THE SET OF COMPLEX NUMBERS: EXPANDING THE REALM OF MATHEMATICS

Abstract

This article explores the set of complex numbers, defined as \( z =
a + bi \), where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit
with \( i^2 = -1 \). Complex numbers extend the real number system, enabling the
solution of equations that have no real solutions. The paper discusses fundamental
properties of complex numbers, including their arithmetic operations, geometric
representation on the complex plane, and the concept of magnitude and argument.
Historical development, from early uses by mathematicians such as Gerolamo
Cardano and Carl Friedrich Gauss to modern applications in engineering, physics,
and computer science, is reviewed. The article highlights the importance of complex
numbers in various fields, emphasizing their role in solving polynomial equations,
analyzing electrical circuits, and modeling physical systems. By examining these
aspects, the paper underscores the integral role of complex numbers in advancing
mathematical and scientific understanding