Math 4312

Daily Calendar - Math 4312 (20575)

What I hear, I forget; what I see, I remember; what I do, I understand.

Session	Date	Read & Study Unit	Discussion Topics	HW problems/Journal entries	Other Info
15	3 - 8	7.2.3 7.2.4	Note on page 165 that al-Khwarizmi had cases for solving quadratic equations, not one formula for all cases like we have today. Al-Samaw'al divided polynomials by polynomials, used negative powers of x if needed, and recognized approximate answers. He was the first to recognize that fractions could be approximated by calculating more and more decimal places through division. [algebraic operations and number operations are parallel] Islamic mathematicians used inductive arguments to prove formulas about the integers: (1) begin with a value for which the result is known to be true; (2) use the result for a given integer to derive the result for the next integer. They only stated a result for particular integers since they did not have ways of saying the general statement - but it is only a short step for modern readers to go from their arguments to a modern complete inductive proof. Study section 7.2.4.	See below. Term paper: By March 25, submit as complete a draft as possible to the Writing Center for feedback. Journal entry: Verify the algebraic identity
14	3 -3	7.1 7.2.1 7.2.2 7.2.3	Golden Age of Islamic World from about 750 to 1050 CE: best period for science and mathematics that greatly influenced learning in Europe, and through translation preserved learning from ancient Greece, Rome and Egypt. One reason for the Golden Age was the establishment of a paper mill in Baghdad; paper was invented in China about 200 BCE. This made it possible for a great many people to get books and learn from them - an important advancement that affected education and scholarship. Translations and research were done at the House of Wisdom, lasted over 200 years. In his arithmetic text, al-Khwarizmi introduces Indian decimal place value system, shows how to write any number in this notation, describes arithmetic algorithms - his work introduces many Europeans to basics of this numeral system. In his algebra book, al-Khwarizmi uses operation called "al-jabr" to solve equations. He used no symbols, only words, and shifts away from geometric meaning of numbers to find numbers satisfying certain properties. He knew rules for multiplying signed numbers. Al-Samaw'al established the algebra of polynomials: introduced negative coefficients, rules for adding, subtracting, multiplying and dividing polynomials. He used a table for multiplying polynomials.	Practice exercises Page 188 # 4, 5, 6, 9, 11
13	3 - 1	6.1 6.2 6.3 6.4 6.5 7.1	More mathematical contributions by the Indians include: algorithms for arithmetic calculations and rules for signed numbers; were familiar with the Pythagorean theorem; had geometric formulas such as for the area of a cyclic quadrilateral; were familiar with the quadratic formula; had methods for solving systems of linear congruences; have the earliest records of combinatorial rules involving the number of possible combinations and the number of possible permutations of n objects; discovered the sine and cosine power series. Islamic mathematicians completed the Indian place value system to include decimal fractions, thus creating the modern Hindu-Arabic decimal place value system (Treatise of Arithmetic, 1172).	Term paper: On March 3, turn in your journal including a preliminary table of contents for your term paper.
12	2 - 24		Test 1 (Chapters 1 - 4)
11	2 - 22	6.1	Class began with review questions for the test. Early Indian mathematical ideas are found in the Sulbasutras. By 600 CE, the Indians used symbols for 1 through 9 in a place value arrangement - the first to use a decimal place value system for the integers (at first, a dot is used for zero). There is no early evidence that it extends to decimal fractions (tenths, hundredths, etc); these were eventually introduced by the Muslims.	Practice exercises Page 159 # 20 - 23
10	2 - 17	5.1 5.2 5.3 5.4	The Chinese were the first to: (1) create a positional decimal numeral system; (2) acknowledge negative numbers; (3) obtain precise values of pi; (4) present Pascal's triangle; (5) be aware of binomial theorem; (6) use matrix methods to solve systems of linear equations; (7) solve systems of simultaneous congruences by Chinese Remainder Theorem; (8) have record of magic squares. They were also aware of the Pythagorean Theorem; could find square roots; and had formulas for areas and volumes of geometric figures. The *Nine Chapters on the Mathematical Art*, a compilation of practical problems and solutions, was used for centuries; over the centuries, commentaries were written to explain or derive rules.	Practice exercises Pages 134 - 135 # 3, 5, 6, 7
