Math 163 History of Mathematics 2004

Instructor: Hans Wenzl, email:

Office: APM 5256, tel. 534-2734

Office hours: MW 1-2, F4-5

Teaching assistants: Aron Lum ( office hours: Monday and Wednesday 11-12 at APM 2250 and Nicholas Slinglend ( office hours: Wednesday 12-2 at APM 2325

Computation of grade: homework 15%, midterm 20%, term paper 25%, final 40%.

Dates of exams: midterm: May 7

Course book: An introduction to the history of mathematics, sixth edition, by Howard Eves. See also literature and websites just above examples of course projects, in particular the book by Burton.

Homework assignments The teaching assistants will tell you the precise deadline for turning in homeworks (but not later than Friday after the session for which the due date is given).

Disclaimer: I will try to get the homework assignment on the net in time. Due to time and other limitations, this may not always be possible. The fact that there is no assignment posted for a particular date does therefore NOT necessarily mean that no homework is due.

due 4/1: 1. Write the numbers 574 and 475 in a) Eyptian hieroglyphics, b) Roman numerals, d) Babylonian cuneiforms AND write in Roman numerals one fourth of MCXXVIII and four times XCIV.

2. Multiply 235 by 41 using Egyptian multiplication.

3. Write 4/17 and n/10 as Egyptian fractions, n=1,2,... 9. Moreover, write 153.125 and 1/11 in Babylonian numbers using the Neugebauer notation.

4. (a) Express (3012)_5 in base 8, (b) For what base is 3x3=10? For what base is it equal to 11? (c) Can 27 represent an even number in any scale? Can 37? Can 72 represent an odd number in any scale? Can 82?

due 4/8: 2.4(a)-(d), 2.7(a)-(e), 2.11(a)-(c), 2.12(a)-(b)

due 4/15: 3.2(a)-(e), 3.4(a)-(d), 3.5(a), 5.1(a),(d),(e). Read chapters 3.1-3, 3.6, 4.1, 4.2 in book.

Also: Write down the (approximate) years of birth and death, places where they lived, as well as up to three of the major achievements of Thales, Pythagoras, Aristotle, Euclid and Eratosthenes. You need not explain their achievements.

due 4/22: read Sections 4.2, 4.8, 5.1-4, 5.7; do 4.7(a),(b) or 4.4(a),(b), 4.13(e)-(i), 5.4(a),(b),(c), 5.7(a),(b)(read p.540),(c)

due 4/29: 6.1(a),(b),(c), 6.7(a)(don't worry about the number of solutions),(b), 6.9(a),(b1) (remark: 6.9(a) is tricky and was done in class on Friday), 6.15ab (for (b): let x be the sum of the four numbers, say a, b, c, d; then the given numbers are x-a, x-b, ... ; show that the sum of the four given numbers must be 3x), read Sections 6.8-10

due 5/6: preparation for midterm, need NOT be turned in (but very similar problems may be on the midterm). 7.1(d), 7.3(a),(b), 3.10(a)

1. Make list of major Greek mathematicians not already covered before with approximate lifetime, where they lived, and their major achievements,

2. (a) Name three major papyri with mathematical contents, (b) Describe three geometric problems in the Rhind papyrus. How good was the approximation of the area of the circle? (see Prob 2.13).

3. Give a geometric proof of Pythagoras' theorem (see e.g. Problem 3.5).

4. Describe the Euclidean algorithm and one other important theorem in number theory in Euclid's elements

5. Name a major mathematical work (i.e. book or collection of books) by Pappus, Diophantus, Claudius Ptolemy and Apollonius, and one important mathematical result in each of these works.

