This course will explore the beauty and mystery of mathematics through a study of the patterns and properties of the natural numbers 1,2,3... . We'll discuss various special classes of numbers, like Fibonacci numbers, factorials and binomials and the many ways they arise in mathematics and nature. We'll also investigate the mysterious behavior of the prime numbers and their distribution, and alternative counting systems such as modular arithmetic.
Prerequisites
We will assume no mathematical background beyond high school algebra. Emphasis will be placed on discovery through conjecture and experimentation.
Faculty
| Name | Office | E-mail address |
| Benedict Gross | Science Center 326 | gross@math.harvard.edu |
| Joseph Harris | Science Center 339 | harris@math.harvard.edu |
Teaching Fellows
| Name | Office | E-mail address |
| Laura De Marco | Science Center 321d | demarco@math.harvard.edu |
| Mark Lucianovic | Science Center 431e | markl@math.harvard.edu |
| Elena Mantovan | Science Center 425e | mantovan@math.harvard.edu |
| Nick Rogers | Science Center 333g | nfrogers@math.harvard.edu |
Course Website
www.courses.fas.harvard.edu/~qr28
Textbooks
Ivan Niven, Mathematics of Choice: Or, How to Count Without Counting
Constance Reid, From Zero to Infinity
See the website www.fas.harvard.edu/~ucbooks to order these books online.
Sourcebook Materials
Philip Davis and Reuben Hersh, The Prime Number Theorem.
Benedict Gross and Joe Harris, The Magic of Numbers.
Benedict Gross and Joe Harris, Catalan Numbers.
Paul Hoffman, Archimedes' Revenge.
Malcolm Lines, Think of a Number.
Edgar Allan Poe, The Gold-Bug
Tom Weston, Infinity.
The sourcebook is available in the Science Center basement.
Homework
There will be short homework assignments after each lecture. The homework will be designed to review concepts from the previous lecture, to reinforce topics in the reading and to introduce some of the ideas of the next lecture. All of the homework questions will be posted on the course's website; they will not be handed out in class. Homework is due the lecture after it is assigned. Absolutely no late homework will be accepted.
Exams
There will be two midterm examinations during the semester and a final exam during the finals period. The midterms will be given during class on Friday, October 12th and Friday, November 16th. Notify one of the teaching fellows as soon as possible if you have a conflict with one of the exam times. The final exam will be scheduled by the registrar's office.
Grading Policy
The final grade will be based on the homework, the two midterms, and the final. The homework and each midterm will count for about 20% of the final grade, and the final exam will count for about 40% of the final grade. Minor adjustments may be made to take into account improvement during the semester.
Sections
Section times will be announced in the second week of the course, based upon the students' schedules. These will meet for one hour per week. In section we will review some topics from lecture and explore related issues. Section attendance is required. We will ask you on Friday for the openings in your schedules and over the weekend we will post your section times on the course website.
Schedule
Below is the tentative lecture schedule for the course. It may change somewhat depending on the length of time required for certain topics and student interest.
Wednesday, September 12: The Remarkable Fibonacci numbers
Lines, Chapter 2
Friday, September 14: Infinity
Reid, Chapter aleph0
Weston, Infinity
Monday, September 17: How to count without counting I
Niven, Chapters 1 and 2
Wednesday, September 19: How to count without counting II
Niven, Chapters 1 and 2
Friday, September 21: Probability
Niven, Chapter 5
Monday, September 24: The binomial theorem
Niven, Chapters 2 and 3
Wednesday, September 26: Pascal's triangle
Niven, Chapter 3
Friday, September 28: Catalan numbers
Gross-Harris, Catalan Numbers
Monday, October 1: The Euclidean algorithm
Wednesday, October 3: Diophantine equations I
Friday, October 5: Diophantine Equations II
Wednesday, October 10: Review
Friday, October 12: First midterm exam
Monday, October 15: Prime numbers
Reid, Chapter 3
Wednesday, October 17: Unique factorization
Reid, Chapter 1
Friday, October 19: The fundamental theorem of arithmetic
Monday, October 22: Numbers and number systems
Archimedes, The Sand Reckoner
Gross-Harris, Chapter 1
Wednesday, October 24: Modular arithmetic I
Lines, Chapter 7 (pp. 60--64)
Reid, Chapter 9
Gross-Harris, Chapters 2, 3
Friday, October 26: Modular arithmetic II
Reid, Chapter 9
Gross-Harris, Chapter 4
Monday, October 29 : Fermat's little theorem I
Gross-Harris, Chapter 5 (pp. 53--63)
Wednesday, October 31: Fermat's little theorem II
Lines, Chapter 7 (pp. 64-66)
Gross-Harris, Chapter 5 (pp. 63--67)
Friday, November 2: Computing powers modulo p
Gross-Harris, Chapter 5 (pp. 67--71)
Monday, November 5: Computing roots modulo p
Gross-Harris, Chapter 6
Wednesday, November 7: Euler's theorem
Gross-Harris, Chapter 7 (pp. 85--94)
Friday, November 9: Computing powers modulo m
Gross-Harris, Chapter 7 (pp. 94--98)
Wednesday, November 14: Review
Friday, November 16: Second midterm exam
Monday, November 19: Computing roots modulo m
Gross-Harris, Chapter 7 (pp. 98--100)
Wednesday, November 21: How to build codes I
Reid, Chapter e (pp. 159--163)
Monday, November 26: How to build codes II
Lines, Chapter 9
Wednesday, November 28: Distribution of primes I
Reid, Chapter e
Friday, November 30: Distribution of primes II
Monday, December 3: Distribution of primes III
Hoffman, Chapter 3
Wednesday, December 5: Distribution of primes IV
Hoffman, Chapter 3
Friday, December 7: Return to Fibonacci numbers
Lines, Chapter 2
Monday, December 10: Surprise Lecture
Wednesday, December 12: Review