The reality of Complex numbers

Abstract

What is a complex number? The question may be well solved today, but about 400 years ago it would have baffled many mathematicians. For example, how was it possible to define the square root of a negative number, which was an "impossible" quantity at the time, but they were all real quantities in mathematics. Complex analysis is now recognized as a fundamental part of mathematics because of its many applications in physics and engineering as well as its connections to other branches of mathematics. By solving some equations of more than one degree, complex numbers were produced. Most mathematicians focused on real analysis and physics applications until the nineteenth century. Cauchy was the first figure to make a considerable effort in a challenging examination. He calculated a number of complex integrals using his integral formula, establishing the foundation for defining the operations and characteristics of complex analysis. Gauss furthered the development of the complex number geometrical theory. The first part of this essay focuses on the history discovery of mathematical and geometrical aspects of complex numbers. The topological features of the complex plane are then carefully understood after that. The beautiful characteristics of complex functions and series are finally compiled based on the topology of the complex field.