical rigor.
Every experienced instructor knows that curious students always ask many questions. This book is written for them, the inquisitive and demanding readers who are seeking real challenge. Questions should always be encouraged and welcomed; as Francis Bacon (15611616) put it, Who questions much, shall learn much, and retain much. In this book we answer many: What are the foundations of mathematics? Why did the Sumerians and the Babylonians chose sexagesimal arithmetic? What is a real number? What is the meaning of irrational powers? What is metric geometry? Why is the Pythagorean Theorem important in Archimedes approximation of ? How much did the ancient Greeks know about conics? Why do we have different approaches to exponentiation?
One of the primary goals of this book is to offer an honest and in-depth text for the readers. Its appeal rests in the clarity of the gradually and carefully built up material and the transparency of the explanations; the emphasis on interconnections among seemingly unrelated topics (in algebra, geometry, number theory, etc.); correct and unglossed answers to many fundamental questions that the student may ask; and intriguing historical notes based on most recent scholarship.
Throughout the entire book we insist on elementary approach, and leisurely pace, taking many side tours when opportunities arise. The text is sprinkled with a variety of thought-provoking examples, often inspired by problems posed in mathematical contests around the world.
There are over 150 challenging exercises at the end of the sections. A solutions manual can be found in the authors website:
https://math.camden.rutgers.edu/faculty/gabor-toth/
- (1)
Various parts of this book have been used by the author in his mathematics contest-training course for high school students in the Princeton Campus of the Art of Problem Solving Academy. A contest preparation course for these students should cover only parts of Sections , and should focus on problem solving strategies without much theoretical material or proofs. Within the main text in these sections, there are a total of 123 worked out and challenging examples, and, in addition, these sections end with 71 additional exercises. These should provide enough material for a one-semester course.
- (2)
The latter part of the book can also be adopted for a senior capstone course in mathematics for advanced undergraduate students. In this capacity, the author used various parts of the text in the last 30 years as material for the capstone unit Mathematics Seminar at Rutgers UniversityCamden for graduating seniors. A typical college course should essentially cover Chapters are written for college seniors.
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