Números complexos e aplicações

Abstract

This work presents a systematic study of complex numbers, ranging from their algebraic construction to some of their practical applications. It begins with formal definitions, properties, and geometric and trigonometric representations, emphasizing the role of complex numbers in extending the real number system. Subsequently, applications in engineering are discussed, particularly in the analysis of alternating current electrical circuits through phasor theory, which illustrates the importance of complex numbers in modeling oscillatory phenomena. In the context of trigonometry, the study demonstrates how the polar representation of complex numbers enables the proof of fundamental identities, such as addition formulas, while also providing an elegant method for calculating powers and roots. Finally, Cardano’s formulas for cubic and quartic equations are presented and examined. The work concludes that complex numbers, beyond extending the numerical structure of mathematics, possess an interdisciplinary character that bridges theory, practical applications, and the history of mathematics, thereby consolidating their role as an indispensable tool in scientific and technological development.