Mathematics 629, Spring, 2015

Last updated Fri 15 May 15

Here is the (tentative) course handout (syllabus). The first thing a student should do is to acquire the book by Stillwell. Be sure to get the current (third) edition. I did not list an ISBN because, according to Springer's web page, the book is available in hardback, paperback, and e-book formats (but the paperback costs as much as the hardback).

New feature: Jump down to current week's material (or close to it)

ANNOUNCEMENTS

• Contrary to the Wikipedia page I recommended, it seems that Allen tacitly defines "normal" to mean "slope of the normal", and similarly for "tangent". Only with this understanding do his examples and homework problems make sense. (That is the understanding my 2003 students came up with, before I found the Wikipedia page.) "Subnormal" and "subtangent" DO refer to lengths (possibly signed?).
• TAMU's spring break is next week. Therefore, the Calculus Week homework may be submitted any time up to March 23.

LINKS

The History of Mathematics by G. Donald Allen ©

• interface with frames
• interface without frames

Old course pages:

• Fulling 06
• Fulling 03
• Geller
• Allen

Home pages: Fulling ._._. Do ._._. eCampus ._._. TAMU Library ._._. Math Dept ._._. University

e-mail: fulling@math.tamu.edu
Grader: dothanh@math.tamu.edu
(Please send homework to both of us, but especially to Ngoc Thanh Do.)

Some other pertinent links

WEEKLY COURSE MATERIALS

Week 1 (the beginnings of mathematics)

• Allen readings: Chapters 1 and 2
• Stillwell readings: Chapter 1 (If you don't have the book yet, don't worry -- you will easily be able to catch up later.)
• Additional readings: See the "lecture".
• "Lecture"
• Written assignment: Just this -- Send me an e-mail message telling about yourself:
  • Where you are. Do you expect to be on the TAMU campus at all during this semester?
  • What your "situation" is: Do you work full-time, part-time, or not at all? Doing what? What degree (if any) are you working toward, and how far along are you in the degree plan? Any "complications" I should know about?
  • (optional) Any other personal information you feel like mentioning.
  • What library resources are available to you.
  • Have you taken a Web-based or distance-learning course before?
  • What you hope to get out of this course.
  I hope that you will write this self-introduction in a form that will be suitable for distribution to the other students in the class. If some or all of it is for my eyes only, please so indicate. Whenever you're ready, you can repost it to the "Introductions" forum on our Discussion Board in eCampus. (Use the "Week1" forum for any comments and questions on the first week's material, and so on.)

Week 2 (Egypt, Babylon, and an overview of Greece)

• Allen readings: Chapter 3 and Sections 4.1 and 4.14-4.16
• Stillwell readings: Chapter 2
• "Lecture"
• Written assignment:
  1. From the exercises on Egyptian mathematics:
     • 2 (multiplication)
     • 4 (unit fractions)
     • 5 (false position)
     • 8 [3 persons; the 4 is a misprint] (proportional distribution)
     • 11 (pedagogy by example)
  2. (W) From the exercises on Babylonian mathematics (same web page but farther down):
     • 1 [1265 only] (sexagesimal)
     • 3c (multiplication)
     • 7 (linear systems)
     • 11 [first sentence only; the rest is a misprint] (cubic)
     • 12 (interpolation)

Week 3 (the early and classic Greek periods)

• Allen readings: Sections 4.3-4.8 (Postpone Euclid to next week if necessary. There was no convenient division point.)
• Stillwell readings: Chapter 3
• Additional readings:
  • In Meno type "inquisitive" into your Web browser's text search window to get quickly to the scene with the slave boy.
  • (optional) Allen's link to an article on Thales by D. V. Panchenko is broken, but the journal reference is Configurations 1 (No. 3) (1993) 381-414. For permission to access it on the web you probably must go through the TAMU library site.
• "Lecture" with addendum on the trisectrix
• Written assignment: Exercises from Stillwell unless stated to be from Allen. Stillwell's problems tend to come in pairs, where the first is a lemma for the second.
  1. 1.2.3-4 (parity in Pythagorean triples)
  2. 1.3.1-2 (Euclid's formula for Pythagorean triples)
  3. Allen Early Greek 5 (irrationality of square roots)
  4. 2.3.2 and Allen Early Greek 12 (constructing square roots)
  5. 2.3.3-4 (pentagon)
  6. 3.2.3 (pentagonal numbers)

