History of Mathematics

Laboratori de Matemàtiques I

Professor: Joe Miró Julià
Office: 168, Tel.: 971 172 982
e-mail: joe.miro (arrob) uib.es

Course documents

Syllabus

NEW Final report or presentation

Initially, part of the grade was to be awarded by writing a report on a subject of history of mathematics. Given that several of you have commented that you liked the experience of researching and presenting the subject of mathematics in the middle ages, I am offering the oportunity of changing the final essay by a presentation similar to the one you made. These are the conditions:

Teams must be of three people maximum
The subject must be on current trends in mathematics
People can choose between a personal report or a team presentation, but at least there must be three presentations. If only one or two teams appear, you all must write a report.
The presentation slides must be written in English, but you can choose the language for the oral presentation. You must also hand in a short report that might be written in any language.
The oral presentations should last around 40-45 minutes.

Possible subjects

The subjects for the presentations are about current trends in mathematics. You can choose one of the following. I list many parts for every subject. These are only examples of material to cover: there is no need to cover them all! If you want to research a subject not on the list, consult me first.

Mathematics of computation: Turing, turing machines, Turing-Church thesis, Godel's incompleteness theorem, theoretical computer science, numerical analysis.
Chaos theory: Non-linear systems, fractals, Mandelbrot, Julia sets
Cryptography: Cryptographic methods, deciphering the Enigma, RSA algorithm, trap functions, zero-knowledge functions.
Finance mathematics: Actuarial mathematics, stok options, Black-Scholes equation, mathematics of risk
Computer aided proofs: the four color theorem.

Homework assignments

Read the document on Greek numbers or the Archimedes' classic The Sand Reckoner and explain briefly the main differences between greek and roman numerals or how Archimedes' way of obtaining large numbers differed from ours.
Analyzing the state of mathematics before Euclid and after the Elements appeared, write a 'Story of Euclid' explaining how he created the Elements.
Look at all 48 propositions of Book I of the Elements and you create a dependency chart: show which propositions depend on which others and on the 5 postulates (don't bother with the definitions and common notions). Once you have created the chart, analyze it and get some conclusions about which results were considered fundamental by Euclid, and which other were simply stepping stones toward the important ones.
(For extra credit) Look at Proposition IX.14. This is not the theorem of unique decomposition of prime factors as we know it. Why not?
Reproduce Archimedes' method of finding the value of pi through inscribed and circumscribed 96-gons using pocket calculators, spreadsheets, computer programs or anything else you need. Compare your results to his. ¿How long did you take?
Mersenne stated in the preface to his Cogitata Physica-Mathematica (1644) that the numbers 2ⁿ-1 were prime for n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257 and were composite for all other positive integers n < 257. Up to 19, this result was known, and it is also easy to prove that if n is not prime 2ⁿ-1 is not prime either.
We now know that the correct list is: n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127. That is, Mersenne only made 5 mistakes (two of commision and three of omision). Establish the probability of doing this well by chance. Explain your result.
Create a graphical time line of all the main events of the history of mathematics explained in class.

Web resources

General

History of Maths at University of Western Australia
Teaching Math with original historical sources

Number systems

Pythagoreans, esthetics, music

Greek mathematics

Euclid's Elements
- Definitions, postulates, and common notions
- Propositions
Archimedes
- Archimedes' Home page
- Measurement of a circle
- A simple solution to the cattle problem.
Mathematics from after the Greeks until the Renaissance
- Maths of the Arabs Student presentation
- Fibonacci Student presentation
- Maths in Europe by 1200 Student presentation
- Fibonacci
Logarithms
- History of logarithms by Xavier Lefort
- Another account of the history of logarithms
Newton, Leinbniz and calculus
Leonhard Euler
- Mac Tutor Biography
- Biography from `A Short Account of the History of Mathematics' by W. W. Rouse Ball.
- Euler's proof that the sum of the inverse of primes diverge.
Carl F. Gauss
Cryptography
- History of cryptography
- The Enigma
- RSA cryptographic method by Rivest, Shamir and Adleman.
- Zero-knowledge proofs