Ten Ways of Looking at Real Numbers
Department of Math/CSci/Physics
Fairleigh Dickinson University
Handbooks and encyclopedias
Varieties of Mathematical Text
A Web-based hypertext of mathematics
A design paper describes goals, organization, technology issues, etc.
First release in June, 2005
This presentation discusses some of the pages on the real numbers
We aim for a “comprehensive introduction" to the real numbers.
Real numbers are everywhere dense in mathematics.
Real numbers have different meanings in different contexts.
Text on Real Numbers
What to include on a text on real numbers?
Foundations and construction of real numbers
Characterizations of the real numbers by structure.
Extensions and substructures of the real numbers
Real numbers classified in different ways.
Real numbers are a system related to other systems.
#1. Foundations of the Real Number System
How to define real numbers and their basic operations from rationals or integers
Presentation of three methods
Dedekind cuts of rational numbers
Equivalence classes of Cauchy sequences of rational numbers
Base-10 digit sequences
#1. Foundations of the Real Numbers
Discussion of foundations from John Conway, "On Numbers and Games"
#2. Real Numbers as a Linear Order
The real numbers form the unique linear order that is:
separable (countable dense subset)