Week 4 (the late Greek period)

• Allen readings: Sections 4.9-4.13
• Stillwell readings: Chapter 4
• "Lecture"
• Written assignment:
  1. Allen Classical/Hellenistic 2(a,d) (translating Euclid)
  2. Allen Classical/Hellenistic 3. Have we seen this proposition before? (Eudemus)
  3. Allen Classical/Hellenistic 4(a) (Ionian numerals)
  4. Allen Classical/Hellenistic 6 (parabola)
  5. Allen Classical/Hellenistic 16 (mean proportion)
  6. 2.4.3-4 (reflection = shortest path)
  7. 4.3.2-3 (volume of pyramid)

Week 5 (Asia, the Islamic period, and the medieval period)

• Allen readings: Chapters 5 and 6 (Recall that he had his say on China and India in Chapter 1.)
• Stillwell readings: Chapter 5
• "Lecture" with addendum on the Chinese remainder theorem
• Written assignment
  1. 5.3.1-2 (Euclidean algorithm)
  2. 5.4.5 OR 5.4.6, depending on whether you feel "familiar with hyperbolic functions" (If not, change a sign somewhere and relate the Diophantus formulas to the well known trigonometric addition formulas.)
  3. 5.6.1-2 (Brahmagupta triangle decomposition)
  4. Allen Islamic 2 (Thabit's theorem)
  5. Allen Islamic 3 (sum of cubes)
  6. Allen Medieval 2 (Fibonacci solution)

Week 6 (the renaissance period)

• Allen readings: Chapter 7 and Sections 8.1-8.4
• Stillwell readings: Chapter 6
• "Lecture" with addendum on cubic equations
• Written assignment:
  1. Allen Renaissance 1 and 2 (solving cubics)
  2. Stillwell 6.4.1-5 (cube root of 2)
  3. Stillwell 6.6.2 (Leibniz-DeMoivre formula) (Assume the result of 6.6.1.)
  4. Stillwell points out that the Leibniz-DeMoivre formula is "analogous to the Cardano formula for cubics". But 3 is not of the form 4m+1! Harmonize these two observations.

Week 7 (the transition, or precalculus, period)

• Allen readings: Chapter 8 (remainder) and Section 9.1
• Stillwell readings: Chapters 7 and 11 and Sections 8.9 and 9.1-9.3
• "Lecture"
• Written assignment:
  1. Do the experiment on the logarithmic distribution of random data
  2. Answer at least one of the questions on island ecology or estimating compound interest.
  3. Stillwell 7.2.1-4 (conic sections)
  4. Stillwell 7.5.1 (quartic equation)

Week 8 (the creation of calculus)

• Allen readings: Chapter 9 (remainder) (not the 400 pages in German!)
• Stillwell readings: Chapters 9 (remainder), 10, and 13
• Special reading: J. V. Grabiner, "The Changing Concept of Change: The Derivative from Fermat to Weierstrass", Mathematics Magazine 56 (1983) 195-206. (As always, if you have trouble accessing this, let me know.)
• "Lecture"
• Written assignment: The first four problems are from Allen and refer to the "early calculus" period that was covered in last week's reading. These exercises in primitive calculus should be approached in the same spirit as those in Egyptian fractions: we want to gain some feeling for how people at the dawn of calculus thought. Note that to them "tangent" and "normal" did not mean certain lines, but rather certain numbers (lengths of line segments). See the Wikipedia page "Subtangent" for clear definitions of these terms along with "subtangent" and "subnormal". (WRONG -- SEE ANNOUNCEMENT ABOVE.)
  1. 3 (subnormal)
  2. 5(b) (adequality)
  3. 6(a) (extrema)
  4. 7 (de Sluse)
  5. OK, now let's work in the other direction and substitute a streamlined modern treatment of the wave equation for the cumbersome discussion on pp. 262-263 of Stillwell. Follow steps here.

