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 civilisation but also of everyday life. Even those who have avoided mathematics for years will find the historical stories both inviting and gripping.The book's lessons begin with the creation and spread of number systems, from the mathematical development of early civilisations in Babylonia, Greece, China, Rome, Egypt, and Central America, to the advancement of mathematics over time and the roles of famous figures such as Descartes and Leonardo of Pisa (Fibonacci). Before long, it becomes clear that the simple origins of algebra evolved into modern problem solving. Along the way, the language of mathematics becomes familiar, and students are gradually introduced to more challenging problems. Paced perfectly, Amy Shell-Gallasch and J. B. Thoo's chapters ease students from topic to topic until they reach the twentieth century.By the end of Algebra in Context, students using this textbook will be comfortable with most algebra concepts, including:Different number bases Algebraic notation Methods of arithmetic calculation Real numbers Complex numbers Divisors Prime factorization Variation Factoring Solving linear equations False position Solving quadratic equations Solving cubic equations nth roots Set theory One-to-one correspondence Infinite sets Figurate numbers Logarithms Exponential growth Interest calculations
Preface Introduction Part I 1. Number Bases 1.1. Base 6 1.2. Base 4 2. Babylonian Number System 2.1. Cuneiform 2.2. Mathematical Texts 2.3. Number System 3. Egyptian and Roman Number Systems 3.1. Egyptian 3.1.1. History 3.1.2. Writing and Mathematics 3.1.3. Number System 3.2. Roman 3.2.1. History 3.2.2. Number System 4. Chinese Number System 4.1. History and Mathematics 4.2. Rod Numerals 5. Mayan Number System 5.1. Calendar 5.2. Codices 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 7. Exercises Part II 8. Addition and Subtraction 9. Multiplication 9.1. Roman Abacus 9.2. Grating or Lattice Method 9.3. Ibn Labban and Chinese Counting Board 9.4. Egyptian Doubling Method 10. Division 10.1. Egyptian 10.2. Leonardo of Pisa 10.3. Galley or Scratch Method 11. Casting Out Nines 12. Finding Square Roots 12.1. Heron of Alexandria 12.2. Theon of Alexandria 12.3. Bakhshali Manuscript 12.4. Nicolas Chuquet 13. Exercises Part III 14. Sets 14.1. Set Relations 14.2. Finding 2n 14.3. One-to-One Correspondence and Cardinality 15. Rational, Irrational, and Real Numbers 15.1. Commensurable and Incommensurable Magnitudes 15.2. Rational Numbers 15.3. Irrational Numbers 15.4. I Is Uncountably Infinite 15.5. card(Q), card(I), and card(R) 15.6. Transfinite Numbers 16. Logic 17. The Higher Arithmetic 17.1. Early Greek Elementary Number Theory 17.1.1. Pythagoras 17.1.2. Euclid 17.1.3. Nicomachus and Diophantus 17.2. Even and Odd Numbers 17.3. Figurate Numbers 17.3.1. Triangular Numbers 17.3.2. Square Numbers 17.3.3. Rectangular Numbers 17.3.4. Other Figurate Numbers 17.4. Pythagorean Triples 17.5. Divisors, Common Factors, and Common Multiples 17.5.1. Factors and Multiples 17.5.2. Euclid's Algorithm 17.5.3. Multiples 17.6. Prime Numbers 17.6.1. The Sieve of Eratosthenes 17.6.2. The Fundamental Theorem of Arithmetic 17.6.3. Perfect Numbers 17.6.4. Friendly Numbers 18. Exercises Part IV 19. Linear Problems 19.1. Review of Linear Equations 19.2. False Position 19.3. Double False Position 20. Quadratic Problems 20.1. Solving Quadratic Equations by Completing the Square 20.1.1. Babylonian 201.2. Arabic 201.3. Indian 20.1.4. The Quadratic Formula 20.2. Polynomial Equations in One Variable 20.2.1. Powers 20.2.2. nth Roots 20.3. Continued Fractions 20.3.1. Finite Simple Continued Fractions 20.3.2. Infinite Simple Continued Fractions 20.3.3. The Number 21. Cubic Equations and Complex Numbers 21.1. Complex Numbers 21.2. Solving Cubic Equations and the Cubic Formula 22. Polynomial Equations Relation between Roots and Coefficients Viète and Harriot 22.3. Zeros of a Polynomial 22.3.1. Factoring 22.3.2. Descartes's Rule of Signs 22.4. The Fundamental Theorem of Algebra 23. Rule of Three 23.1. China 23.2. India 23.3. Medieval Europe 23.4. The Rule of Three in False Position 23.5. Direct Variation, Inverse Variation, and Modeling 24. Logarithms 24.1. Logarithms Today 24.2. Properties of Logarithms 24.3. Bases of a Logarithm 24.3.1. Using a Calculator 24.3.2. Comparing Logarithms 24.4. Logarithm to the Base e and Applications 24.4.1. Compound Interest 24.4.2. Amortization 24.4.3. Exponential Growth and Decay 24.5. Logarithm to the Base 10 and Application to Earthquakes 25. Exercises Bibliography Index
""The book was clearly written with extreme care, and for such a large first edition... I fully expect to make frequent use of the text as a resource for material to incorporate in my current classes, including mathematical content courses for elementary school teachers.""
