### Dylan Stover and Dr. Ivona Grzegorczyk

## Abstract

We present an introduction to higher-level mathematics by studying various subsets of integers, rational and real numbers. For example, we consider sets of natural numbers, even numbers, numbers divisible by a given number ‘m’, fractions, rational and irrational numbers, etc. We define operations on these sets (including addition, multiplication, subtraction, and division) and study their properties. We introduce concepts of groups, rings, and fields, and the operations in this context. We introduce primality to study integers modulo given numbers and identify differences in these sets (for example, the existence of inverse elements, zero divisors, the existence of solutions of polynomial equations, etc.). The goal of his lesson is to improve students’ comprehension of numbers beyond the traditional scope of the algebra curriculum. Many of these sets and their properties are used to build cryptographic ciphers and to code and decode secret messages. Other applications include advanced number theory and advanced algebra problems, that are used not only in mathematics and computer science, but in chemistry and biology as well.

Good job, Dylan!

Congratulations, Dylan!

The lesson sounds so useful, nice project.

Great topic! It is true that many students do struggle holistically trying to understand the subsets of the real numbers. This is a very innovative way of introducing algebra to the students and eventually pique their interests in modular arithmetic, cryptography, and encryption.