Spring 2005

MTHS 638
Numbers: Their Origins and Modern Development

Reid,James D.

01/25/2005 - 05/03/2005
Tuesday 05:30 PM - 08:00 PM

Science Tower 139

This course will consist of parallel discussion--of the historical genesis of our number systems on the one hand, and a careful development of the modern view of these numbers on the other. We will examine early ideas of number, the introduction of the concept of zero and the consequences of this idea; the confusion over the meaning of negative numbers and the resolution of this confusion; fractions and ratios; irrational numbers; complex numbers, etc. Included will be a study of the axiomatization of the natural numbers (Peano), the subsequent treatment of the negative numbers and fractions, the constructions of the real numbers due to Dedekind and Cauchy, and the Gauss-Argand view of the complex plane. Along the way, we will pause to visit with many numbers of interest such as π, e, and Euler's constant γ. As a bonus, we will include a discussion of modular arithmetic, the p-adic numbers, their relationship and some of their applications to accessible problems. The course will end with a survey of Cantor's work on set theory, infinite (cardinal and ordinal) numbers, and a discussion of the continuum hypothesis.

The course text is MATH THROUGH THE AGES (Expanded Edition), William P. Berlinghoff and Fernando Q. Gouveu, Oxton House Publishers and The Mathematical Association of America, 2004.

James Reid (B.S., M.A. Fordham University; Ph.D. University of Washington) is professor of mathematics, emeritus. He has published extensively in the field of Abelian Groups, including recent papers "Quotient divisible groups, omega-groups, and an example of Fuchs. Abelian groups, rings, modules, and homological algebra", 265-273, Lect. Notes Pure Appl. Math., 249, Chapman & Hall/CRC, Boca Raton, FL, 2006; "Endomorphism Rings of Free Modules," Rocky Mtn. J. Math., 2002; and "Some Matrix Rings Associated with ACD Groups," Proc. International Conference on Abelian Groups and Modules, Dublin, 1998.


Consent of Instructor Required: No

Format: Seminar

Level: GLSP Credits: 3 Enrollment Limit: 18

Texts to purchase for this course:
William P. Berlinghoff, MATH THROUGH THE AGES, Expanded Edition (The Mathematical Association of America), Hardcover


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