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 Sections	Discussion Topics	HW problems/Journal entries	Other Info
29	5 - 5		FINAL EXAM May 5, 10 AM - 12:30 PM Turn in your journal at exam time.
28	4 - 28		Term paper presentations by Hamilton, Iride, Xu, Kristine, Tan, Tu and Benjamin.	Journal Entry: Include short summaries of each presentation in your journal.
27	4 - 26		Term paper presentations by Jennifer, Laura, Anjali, Joseph, and Juan.	Journal Entry: Include short summaries of each presentation in your journal.
26	4 - 21		Test 2 (Chapters 5, 6, 7, 8, 9)
25	4 - 19	11.4 11.5	Note that modern symbolic algebra was essentially complete by 1637 with Descartes' "La Geometrie." Both Newton and Leibniz understood and applied the inverse relationship between differentiation and integration, that is, they were familiar with the Fundamental Theorem of Calculus. Both discovered essentially the same rules and procedures of today's calculus: determine maxima and minima, draw tangent lines, find curvature of a curve, determine arc length, substitution method and integration by parts. But they differed in their approaches: Newton through the ideas of velocity and distance; Leibniz through ideas of differences and sums. Also Leibniz's notation and method of differentials was easier to work with and this helped analysis (advanced calculus) progress faster on the European continent. Read about the controversy between Newton and Leibniz on page 323.	See below.	A copy of "La Geometrie" is available online at the Cornell University library: www.cornell.edu.
24	4 - 14	11.1.3 11.2.1 11.2.2 11.2.3 11.2.4 11.2.5 11.3.2 11.3.3	With development of analytic geometry, all sorts of new curves and solids could be constructed: any algebraic equation determined a curve, and a new solid could be formed by rotating a curve around any line in the plane. Hudde developed a general algorithm to construct tangents to curves given by polynomial equations. Cavalieri developed some formulas for the area under y=x² and under y=x^k . Torricelli showed that if you rotate a hyperbola from x=a to infinity, the infinitely long solid has finite volume - he used cylindrical shells. Fermat would find areas under curves y=p/x^k by dividing up the x-axis and summing up areas of inscribed and circumscribed rectangles - he derived some integration formulas. Wallis was the 1st to explain fractional exponents and find the area under curves y=x^p/q. Mercator developed a power series (infinite polynomial) formula for log(1+x) by which logs could be calculated. Barrow and Gregory related the tangent problem to the area problem - idea of the Fundamental Theorem of Calculus. Pascal related area under the sine curve to the cosine curve. Why are Newton and Leibniz considered the co-inventors of the calculus?	Practice exercises Page 326 # 36, 42
23	4 - 12	11.1.1 11.4.1	Newton generalized the binomial theorem to powers n that were not positive integers: the kth term of (a+b)ⁿ has coefficient [(n)(n-1)(n-2)...(n-k+1)] / (k!)] and this is multiplied times a^n-kb^k ; but this lead to "infinite polynomials" that are now called power series. In modern calculus, to find the extrema of a function we learn to solve f '(x)=0 that is we find the points that have horizontal tangent lines. Fermat's solution for the maximum product of f(x)=bx-x² agrees with our modern solution but his reasoning is very different.	Practice exercises # 13, page 253, hint: (# earls) = (# dukes)(# of earls per duke) = (D dukes) (2D earls per duke) = 2D² .... Page 325 # 22, 23, 24
22	4 - 7	10.2.2 10.3 10.4	Descartes was familiar with the modern factor theorem. Pascal's triangle may be written in a tabular format of rows and columns. The combination notation stands for the kth entry in the nth row. Pascal found many patterns among the numbers in the table and these are identities when expressed using the combination notation. He also laid the foundation for the theory of probability. Newton generalized the binomial theorem and binomial coefficients to noninteger values of k. From the time of Euclid, perfect numbers were known to be related to what are now called Mersenne primes, primes that can be written in the form 2^p-1 where p is prime. Note that 2¹¹-1 =2047=2389. Fermat* proved that if n is not prime, then 2ⁿ-1 is not prime. Fermat's Little Theorem is an important result in number theory.	Journal Entry: Use Newton's generalized method of binomial coefficients to compute C(1/2,1), C(1/2,2), C(1/2,3) and C(1/2,4).
21	4 - 5	9.3.2 10.1 10.2	Some steps in the development of analytic geometry can be identified: position of a point by suitable coordinates in surveying by Egyptians and Romans, and in map making by Greeks; geometric interpretation of relations among coordinates was used by Apollonius and Menaechmus; Oresme represented some laws by graphing dependent variable against independent variable and introduces equation of a straight line. Descartes would begin with a set of points and then find the equation; Fermat would start with an equation and then find the set of points. How would you use analytic geometry to prove that the midpoints of a quadrilateral are the vertices of a parallelogram? Descartes used the idea of a normal to a circle to describe a tangent line. The invention of logarithms as an aid to calculation is attributed to John Napier (1550-1617) who also popularized the use of the decimal point.	See below.
