Instructor: Dr. Peter Tannenbaum
Office: Peters Business Building 343
e-mail: petert@csufresno.edu

Textbook:
The Book of Numbers, by J. H. Conway and R. Guy (Springer, 1996)

Additional Readings:
o From Zero To Infinity, by Constance Reid (MAA, 1992)
o Numbers: The Universal Language, by Denis Guedj (Abrams, 1997)
o Powers of Ten, by Philip Morrison (Scientific American Books, 1982)
o The Number Sense: How the Mind Creates Mathematics, by Stanislas
Dehaene (Oxford University Press, 1997)


COURSE MANIFESTO

Numbers can be studied for two different reasons-as the building blocks of larger mathematical structures (arithmetic, algebra, calculus, etc.), or for their own intrinsic interest. In this course we will emphasize the latter approach.

This is a course about numbers-their history, their lineage, their uses, and their place in the fabric of life. We will discuss the different types of numbers in school mathematics -integers, fractions, irrational numbers and imaginary numbers- as well as special famillies of numbers within the various types. The goal of the course is to develop in the prospective mathematics teacher a familiarity and appreciation for the wonders of the numerical world, and how this appreciation can enrich our view of the real world.

COURSE OUTLINE

1. HISTORICAL AND CULTURAL ASPECTS OF NUMBER SYSTEMS
o The importance of numbers in different cultures
o Roman, Egyptian, and Hindu-Arabic numbering systems
o The special role of numbers in the English language

2. NUMBERS AND HUMANS
o Number Sense in Children
o Numerical prodigies
o Arithmetic and the Brain

3. MAGNITUDE
o Orders of magnitude in the physical universe
o The language of the very large and the very small
o The art of estimation
o Estimating the estimation: Error Bars.
o Powers of Ten

4. DIVISIBILITY PROPERTIES OF THE INTEGERS
o Divisibility by 2,3,4,5,6,7,8,9, 10, and 11
o Residue classes and Modular Arithmetic
o The Game of Juniper Green

5. FIGURATIVE NUMBERS
o Square Numbers
o Triangular Numbers
o Pentagonal and Hexagonal Numbers
o Polygonal Numbers
o Tetrahedral and Square Pyramidal Numbers

6. COMBINATORIAL NUMBERS
o Factorials
o Arrangement Numbers (Permutations)
o Choice Numbers (Combinations)
o Properties and Identities of Choice Numbers

7. PRIME NUMBERS
o The Sieve of Erathostenes
o Modular Arithmetic with Prime Numbers
o Mersenne Primes
o Perfect Numbers
o Fermat Primes
o The Prime Number Theorem

8. REVISITING FRACTIONS
o Fractions and their Decimal Expansions
o Irreducible Fractions with a Given Denominator
o The Euler Phi Function
o The Structure of Decimal Expansions for a Fixed Denominator

9. IRRATIONAL NUMBERS
o Square Roots and Pythagoras' "Other" Theorem.
o The Golden Ratio
o Pi
o e
o log10 2

10. IMAGINARY NUMBERS
o A Brief History of i.
o Why do we need complex numbers?
o The arithmetic of complex numbers.
o The geometry of complex numbers.