Abstract:
All over the world, the teaching of Mathematics for Secondary Schools is conditioned by questions on "What Mathematics is due to be taught?", "How to teach Mathematics?" or "Should Mathematics be useful? In what sense?"
In this research study we refer, in the Introduction and Chapter I, that the concept of number is one of the most important in Mathematics that runs from the basic levels of Mathematics teaching to University instruction. It is during this path that we naturally find the Complex Numbers: usually as a generalization process beyond real numbers. We agreed, in particular, that Complex Numbers may be particularly important in relation to the questions that were posed in the last paragraph.
Chapter II deals with the national curricula for Secondary Schools on the teaching of Complex Numbers through the past 50 years and we also analyzed questions on exams related to Complex Numbers.
In Chapter III we studied the History of Complex Numbers and we came to acknowledge the reasons, both utilitarian and philosophical, related to its evolution. We point out to the doubts and the difficulties felt by the authors who built, bit by bit, the concept of Complex Number without whom the Theory of Analytical Functions would never be reached in 1825. We also recognized the pedagogical importance of acknowledging errors and faults committed by well known mathematicians.
Chapter IV offers an evaluation grid for schooltexts, both for existence and logical coherence, on teaching Complex Numbers in Portuguese schools.
In Chapter V we approach the use of History of Mathematics on teaching Mathematics by suggesting some school activities for teaching Complex Numbers.
The aim of the conducted research on the actual teaching of Complex Numbers was the clarification, through a didactical-historical systematic study, of the various implications on the richness of accumulated knowledge by centuries of History. We finally concluded that the theme of Complex Numbers is perfectly justified in the National Curricula for Secondary Schools because:
 -it refers both to abstract thinking and intuitive capacities;
 -it organizes and it relates previous knowledge to actual mathematical knowledge, by involving pupils on discovery activities;
 -it allows both teachers and pupils to deal with fundamental questions on "What is a number?", "What are numbers good for?", "Who invented numbers?" or "How were numbers invented?"

Os números imaginários: (um estudo sobre) a sua "realidade" (pdf) »»