6. Review doing calculations in and de/encoding from/into ancient mathematical number systems such as the Babylonian, Egyptian and Greek ones. Example: Write 2/5 in Egyptian notation using Fibonacci's method

7. Construct the geometric mean of two line segments with lengths a and b.

A few more review problems will be posted by some time early next week.

due 5/13: 7.6(a), 7.8(a),(c),(e), 7.11(a)

due 5/20: 8.4(b) 8.14 (d) (dom't worry about Viete's method),(e) 8.15(a)

due 5/27: 9.8(a),(c),(d), 10.3(a),(c),(d), read: 8.8, (9.2), 9.3, 9.6, 9.7,

due 6/3: TERM PAPER moreover, read 11.9, 11.10, do 14.24(a),(b),14.25(b),(c), 14.26(c), 14,27(a)

Special office hours for final: Sunday 3-5pm (Nick) Monday 1-2 at my office.

Final: The final will take place on Monday, June 7, from 3-6pm in the usual class room. The same rules apply as for the midterm: You will be allowed to use one hand-written normal size cheat-sheet, but no other books or calculators. Problems may include calculations in ancient notations, some historical questions and problems similar to homework problems. The mathematical problems will be easier than the most difficult homework problems - however you are expected to understand and possibly use the results of those homework problems.

Material The material will primarily be taken from what we covered after the midterm, i.e. after Greek mathematics. Exceptions may be material which may be related to our more recent material such as Euclidean algorithm, calculations with respect to certain bases (e.g. basis 60, or more generally for congruences modulo some number n). Here are some problems which should give you an idea what you can expect on the final (this does not mean that exactly the same problems will appear, or that there could not appear different problems):

1. Name some important Arabic (including Persian), Hindu, Chinese mathematicians, with some of their most important achievements (e.g. books or papers). Same questions about European mathematicians of a certain period, e.g. between 14-th and 16-th century, 17-th century, 18th century (only as far as covered in class).

2. When and where did Newton and Leibniz publish their work on calculus? Explain why it came to a controversy.

3. Use the Cardano-Tartaglia formula to find a solution of the cubic equation x^3 + 24x=16.

4. Compute all integer solutions of the equation 6x + 15y= 21.

5. Let F_n be the n-th Fibonacci number, defined inductively by F_1=F_2=1 and F_{n+1}=F_n+F_{n-1}. Prove by induction that F_n^2=F_nF_{n+1}-F_nF_{n-1}.

6. Use Fermat's little theorem to determine the remainder of 15, taken to the its 33rd power modulo 17 (i.e. compute the remainder of 15^{33} after division by 17).

UNDER CONSTRUCTION ! contents may change!

Term paper: An important part of your grade will depend on your term paper. Deadline will be the TA-section in the last week of classes in this term. This is an involved project, so GET STARTED SOON (before the midterm)! Here are the steps you have to take for your project:

1. Find a topic and a primary reference. It is wise to consult me or your TA about your choice of a topic. See also below for some suggestions. Once we have agreed on a topic then you should begin to collect references for your paper. There must be at least one primary reference. This means that if you are writing about, say, Descartes you should quote material by Descartes or something by a contemporary of Descartes. Putting material from standard history books in your own words will not be enough. Translations of original material will be OK. Check with us and we will let you know if you are on the right track.

2. After passing the first two hurdles (topics and primary reference) you should do your research. Make an outline or a synopsis of your proposed paper and then get it approved without going on. We will make some suggestions if we don?t think that the paper will be satisfactory.

3. The rest is up to you. The average size of the papers in earlier classes was about 12 pages. The paper due is before the last day of class. Remember that all quotations and factual statements must be backed by references. At least one reference must be primary and important to the paper. You may turn in your paper by email. I have put several books on reserve, such as Burton, David: The history of mathematics, fourth edition, Smith, David E.: A source book in mathematics, Heath, Thomas L.: Euclid's elements Midonick, H.: The treasury of Mathematics. A wealth of information can also be found at the following website MacTutor . Here is a link to Euclid's elements

Here are a few topics that have made good papers in the past:

The Pythagoreans.



Diaphantes and Fermat?s Last Theorem

Mathematics of Ancient China

Mathematics of Ancient India

Mayan Mathematics


Cardano and Gambling




Pascal and computing

Twentieth century logic