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:
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. PAPERBACK. Book Condition: New. 0130190748 New Condition. Bookseller Inventory # NEW6.0042651
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Book Description Pearson, 2001. Paperback. Book Condition: New. Bookseller Inventory # P110130190748
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