Introduction to Pure Mathematics for Advanced High School Students consists of a series of lessons in Logic, Set Theory, Abstract Algebra, Number Theory, Real Analysis, Topology, Complex Analysis, and Linear Algebra.
Lesson 5 from this series focuses on properties of the real numbers. In this lesson we will learn about ordered fields, why the rational numbers are not enough for real world applications, upper and lower bounds, the completeness property of the reals, the Archimedean property, and the density theorem.
The author recommends covering lessons 2, 3, and 4 before beginning this lesson.
Two PDF files are included with your purchase. The first PDF file consists of notes followed by a problem set. The problems are broken down into five levels of difficulty. The second PDF file consists of solutions to all the problems.
This series is intended to give high school students who are advanced in math an honest and rigorous introduction to pure mathematics. It is unfortunate that most students will not be exposed to real theoretical mathematics until their sophomore year of college and after they have already declared themselves math majors. My hope is that students wishing to explore real theoretical mathematics before committing to becoming a math major will now have the opportunity to do so. And of course I hope that they enjoy experiencing pure mathematics as much as I enjoy writing about it.