Complex numbers and applications in proving real identities

Abstract

The rapid development of mathematics is largely due to the creation of complex numbers. Although complex numbers are just strange and seemingly meaningless notations for most people, complex numbers play important roles in engineering fields. A better understanding of complex numbers' geometric and algebraic structure enables us to study complex analysis better. Complex numbers convince scientists that our world is magical, full of wonderful insights, and even miraculous. In this paper, I first review several basic properties of complex numbers. The set of complex numbers is a group under the operation of the multiplication and under the operation of addition. Then I visualize complex numbers by building a bijective connection between complex numbers and points on the complex plane. I also give several alternative forms of complex numbers, namely the trigonometric and general forms. By invoking the arithmetic properties of complex numbers, I prove two trigonometric identities.