Week 9 (complex numbers)

• Allen readings: none
• Stillwell readings: Chapters 14 and 15 and Sections 12.8, 16.1-16.3, and 16.7
• "Lecture"
• Written assignment:
  1. 14.6.1-2 (complex conjugates)
  2. 15.1.1 (n roots of an nth degree equation). The de Moivre theorem in qustion is formula (2) on p. 281, discussed in exponential form on pp. 315-316.
  3. 15.1.2 (double point). Suggested strategy:
     1. Sketch the cubic curve (or plot it with a program such as Maple). Observe that it crosses itself at the origin. (The graph looks like Fig. 7.1 rotated, except that instead of having a vertical asymptote, it behaves like |y| ~ x^3/2 at infinity.)
     2. Use implicit differentiation to show that the slopes at the origin are plus and minus 1.
     3. Observe that any nonvertical line that passes close to the origin but not exactly through it has its full Bezout quota of 3 intersections, even before considering complex or infinite points. (Don't overlook faraway intersections with the |y| ~ x^3/2 curves.)
     4. [side issue] Make an educated guess about where the missing intersections are if the line is either (a) vertical or (b) so far to the left that it misses the loop of the curve.
     5. Describe what happens to the intersections as the line moves to pass through the origin. Special attention is needed when |slope| = 1.
  4. 16.3.1-3 (Cauchy-Riemann equations)

Week 10 (modern geometry)

• Allen readings: none
• Stillwell readings: Chapters 17 and 18
• "Lecture"
• Written assignment:
  1. 17.1.4-5 (tractrix)
  2. 17.2.5 (parametric equations of tractrix) (Start with the parametric equations as given.)

     There are many more tractrix problems throughout these two chapters. I know better than to assign them all, but after this hands-on experience you can at least go back and read them with more appreciation.

  3. 17.7.1 (additivity of angle excess) THIS ONE ACTUALLY APPEARS IN SECTION 17.6.
  4. 18.2.1-2 (spherical circles and the negative-curvature analog)
  5. Within the Klein disk model (cf. Fig. 18.7) draw a sketch showing that Legendre's axiom (see "lecture") is false in hyperbolic geometry. (If you can't scan a sketch, describe in words what you would add to Fig. 18.7.)

Week 11 (modern algebra)

• Allen readings: Chapter 11
• Stillwell readings: Chapters 19 and 20 (except Sec. 20.7) and Sections 21.7-8
• "Lecture" with addendum on linear algebra
• Written assignment:
  1. 19.4.1-3 (Cayley's theorem)
  2. 20.5.2-3 (quaternion norm)
  3. 20.5.4 (dot and cross products) (NOTE: "Real and imaginary parts" does not refer to the Cayley matrix representation at bottom of p. 426, because the elements of matrix j are not imaginary. As indicated on p. 437, the αI term is considered real and all the rest (multiples of the three quaternion units by real coefficients) is considered imaginary.)
  4. Allen 2 (multiplication mod p)
  5. 19.1.1 (inverses mod 5)

Week 12 (analysis and topology)

• Allen readings: Chapter 10
• Stillwell readings: Chapters 22 and 23 (except Secs. 23.1, 23.2, 23.4)
• "Lecture" with addendum on Lie groups and algebras
• Written assignment: none

Week 13 (set theory and combinatorics)

• Allen readings: Chapter 12
• Stillwell readings: Chapter 24 and Secs. 25.Preview, 25.1, and 25.9. Look briefly at Secs. 25.7-8 for better understanding of remarks in Sec. 24.8.
• "Lecture"
• Written assignment:
  1. 24.1.2-3 (countability of rationals)
  2. 24.1.4 (countability of algebraic numbers)
  3. 24.4.2-4 (nonmeasurable sets)
  4. 24.5.3 (cardinality of a power set)
  5. 24.6.2-3 (Turing halting problem)
  6. Allen 3 (Dedekind arithmetic)
  7. Allen 4 (describable numbers paradox)

Week 14 (modern education: the math wars)

• Allen readings: none. Instead, we will read another educational debate:
  • Klein ©
  • Schoenfeld ©
  • Moursund
• Stillwell readings: none
• "Lecture"
• Written assignment: none