CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / Este trabalho apresenta os nÃmeros complexos com um enfoque que julgamos ser adequado para os professores do ensino mÃdio. O objetivo do trabalho à fornecer mais um texto sobre o tema e auxiliar os professores do ensino mÃdio em suas aulas. Iniciamos o trabalho com uma definiÃÃo de nÃmeros complexos que contempla o rigor matemÃtico necessÃrio e busca manter a simplicidade exigida para esse nÃvel de ensino. Utilizamos a representaÃÃo geomÃtrica de um nÃmero complexo sempre que possÃvel para motivar e simplificar as definiÃÃes e demonstraÃÃes. Abordamos as fÃrmulas trigonomÃtrica e de Moivre ressaltando a sua importÃncia. Apresentamos a deduÃÃo da fÃrmula da raiz n-Ãsima de um nÃmero complexo . No penÃltimo capÃtulo, abordamos alguns assuntos que nÃo sÃo contemplados nos livros didÃticos de matemÃtica do ensino mÃdio que sÃo a fÃrmula de Euler, a qual grande parte das aplicaÃÃes dos nÃmeros complexos existe devido a essa grande descoberta. O logaritmo complexo, cuja teoria explica como se calcular logaritmos de nÃmeros negativos ou complexos e tambÃm tratamos sobre potÃncias complexas, de tal maneira a explicar como se calcular potÃncias de nÃmero quando a base e o expoente sÃo nÃmeros complexos. Finalmente, encerramos este trabalho fazendo uma anÃlise de alguns livros didÃticos do ensino mÃdio. / This paper presents the complex numbers with an approach that we think to be appropriated for high school teachers. The aim is to provide one more text on the subject and assist high school teachers in their classes. We started working with a definition of complex numbers which includes the mathematical rigor necessary and seeks to maintain the simplicity required for this level of education. We used the geometric representation of a complex number, wherever possible, to motivate and simplify the definitions and demonstrations. We discussed the trigonometric and Moivre formulas emphasizing their importance. We presented the deduction of the formula for n-th root of a complex number. On the penultimate chapter, we discussed some issues that are not covered in mathematics textbooks from high school such as Eulerâs formula, which the majority of applications of complex numbers exists because of this great discovery. The complex logarithm,whose theory explains how to calculate logarithms of negative or complex numbers and we also worked on complex powers, in such a way to explain how to calculate power number when the base and the exponent are complex numbers. Finally, we concluded this paper by analyzing some high school textbooks.

This thesis investigates the formal aspects of the bath-gymnasium complex plan type which was developed in Asia Minor during the Roman era in relation to the development of the the imperial thermae in Rome, the capital city of the Roman Empire. Close resemblances in the architectural configuration of bath-gymnasium complexes and imperial thermae are analyzed in order to provide complementary insight concernin the evolution of Roman bath architecture and bathing tradition. The comparative investigation of the formal aspects of the plan types reveals the contribution of Asia Minor and its role in influencing the architectural developments in the capital. The thesis concentrates on the development of the bath-gymnasium complex plan type in Asia Minor and the imperial thermae in Rome in order to elucidate the outcomes of mutual influence in criss-crossing Greek and Italic features. Crucial to this investigation ,s the understanding of the multiple effects of historical processes such as Hellenization, Romanization and urbanization that were synthesized in the bath architecture of the capital and the provinces. Also, the symbolic, cosmological, and political aspects of Roman bath architecture are highlighted ,n this thesis.