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Available for adoption from JOHNS HOPKINS UNIVERSITY PRESS Algebra in Context Introductory Algebra from Origins to Applications Amy Shell-Gellasch, Montgomery College J. B. Thoo, Yuba College 552 pp., 28 b&w illus., 12 line drawings, 9 maps HC 978-1-4214-1728-8 September 2015 $99.50 An engaging new approach to teaching algebra that takes students on a historical journey from its roots to modern times. This book’s unique approach to the teaching of mathematics lies in its use of history to provide a framework for understanding algebra and related fields. With Algebra in Context, students will soon discover why mathematics is such a crucial part not only of civilization but also of everyday life. Even those who have avoided mathematics for years will find the historical stories both inviting and gripping. Features: — There is no other text that covers the material in college algebra that use the history of mathematics as the vehicle to impart this material —Classroom tested —Keeps students engaged with a textbook that includes math woven together with illustrations throughout history —Exercises included at the end of every section —An innovative alternative to teaching standard quantitative literacy Request an exam copy: http://bit.ly/JHUPExamCopies press.jhu.edu Contents Prefaceix Introductionxv Part I Numeration Systems 1 1 Number Bases 1.1 Base 6 1.2 Base 4 3 5 8 2 Babylonian Number System 11 2.1Cuneiform 2.2 Mathematical Texts 2.3 Number System 12 13 15 3 Egyptian and Roman Number Systems 21 3.1Egyptian 21 3.1.1 History 21 3.1.2 Writing and Mathematics 22 3.1.3 Number System 24 3.2Roman 27 3.2.1 History 27 3.2.2 Number System 29 4 Chinese Number System 4.1 History and Mathematics 4.2 Rod Numerals 35 35 37 5 41 42 44 44 49 Mayan Number System 5.1Calendar 5.2Codices 5.3 Number System 5.4 Native North Americans 6 Indo-Arabic Number System 6.1 India 6.1.1 History 6.1.2 Mathematics 6.2 The Middle East 6.2.1 History 6.2.2 Mathematics 6.3 Number System 6.3.1 Whole Numbers 6.3.2 Fractions 51 51 52 53 55 55 59 60 64 66 7 Exercises 73 Part II Arithmetic Snapshots 87 8 Addition and Subtraction 89 9Multiplication 9.1 Roman Abacus 9.2 Grating or Lattice Method ¯ and Chinese 9.3 Ibn Labban Counting Board 9.4 Egyptian Doubling Method 95 95 97 98 100 10Division 10.1Egyptian 10.2 Leonardo of Pisa 10.3 Galley or Scratch Method 105 105 107 113 11 Casting Out Nines 117 12 Finding Square Roots 12.1 Heron of Alexandria 12.2 Theon of Alexandria a i Manuscript 12.3 Bakhsh¯l¯ 12.4 Nicolas Chuquet 121 122 124 129 131 13 Exercises 135 Part III Foundations 141 14 Sets 143 14.1 Set Relations 146 14.2 Finding 2n152 14.3 One-to-One Correspondence and Cardinality 154 15 Rational, Irrational, and Real Numbers 159 15.1 Commensurable and Incommensurable Magnitudes 162 15.2 Rational Numbers 163 15.3 Irrational Numbers 168 15.4 I Is Uncountably Infinite 174 15.5 card(Q), card(I), and card(R) 177 15.6 Transfinite Numbers 179 16Logic 183 17 The Higher Arithmetic 197 17.1 Early Greek Elementary Number Theory 198 17.1.1 Pythagoras 199 17.1.2 Euclid 200 17.1.3 Nicomachus and Diophantus 202 17.2 Even and Odd Numbers 203 17.3 Figurate Numbers 207 17.3.1 Triangular Numbers 207 17.3.2 Square Numbers 208 17.3.3 Rectangular Numbers 210 17.3.4 Other Figurate Numbers 213 17.4 Pythagorean Triples 214 17.5 Divisors, Common Factors, and Common Multiples 220 17.5.1 Factors and Multiples 220 17.5.2 Euclid’s Algorithm 223 17.5.3MultipleS 233 17.6 Prime Numbers 238 17.6.1 The Sieve of Eratosthenes240 17.6.2 The Fundamental Theorem of Arithmetic 242 17.6.3 Perfect Numbers 248 17.6.4 Friendly Numbers 252 18 Exercises 253 Part IV Solving Equations 265 19 Linear Problems 19.1 Review of Linear Equations 19.2 False Position 19.3 Double False Position 267 270 273 284 20 Quadratic Problems 301 20.1 Solving Quadratic Equations by Completing the Square 302 20.1.1Babylonian 304 20.1.2Arabic 312 20.1.3Indian 320 20.1.4 The Quadratic Formula 326 20.2 Polynomial Equations in One Variable 335 20.2.1Powers 337 20.2.2nth Roots 345 20.3 Continued Fractions 356 20.3.1 Finite Simple Continued Fractions 359 20.3.2 Infinite Simple Continued Fractions 360 20.3.3 The Number 0 369 21 Cubic Equations and Complex Numbers 21.1 Complex Numbers 21.2 Solving Cubic Equations and the Cubic Formula 375 377 395 22 Polynomial Equations 415 22.1 Relation between Roots and Coefficients 416 22.2 Viète and Harriot 419 22.3 Zeros of a Polynomial 424 22.3.1 Factoring 424 22.3.2 Descartes’s Rule of Signs 434 22.4 The Fundamental Theorem of Algebra 436 23 Rule of Three 23.1China 23.2 India 23.3 Medieval Europe 23.4 The Rule of Three in False Position 23.5 Direct Variation, Inverse Variation, and Modeling 439 439 442 446 447 450 24Logarithms 461 24.1 Logarithms Today 470 24.2 Properties of Logarithms 472 24.3 Bases of a Logarithm 475 24.3.1 Using a Calculator 476 24.3.2 Comparing Logarithms 478 24.4 Logarithm to the Base e and Applications 480 24.4.1 Compound Interest 484 24.4.2Amortization 491 24.4.3 Exponential Growth and Decay 494 24.5 Logarithm to the Base 10 and Application to Earthquakes 499 25 Exercises 505 Bibliography 521 Index Request an exam copy: http://bit.ly/JHUPExamCopies pr e s s . j h u . e d u