20	3 - 31	9.1.3 9.3.1 10.1.2 10.2	Ferrari (student of Cardano) found solution to general quartic equation. Abel (1802-1829) proved the roots of a 5th or higher degree equation cannot be expressed by means of radicals in terms of coefficients of the equation. Hermite (1822-1901) gave solution of general quintic by means of elliptic functions. Later solutions to general equation of higher degree are found using Fuchsian functions. Stevin (1548-1620) in his "The Art of Tenths" popularized decimal fractions with well-thought out notation. Girard (1595-1632) gave complete relationship between roots of an equation and the coefficients of the equation: coefficients correspond to "factions". Two fathers of analytic geometry are Descartes (1596-1650) and Fermat (1601-1665).	Journal Entry: Use Girard's method of factions to write the polynomial equation with solutions 1, 2, -3, 4, and 5.
19	3 - 29	9.1.3 9.1.4 9.1.5	Note that the terms of the Fibonacci sequence converge to the Golden Ratio. Italian mathematicians were able to solve cubic and quartic equations algebraically. Some of the noted individuals are: del Ferro and Tartaglia (both solved the cubic with no quadratic term); Cardano published the solution to the cubic in his "Ars Magna" in which he also considers negative roots and complex numbers; Bombelli in his "Algebra" contributes to algebraic notation and presents examples of the four arithmetic operations with complex numbers including notion of conjugate for division. The solutions were in the form of rules of procedure; no formulas were written down like the quadratic formula of today. Viete shows how to transform a cubic so it has no quadratic term, wrote 1st book in Western Europe using all 6 trig functions, introduces symbols for constants, replaces search for solutions to equations with theory of equations.	Practice exercises Page 253 # 13, 17, 18, 21 Journal Entry: In the Cardano-Tartaglia formula for solving a cubic equation with no quadratic term, x = a-b. The formula for a was derived in class. Show the steps to find the formula for b.
18	3 - 24	9.1.1 9.1.2 9.1.3	The nth term of the Fibonacci sequence is given by [(1+√5)ⁿ - (1-√5)ⁿ ]/(2ⁿ√5)). Levi ben Gerson in his "Art of the Calculator", 1321, uses an inductive style of proof to show how to find the number of permutations and the number of combinations of n objects taken k at a time. In 14th century Italy, the abacists teach the children of middle-class merchants the Hindu-Arabic numeral system and their methods of calculation. About 1450, printing was invented. Regiomantanus (1436-1476) gave Europe 1st study of plane trig independent of astronomy. Frenchman Chuquet introduces Europe to negative numbers, uses positive, negative and 0 exponents. Nicole Oresme (1323 - 1382) makes 1st use of fractional exponents, locates points by coordinates, finds sum of an infinite series. Christoff Rudolff wrote 1st comprehensive German algebra that uses + and - for addition and subtraction, and radical sign. German Michael Stifel (1486-1567) uses symbols of Rudolff, foreshadows invention of logarithms, combines cases of quadratic equation into one, represents unknown with a letter. Robert Recorde (1510-1558) wrote 1st English algebra, uses = for first time.	Practice exercises Page 252 # 1, 8 Journal Entry: Calculate the 20th, 25th and 50th terms of the Fibonacci sequence.
16	3 - 22	8.1.1 8.1.2 8.3.1	The Roman empire collapsed in 476. For five centuries the general level of culture was low in Europe (Dark Ages/Medieval Europe). At turn of the millenium, through work of translators, mathematical tradition of Greeks and Islamic world became known in Western Europe. In Spain, Islamic manuscripts were translated into Spanish then into Latin. Abraham bar Hiyya: wrote text to help with measurement of fields; used 3 1/7 for pi; measured areas of segments of circles that uses a table of chords (1st time in Europe there appears a trigonometric table). Leonardo of Pisa (Fibonacci). His greatest work was the *Liber abacci; it introduces Hindu-Arabic numerals and the rules for computing with them; it contains many problems that are now standard in algebra texts such as mixture problems, motion problems, container problems, Chinese-remainder problem, problems solvable by quadratic equations and systems of equations, and the rabbit problem (Fibonacci sequence*). See pages 202 - 203.	Practice exercises Page 210 # 1, 2, 3, 7, 15 Term paper: Turn in completed term paper by April 14.
15	3 - 10	7.2.3 7.2.4 7.2.5 7.3 7.4 7.5 7.6	There are examples in the book of the inductive type of argument and calculations used by al-Haytham to find the sum of the fourth powers of the first n integers. Al-Karaji also used geometry to prove an identity for the sum of the cubes of the first ten integers. The new Hindu-Arabic numeral system versus the old Roman numerals and abacus method of computation led to a rivalry: the abacist versus the algorist. Al-Baghdadi was able to prove a result on the density property of the irrational numbers. Omar Khayyam was the first to systematically classify and solve all types of cubic equations by the method of intersecting conic sections. Islamic mathematicians also studied combinations, permutations, Pascal’s triangle, the Binomial theorem and amicable numbers. They also used the method of exhaustion to find volumes of solids. While the Hindus developed the sine function, the Arabs added the other five trig functions, and had highly accurate trig tables. They also tried to prove Euclid’s fifth postulate – this eventually was studied in Europe and led to discovery of non-Euclidean geometry.
15	3 - 8	7.2.3 7.2.4	Note on page 165 that al-Khwarizmi had cases for solving quadratic equations, not just 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.