In this lecture we introduce the sets of natural numbers, integers, and rational numbers. Although the construction of the real numbers from the rationals will be presented in a future video, we present the axioms of an Archimedean Ordered Field which are satisfied by both the rationals and the reals. These axioms are the fundamental algebraic properties of the real number systems that will be assumed throughout our proofs in the following lectures. We further illustrate the main structure of the lectures that follow by providing a definition of the absolute value of a real number and proving that it satisfies the triangle inequality.

This course is taught by Jason Bramburger.