Suzuki, Jeff A History of Mathematics

ISBN 13: 9780130190741

A History of Mathematics

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9780130190741: A History of Mathematics

Blending relevant mathematics and history, this book immerses readers in the full, rich detail of mathematics. It provides a description of mathematics and shows how mathematics was actually practiced throughout the millennia by past civilizations and great mathematicians alike. As a result, readers gain a better understanding of why mathematics developed the way it did. Chapter topics include Egyptian Mathematics, Babylonian Mathematics, Greek Arithmetic, Pre-Euclidean Geometry, Euclid, Archimedes and Apollonius, Roman Era, China and India, The Arab World, Medieval Europe, Renaissance, The Era of Descartes and Fermat, The Era of Newton and Leibniz, Probability and Statistics, Analysis, Algebra, Number Theory, the Revolutionary Era, The Age of Gauss, Analysis to Mid-Century, Geometry, Analysis After Mid-Century, Algebras, and the Twentieth Century. For teachers of mathematics.

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The Mathematics of History

The author of a text on the history of mathematics is faced with a difficult question: how to handle the mathematics? There are several good choices. One is to give concise descriptions of the mathematics, which allows many topics to be covered. Another is to present the mathematics in modern terms, which makes clear the connection between the past and the present. There are many excellent texts that use either or both of these strategies.

This book offers a third choice, based on a simple philosophy: the best way to understand history is to experience it. To understand why mathematics developed the way it did, why certain discoveries were made and others missed, and why a mathematician chose a particular line of investigation, we should use the tools they used, see mathematics as they saw it, and above all think about mathematics as they did.

Thus to provide the best understanding of the history of mathematics, this book is a mathematics text, first and foremost. The diligent reader will be classmate to Archimedes, al Khwarizmi, and Gauss. He or she will be looking over Newton's shoulders as he discovers the binomial theorem, and will read Euler's latest discoveries in number theory as they arrive from St. Petersburg. Above all, the reader will experience the mathematical creative process firsthand to answer the key question of the history of mathematics: how is mathematics created?

In this text I have emphasized:

  1. Numeration, computation, and notation. Notation both limits and guides. Limits, because it is difficult to think "outside the notation"; guides, because a good system of notation can suggest relationships worthy of further study. As much as possible, I avoid the temptation to "translate" a mathematical result into modern (mathematical) language or notation, for modern notation brings modern ways of thinking. In a similar vein, the means of computing and expressing numbers guides what discoveries one may make, and ultimately influences the direction taken by mathematicians.
  2. Mathematical results in their original form with their original arguments. The most dramatic change in mathematics over the past four millennia has been the standard of proof. What was acceptable to Pythagoras is no longer be acceptable today. The binomial theorem was, when Newton proposed it, nothing more than a conjecture, and while most of Euler's results on series summation are accepted today, his methods are not. But to provide a modern proof of a result would be historically inaccurate, while to omit how the these results were supported would be to neglect a vital part of the history of mathematics. To protect modern sensibilities, I will distinguish between proofs, where the term is used in the modern sense, and demonstrations, which contain some elements no longer acceptable in a modern context, and if a non-obvious result is presented without a proof, it will be labeled a conjecture.
  3. Mathematics as an evolving science. The most important thing we can learn from a history of mathematics is that mathematics is created by human beings and not by semimythical demigods. The ideas of Newton, Euler, Gauss, and others originated from the mathematics they knew and the problems they saw around them; they made great contributions, but they also made mistakes, which were faithfully replicated (or caustically reviewed) by fellow mathematicians. Finally, no mathematical idea is born, fully mature and in the modern form: we will follow, where space and time permit, the development of mathematical ideas; through their birth pains, their early, formative years, and onwards to their early maturity and final, modern form.

To fully enter into all of the above for all of mathematics would take a much larger book, or even a mufti-volume series: a project for another day. Thus, to keep the book to manageable length, I have restricted its scope to what I deem "elementary" mathematics: the fundamental mathematics every mathematics major and every mathematics teacher should know. This includes numeration, arithmetic, geometry, algebra, calculus, real analysis, and the elementary aspects of abstract algebra, probability, statistics, number theory, complex analysis, differential equations, and some other topics that can be introduced easily as a direct application of these "elementary" topics. Advanced topics that would be incomprehensible without a long explanation have been omitted, as have been some very interesting topics which, through cultural and historical circumstances, had no discernible impact on the development of modern mathematics.

Using this book

I believe a history of mathematics class can be a great "leveler", in that no student is inherently better prepared for it than any other. By selecting sections carefully, this book can be used for students with any level of background, from the most basic to the most advanced. However, it is geared towards students who have had calculus. In general, a year of calculus and proficiency in elementary algebra should be sufficient for all but the most advanced sections of the present work. A second year of calculus, where the student becomes more familiar and comfortable with differential equations and linear algebra, would be helpful for the more advanced sections (but of course, these can be omitted). Some of the sections require some familiarity with abstract algebra, as would be obtained by an introductory undergraduate course in the subject. A few of the problems require critiques of proofs by modern standards, which would require some knowledge of what those standards are (this would be dealt with in an introductory analysis course).