9	2 - 15	2.2 3.3.3 4.3 5.1	One can prove that the sum of the angles in a triangle is 180^o using a geometric proof involving parallel lines, alternate interior angles and corresponding angles. (Greek) Ptolemy wrote the *Almagest, a work studied, commented upon and extensively criticized, but never replaced for 1400 years, in which he used ideas from plane geometry and spherical geometry and devised ways to perform extensive numerical calculations. The basic element in his trigonometry was the chord* subtending a given central angle in a circle of fixed radius, see the formula and figure 3.19 on page 89. He produced a table listing angles and the values of the chords of the angles (equivalent of a sine table). Hypatia (355-415 CE) is the only Greek woman for which there is substantial indication of math accomplishments. Her death effectively ended the Greek mathematical tradition of Alexandria. The "oracle bones" are the source of current knowledge of early Chinese number systems, about 1600 BCE. The Chinese used a multiplicative positional decimal numeral system. See pages 117-118.	Practice exercise Page 134 # 1
8	2 - 10	2.2.4 3.1 3.2 3.3	In book VII of the Elements, Euclid describes the Euclidean algorithm for finding the GCF of two numbers. Archimedes (287 - 212 BCE) is considered by some as the greatest mathematician of antiquity; his mathematical writings resemble modern journal articles; his use of the method of exhaustion foreshadows modern methods of integral calculus; he found a good of approximation of pi, 223/71 < pi < 22/7; he discovered elementary properties of centroids and applied mathematics to hydrostatistics. Erathostenes is noted for his prime number sieve. Apollonius (262 - 190 BCE) wrote a book on the Conics that superseded all others; he brought the study of conics to the modern point of view; he was apparently the first to show that from a double-napped cone come all conics; he introduced the names ellipse and hyperbola. Hipparchus perhaps introduced into Greece the division of a circle into 360 degrees; he developed a table of trig sines from 0 to 90 degrees. Heron (150 BCE - 250 CE) in one of his works describes a method for square roots that was known to the Babylonians. Diophantus may have been the first to take steps toward algebraic notation by adopting abbreviations for unknown quantities and arithmetic operations.	Practice Exercise Page 114 # 1 Term paper: By March 3, turn in a preliminary table of contents for your term paper (chapter or section titles).
7	2 - 8	2.1.2 2.2 2.2.5	(1) Aristotle's (384-322 BCE) rules of attaining knowledge by beginning with axioms and using demonstrations to gain new results became the model for mathematicians to the present day. See page 36. (2) Euclid's (325 -256 BCE) Elements, written about 2300 yrs ago is the most important mathematical text probably of all time. There are simply definitions, axioms, theorems and proofs - no numbers are used in it except for a few small positive integers. It is recognized for its completeness and organization. (3) Euclid's fifth postulate (the parallel postulate) has caused immense controversy. It is equivalent to the more familiar Playfair's axiom. See pages 42-43. (4) Books VII - IX of the Elements deal with number theory. Proposition IX-20 is the famous "there are infinitely many primes." See page 58. (5) Book XII uses the method of exhaustion. See page 60. (6) In Book XIII, Euclid constructs the edges of the five regular solids, and demonstrates that there are no regular polyhedra except for these five. (7) Archimedes and Apollinius probably received mathematical training from students of Euclid. See page 62.	Practice Exercises Page 64 # 19, 25
6	2 - 3	2.2.2 2.2.6 3.1.1	Some mathematical accomplishments of the Greeks include: (1) The Greeks devised geometric processes for carrying out algebraic operations: "geometric algebra". (2) A regular polyhedron has faces that are congruent regular polygons. There are only five different regular polyhedra; Plato gave a description of these. (3) Three famous straight-edge and compass construction problems studied by the Greeks are: squaring the circle, duplicating the cube, trisection of an angle. These were not proved impossible until 1880 (over 2000 yrs later), but in the mean time, the search led to many mathematical discoveries. (4) Archimedes used the perimeter of inscribed and circumscribed regular polygons to better and better approximate pi (showed between 223/71 and 22/7): related to the Method of Exhaustion.	Study the solution given in class to problem 29, page 27. Do you understand it? Term paper: By February 10, turn in 10 references on your term paper topic. At least six of these must be books or articles (give author, title, journal, pages), and at most four may be web sites (give title and URL).
