Completion of the Real Numbers and Its Implications

The goal of this paper is to explore the completion of the set of real numbers and its overall importance as it pertains to teaching modern mathematics and real-world applications. Cauchy sequences and Dedekind cuts both have advantages and disadvantages when utilized to complete the reals. Both methods help to provide a stronger foundation to the real number system which is very significant to modern mathematics today. Given that the set of real numbers is a complete field, it has a number of implications. It is highly imperative that modern-day students of mathematics have a strong understanding of the properties and axiomatic consequences of this set. By the end of the paper, we hope to develop a clear and thorough comparison of each completion of the real numbers, as well as relate them to real-world applications and education.