Name: MthEd/Math 300 Williams Fall 2011 Midterm Exam 2 Closed Book / Closed Note. Answer all problems. You may use a calculator for numerical computations. Section 1: For each event listed in the first column, circle the time period in which it occurred: M for Medieval Europe, R for Renaissance Europe, 16 for 1600's. One point each. Event Medieval Europe Renaissance Europe 1600's 1. Introduction of decimal fractions to Europe R 2. Development of formulas for solution of 3 rd and 4 th degree polynomial equations R 3. First accounts of analytic geometry 16 4. First published treatment of trigonometry as a mathematical subject separate from astronomy R 5. Fra Luca Pacioli s prediction that the cubic could not be solved. R 6. Publication of The Analytic Art R 7. First coherent European treatment of complex numbers R 8. Publication of work claiming that the orbits of planets were elliptic. R 9. The development of calculus. 16 10. Development of logarithms R 11. Publication of Liber Abaci M 12. Various mathematicians develop methods for finding maxima and minima, slopes, and areas under curves 16 13. Introduction of Hindu-Arabic numerals and algorithms to Europe. M 14. Copernicus introduces heliocentric model of solar system. R 15. Ars Magna, or The Great Art, is published. R Section 2: For each mathematical advance listed in the left column, write the name of the
mathematician or mathematicians responsible in the blanks. There may be two or more correct answers for some questions; if so, put two names. One point each. 16. Developed logarithms as a computational tool to aid in astronomy. 17. Introduced Hindu-Arabic numeration and algorithms to Europe 18. Gave a solution of fourth-degree polynomials by radicals (a quartic formula) 19. Introduced decimal fractions and operations to Europe 20. Wrote a book on algebra which in fact gave us the name algebra. 21. Made careful observations of planetary movements that others built on. 22. Created what came to be known as analytic geometry. 23. Published a version of the Fundamental Theorem of Calculus before Newton or Leibniz. 24. Used a very involved method find tangent lines by first finding normals. 25. Was accused by Tartaglia of stealing his solution of the depressed cubic. 26. Published The Analytic Art, introducing a systematics approach to equations. Napier Fibonacci Ferrari Stevin Al-Khwarizmi Brahe Fermat Barrow Descares Cardano Viète (Cardano also OK) Descartes 27. Invented the calculus. Newton Liebniz 28. Wrote the first treatment of trigonometry as a subject separate from astronomy. 29. During the Islamic Empire, wrote a book outlining the solution of cubics by conics. 30. Gave first treatment of arithmetic for complex numbers Regiomontanus Omar Khayyam Bombelli
Section 3: Short Answer. Five points each. 31. Describe two of Omar Khayyam s mathematical contributions. 1. Wrote a book on the solution of polynomial equations (including cubics) using conic sections. 2. Wrote a book on Euclid s Fifth Postulate, in which he tried to prove it from another postulate that he got from Aristotle. 3 points for one of the above, 5 points total for both. Obvious mistakes or misunderstandings could cost a point or two. 32. Describe some of the mathematics discussed in the Sulbasutras. The design and construction of altars, especially finding squares with areas of given circles, and vice-versa, and finding squares with areas of given rectangles, and vice-versa. (3 points)
33. Describe role of the Nine Chapters in the development of Chinese mathematics. Include at least two topics covered in the book. The Nine Chapters played the same role in Chinese mathematical culture as the Elements did in ancient Greece. They summarized a lot of the mathematics that was known at the time they were written, and became a standard textbook for study of mathematics. Commentaries were also written about both the Nine Chapters and the Elements by later scholars. Topics: field measurements, fractions, and areas; percentages and proportions; arithmetic and geometric progressions; square- and cube-roots; volumes of shapes; fair distribution; excess and deficit problems; matrix solutions of linear systems; the Pythagorean theorem. (3 points) 34. Today, a mathematician tries to publish new discoveries as quickly as possible so as to have priority that is, so that everyone knows she was first. Give some (at least two) examples of how that was not necessarily true in the past. In Tartaglia s and Cardano s time, mathematicians kept their methods secret in order to win mathematical contests, rather than publishing. Newton didn t publish his calculus for a long time after developing it. Liebniz published first, even though Newton probably developed the ideas first. Fermat published almost none of his work; his work is known mainly by way of letters he wrote to friends. (Although Gauss was actually talked about after the material covered in this test, I also accepted:) Gauss published only what he thought was finished and complete. Some things he didn t ever publish (e.g. non-euclidean geometry). 3 points for one of the above, 5 points total for both. Obvious mistakes or misunderstandings could cost a point or two.