5	2 - 1	2.1	The geography of a country had a tremendous effect on the accomplishments of its people: think of the Egyptians and the Nile river, of the Babylonians and the Tigres and Euphrates rivers, of the Greeks and the Mediterranean Sea, etc. A new attitude toward mathematics appeared in Greece beginning in the 6th century BCE: it was no longer enough to calculate numerical answers; it was necessary to prove the results were correct. Thales is credited with beginning the Greek mathematical tradition. See page 30. Aristotle took the ideas developed over centuries and first codified the principles of logical reasoning; logical argument according to his methods was the only certain way of attaining scientific knowledge. See page 35. Problems discussed in class: # 4 and 5, page 63; mathematical induction; Pythagorean theorem; sqrt(2) is irrational.	Complete solutions to # 4 and 5 discussed in class. Also, see problems below
4	1 - 27	2.1	Read the introduction on page 29 about the new attitude toward mathematics by the Greeks (800 - 336 BCE). The Pythagoreans were both a religious order and a philosophical school in which "numbers formed the basic organizing principle of the universe... and the basis of all physical phenomena." The Pythagoreans were especially interested in number theory and studied special types of numbers: an amicable pair of numbers, figurate numbers including square numbers, triangular numbers, etc. With their discovery of irrational numbers, they were forced to adjust their mathematical philosophy and this enabled Greeks to develop new theories.	Practice Exercises Page 63 # 3, 4, 6, 7	Check out this cool web site! MathWorld (http://mathworld.wolfram.com/) is a comprehensive and interactive mathematics encyclopedia.
3	1 - 25	1.2	Mesopotamia/Babylonian civilization: (1) had sexagesimal (base-60) place value numeral system; (2) made extensive use of tables in arithmetic computations; (3) treated all fractions as sexagesimal fractions, 0;1,15 represents 1/60+15/60^2; (4) had algorithms for square roots; (5) solved linear and quadratic equations - beginnings of a rhetorical or prose algebra; (6) approximated pi ≈ 3; (7) besides areas of geometric figures, also dealt with volumes of solids; (8) were aware of the Pythagorean theorem and had great interest in Pythagoreans triples; (9) made use of "cut-and-paste" geometry to solve equations (derive quadratic formula), etc.	Practice Exercises Pages 26 - 27 # 18, 29 Term paper: Skim through the index of our book and the index of the MacTutor History of Mathematics Archive, http://www-groups.dcs.st-and.ac.uk/~history/ to help you choose possible topics for your term paper. On Feb. 1, turn in two possible topics for your term paper. Journal entry: By Feb 1, complete and include the analysis of your chosen paper in your journal-see the instructions in the class of Jan 18.
2	1 - 20	1.1	Note: The Egyptians (1) only dealt with unit fractions with the exception of 2/3. (2) used the method of doubling for multiplication and division, (3) knew how to find the areas of rectangles, triangles and trapezoids by normal modern methods; (4) for the area of a circle used 256/81 for the value of Π; (5) had a formula for the volume of a truncated pyramid; (6) knew how to determine the surface area of a hemisphere.	Journal entry: What is the meaning of civilization? Give an example of an ancient civilization. Can you give an example of a modern civilization? Practice Exercises Pages 25-26 # 2 (and check the answer using the Egyptian method of doubling) #5	Check out the services offered by the UHD Writing Center (N925). A brochure of their services is available - ask me for a copy.
1	1 - 18	1.1	1.1.1 What is the significance of the Rhind Mathematical Papyrus? Of the Moscow Mathematical Papyrus? 1.1.2 Egyptians developed two different number systems: hieroglyphic system and hieratic system. Describe each of these systems.	Go through the database/index for some math journals for a couple of years and make a list of the titles of at least 5 articles that are of interest to you. Turn this list in at the Jan 25 class. Some possible math journals are: American Mathematical Monthly, college Mathematics Journal, Mathematics Magazine, Mathematics Teacher, Historia Mathematica. Read the abstract of each of the articles on your list. Choose 1 article from the list, print a copy of it and then analyze the article: What is the major topic? What is the time period it covers? Describe the math content. What type references are used? Etc.	MathSciNet (http://www.uhd.edu/library/ subjects/databases.html) is now available through the UHD library; it is a comprehensive database covering the world's mathematical literature since 1940; the approximately 1800 current serials and journals reviewed in whole or in part are listed in the Abbreviations of Names of Serials; MathSciNet is available 7 days a week. The period between 6 a.m. and 6:30 a.m. U.S. Eastern Time is reserved for maintenance. MathSciNet may be unavailable during this period.