The Historical Roots of Elementary Mathematics

The Historical Roots of Elementary Mathematics

Paperback Dover Books on Mathematics

By (author) Lucas N.H. Bunt, By (author) Philip S. Jones, By (author) Jack D. Bedient

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  • Publisher: Dover Publications Inc.
  • Format: Paperback | 336 pages
  • Dimensions: 157mm x 261mm x 28mm | 844g
  • Publication date: 28 March 2003
  • Publication City/Country: New York
  • ISBN 10: 0486255638
  • ISBN 13: 9780486255637
  • Sales rank: 308,561

Product description

"Will delight a broad spectrum of readers." -- "American Mathematical Monthly "Do long division as the ancient Egyptians did! Solve quadratic equations like the Babylonians! Study geometry just as students did in Euclid's day! This unique text offers students of mathematics an exciting and enjoyable approach to geometry and number systems. Written in a fresh and thoroughly diverting style, the text -- while designed chiefly for classroom use -- will appeal to anyone curious about mathematical inscriptions on Egyptian papyri, Babylonian cuneiform tablets, and other ancient records. The authors have produced an illuminated volume that traces the history of mathematics -- beginning with the Egyptians and ending with abstract foundations laid at the end of the 19th century. By focusing on the actual operations and processes outlined in the text, students become involved in the same problems and situations that once confronted the ancient pioneers of mathematics. The text encourages readers to carry out fundamental algebraic and geometric operations used by the Egyptians and Babylonians, to examine the roots of Greek mathematics and philosophy, and to tackle still-famous problems such as squaring the circle and various trisectorizations. Unique in its detailed discussion of these topics, this book is sure to be welcomed by a broad range of interested readers. The subject matter is suitable for prospective elementary and secondary school teachers, as enrichment material for high school students, and for enlightening the general reader. No specialized or advanced background beyond high school mathematics is required.

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Table of contents

Preface The Greek alphabet 1 EGYPTIAN MATHEMATICS   1-1 Prehistoric mathematics   1-2 The earliest written mathematics   1-3 Numerical notation   1-4 Arithmetic operations   1-5 Multiplication   1-6 Fractions and division   1-7 The red auxiliary numbers   1-8 The 2 ÷ n table   1-9 The leather roll   1-10 Algebraic problems   1-11 Geometry 2 BABYLONIAN MATHEMATICS   2-1 Some historical facts   2-2 Babylonian numerical notation   2-3 The fundamental operations   2-4 Extraction of roots   2-5 Babylonian algebra   2-6 A Babylonian text   2-7 Babylonian geometry   2-8 Approximations to p   2-9 Another problem and a farewell to the Babylonians 3 THE BEGINNING OF GREEK MATHEMATICS   3-1 The earliest records   3-2 Greek numeration systems   3-3 Thales and his importance to mathematics   3-4 Pythagoras and the Pythagoreans   3-5 The Pythagoreans and music   3-6 Pythagorean arithmetica   3-7 Pythagorean numerology   3-8 Pythagorean astronomy   3-9 Pythagorean geometry   3-10 Incommensurable segments and irrational numbers 4 THE FAMOUS PROBLEMS OF GREEK ANTIQUITY   4-1 Introduction   4-2 Hippocrates of Chios and the quadrature of lunes   4-3 Other quadratures   4-4 Hippocrates' geometry   4-5 Duplication of the cube   4-6 The trisection problem   4-7 Hippias and squaring of the circle   4-8 The solutions of the Greek problems 5 EUCLID'S PHILOSOPHICAL FORERUNNERS   5-1 Philosophy and philosophers   5-2 Plato   5-3 Aristotle and his theory of statements   5-4 Concepts and definitions   5-5 Special notations and undefined terms 6 EUCLID   6-1 Elements   6-2 The structure of the Elements of Euclid   6-3 The definitions   6-4 Postulates and common notions   6-5 The meaning of a construction   6-6 The purport of Postulate III   6-7 Congruence   6-8 Congruence (continued)   6-9 The theory of parallels   6-10 The comparison of areas   6-11 The theorem of Pythagoras   6-12 The difference between the Euclidean and the modern method of comparing areas   6-13 Geometric algebra and regular polygons   6-14 Number theory in the Elements 7 GREEK MATHEMATICS AFTER EUCLID. EUCLIDEAN VS. MODERN METHODS   7-1 The span of Greek mathematics   7-2 Archimedes and Eratosthenes   7-3 Apollonius of Perga   7-4 Heron of Alexandria and Diophantus   7-5 Ptolemy and Pappus   7-6 Review of the Greek method   7-7 Objections to the Euclidean system   7-8 The meaning of deduction   7-9 Euclid's system is not purely deductive   7-10 How is geometry built up purely deductively?   7-11 A four-point system 8 NUMERATION AND ARITHMETIC AFTER THE GREEKS   8-1 Roman numerals   8-2 The abacus and tangible arithmetic   8-3 The Hindu-Arabic numerals   8-4 An early American place-value numeration system   8-5 Later developments in positional notation   8-6 Conversions between numeration systems   8-7 Addition and subtraction algorithms in nondecimal bases   8-8 Multiplication alogorithms in nondecimal bases   8-9 "Fractions, rational numbers, and place-value numeration"   8-10 Irrational numbers   8-11 Modern theoretical foundations of arithmetic   8-12 Modern numeration HINTS AND ANSWERS TO SELECTED ANSWERS INDEX