Section 4: Longer Essay Questions. Ten points each. 35. It has been claimed that the main role of the Islamic Empire in the development of mathematics was preserving the mathematics of ancient Greece. Discuss this in light of what we learned in class, giving specific examples of Islamic contributions. It is true that the Islamic Empire preserved the mathematics of ancient Greece, but They also contributed some original mathematics, adding to what was known. In particular, al-khwarizmi added to our knowledge of algebra and arithmetic with two of his books; Hindu-Arabic numeration revolutionized arithmetic computations; (3 points each for two of these examples) Khayyam added wrote on non-euclidean geometry, studying what we now call Saccheri quadrilaterals, and systematized the solution of polynomial equations with the use of conics; Al-Tusi also worked on Euclid s fifth postulate; his work may have informed that of mathematicians in the 1700's Various Muslim mathematicians added to our knowledge of trigonometry.
36. Calculus is clearly one of the most important mathematical developments of modern times, but it emerged from techniques that had been in use before, some for centuries. Describe some of these techniques and the names we associate with them. I won t write out all the details fo teach of these examples, but: The method of exhaustion, as practiced by Eudoxus and Archimedes, was a precursor to methods of integration, i.e., finding areas under curves. Cavalieri s Principle was also used to find areas of certain figures, and was a precursor to methods of integration. Fermat had methods of finding maxima and minima. Finding tangent lines and normal lines was accomplished by Fermat, Descartes, and Barrow. Five points each for any two of these, with appropriate details.
37. How did Newton s and Leibniz s approach to the development of calculus differ (aside from the obvious differences in notation)? (Hint: what ideas did Newton start with, and what ideas did Leibniz start with?) Leibniz began with sums and differences. He found ways to relate them. For example, he found formulas for the sum of many differences, and the difference of sums. This led naturally to work on areas, as well as to the basic idea behind the fundamental theorem of calculus. This led to ways of relating derivatives and integrals. The differences eventually became differentials, the sums became integrals. Newton started with the idea of rate of change. He was primarily thinking about motion along curves, so he saw what we call graphs of functions being generated by motion. He wanted to express instantaneous rates of change of that motion, and relate the movements in the x- and y- directions. 5 points each for these two explanations.
38. About Trigonometry: a. Where does trigonometry have its roots? Astronomy. Angles and arcs needed to be measured and related. b. How do early tables of trig functions differ from ours today (OK, pretend we have tables instead of calculators. Young whippersnappers.) Early tables didn t give ratios of sides of a right triangle, or ratios of x and y coordinates of a unit circle. Instead, for a circle of fixed radius, the table gave the length of the chord cut off by a central angle (eventually these became half-chords). (4 points) c. What does trigonometry have to do with the invention of logarithms? Because astronomers used the law of sines to solve triangles, they often needed to multiply sines by numbers. This was prone to create errors, since sines were given to several digits of accuracy. Napier invented logarithms to reduce multiplication to addition, which was easier and less prone to errors. (4 points)
39. We know that the Pythagorean Theorem and Pascal s triangle both showed up in the mathematical work of several cultures. Briefly discuss two other mathematical ideas or problems that were also significant in more than one culture. Five points each for any two of the following, with supporting details: Indeterminate equations modular arithmetic determining approximations to π quadratic and cubic equations counting, permutations, and combinations arithmetic and geometric series method of false position others, others, others.