Números complexos: história, estrutura algébrica e presença em vestibulares da UFPB (1965-2012)

Abstract

This final project investigates the origin, formalization, and applications of Complex Numbers, emphasizing their historical, algebraic, and pedagogical relevance. We begin by presenting their historical trajectory, from the contributions of Cardano and Bombelli to the consolidation achieved by Euler and Gauss. We then highlight the transformation of Complex Numbers into a legitimate mathematical object. We then discuss the algebraic structure of Complex Numbers, highlighting their field properties and the different forms of representation (algebraic and trigonometric). In the educational context, we conducted a documentary analysis of 87 entrance exams for the Universidade Federal da Paraíba (UFPB), applied between 1965 and 2012, of which 25 questions involving complex numbers were identified. All items were solved and commented on, allowing us to observe the evolution of the approach to the subject over these decades. We found that in the early years, the content was limited to quadratic equations, but from 1980 onward, it expanded to include algebraic operations, properties of conjugates and modules, geometric representations, and applications in progressions and higher-degree polynomials. The results indicate that Complex Numbers, although absent from the Base Nacional Comum Curricular (BNCC), constitute a landmark in the history of Mathematics, as they have been shown to have significant value, from a historical and algebraic perspective, in the teachinglearning process and in exam assessment. Thus, the study reinforces the importance of being included again in the high school curriculum and provides support material for teachers and students, combining theoretical rigor and pedagogical practice.