There is more than enough material in this book for a one-year course covering the full history of mathematics. For shorter courses, some choices are necessary. This book was written with two particular themes in mind, either of whEch are suitable for students who have had at least one year of calculus:

1. Creating mathematics: a study of the mathematical creative process. This is embodied throughout the work, but the following sections form a relatively self-contained sequence: 3.2, 6.4.1, 9.1.5, 9.2.3, 11.6, 13.1.2, 13.4.1, 13.6.1, 17.2.2, 20.2.4, 20.3.1, 22.4.

2. Origins: why mathematics is done the way it is done. Again, this is embodied throughout the work, but some of the more important ideas can be found in the following sections: 3.2, 4.3.1, 5.1, 6.1.2, 9.2.3, 9.4.5, 13.6.1, 15.1, 15.2, 19.5.2, 20.1, 20.2.5, 22.1, 23.3.2

In addition, there are the more traditional themes:

3. Computation and Numeration: For students with little to no background beyond elementary mathematics, or for those who intend to teach elementary mathematics, the following sections are particularly relevant: 1.1, 1.2, 2.1, 3.1, 7.1, 8.1, 8.3.1, 8.3.2, 9.1, 10.1.2 through 10.1.4, 10.2.1, 10.2.6, and 11.4.1.

4. Problem Solving: In the interests of avoiding anachronisms, these sections are not labeled "algebra" until the Islamic era. For students with no calculus background, but a sufficiently good background in elementary algebra, the following sequence is suggested: Sections 1.3, 2.2, 5.4, 7.4, 8.2, 8.3, 9.2, 9.3, 10.2.2, 10.3.1, 11.2, 11.3, 11.5.1, and for the more advanced students, Chapter 16.

5. Calculus to Newton and Leibniz: The history of integral calculus can be traced through Sections 1.4.2, 4.2, 4.3.3, 4.3.4, 5.6, 5.9, 6.1, 6.3, 10.4, 11.4.3, 12.4, 12.3.2, 13.1.2, 13.2, and 13.4.2. Meanwhile, the history of differential calculus can be traced through Sections 5.5.2, 6.6.2, 12.1.1, 13.3, 13.5, 13.6.

6. Number theory: I would suggest Sections 3.2, 5.8, 7.4, 8.3.5, 8.3.6, 10.2.5, 12.2, and all of Chapter 17.

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Book Description Pearson, 2001. Book Condition: New. Brand New, Unread Copy in Perfect Condition. A+ Customer Service! Summary: 1. Egyptian Mathematics. Numeration. Arithmetic Operations. Problem Solving. Geometry. 2. Babylonian Mathematics. Numeration and Computation. Problem Solving. Geometry. 3. Greek Arithmetic. Numeration and Computation. The Pythagoreans. The Irrational. 4. Pre-Euclidean Geometry. Thales and Pythagoras. The Athenian Empire. The Age of Plato. 5. TheElements. Deductive Geometry. Rectilinear Figures. Parallel Lines. Geometric Algebra. Circles. Ratio and Proportion. The Quadratic Equation. Number Theory. The Method of Exhaustion. Book XIII. 6. Archimedes and Apollonius. Circles. Spheres, Cones, and Cylinders. Quadratures. Large Numbers. Vergings and Loci. Apollonius'sConics. Loci and Extrema. 7. Roman Era. Numeration and Computation. Geometry and Trigonometry. Heron of Alexandria. Diophantus. The Decline of Classical Learning. 8. China and India. Chinese Numeration and Computation. Practical Mathematics in China. Indian Problem Solving. Indian Geometry. 9. The Islamic World. Numeration and Computation. Al Khwarizami. Other Algebras. Geometry and Trigonometry. 10. Medieval Europe. The Early Medieval Period. Leonardo of Pisa. The High Middle Ages. Nicholas Oresme. 11. Renaissance. Trigonometry. The Rise of Algebra. The Cossists. Simon Stevin. Francois Viete. The Development of Logarithms. 12. The Era of Descartes and Fermat. Algebra and Geometry. Number Theory. The Infinite and Infinitesimals. The Tangent and Quadrature Problems. Probability. The Sums of Powers. Projective Geometry. 13. The Era of Newton and Leibniz. John Wallis. Issac Barrow. The Low Countries. Isaac Newton. Fluxions. Leibniz. Johann Bernoulli. 14. Probability and Statistics. Jakob Bernoulli. Abraham De Moivre. Daniel Bernoulli. Paradoxes and Fallacies. Simpson. Bayes's Theorem. 15. Analysis. Leonhard Euler. Calculus Textbooks. Mathematical Physics. Trigonometric Series. 16. Algebra. Newton and Algebra. Maclaurin. Euler and Algebra. 17. Number Theory. Euler and Fermat's Conjectures. The Berlin Years. Return to St. Petersburg. 18. The Revolutionary Era. Analysis. Algebra and Number Theory. Probability and Statistics. Mathematics of Society. 19. The Age of Gauss. The Roots of Equations. Solvability of Equations. The Method of Least Squares. Number Theory. Geometry. 20. Analysis to Midcentury. Foundation of Analysis. Abel. Fourier Series. Transcendental Numbers. Chebyshev. 21. Geometry. Analytic and Projective Geometry. Differential Geometry. The Fifth Postulate. Consistency of Geometry. 22. Analysis After Midcentury. Analysis in Germany. The Real Numbers. The Natural Numbers. The Infinite. Dynamical Systems. 23. Algebras. The Algebra of Logic. Vector Algebra. Matrix Algebra. Groups. Algebra in Par. Bookseller Inventory # ABE_book_new_0130190748

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