An Excursion through Elementary Mathematics, Volume III by Antonio Caminha Muniz Neto

Author:Antonio Caminha Muniz Neto
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

© Springer International Publishing AG, part of Springer Nature 2018

Antonio Caminha Muniz NetoAn Excursion through Elementary Mathematics, Volume IIIProblem Books in Mathematicshttps://doi.org/10.1007/978-3-319-77977-5_13

13. Complex Numbers

Antonio Caminha Muniz Neto1

(1)Universidade Federal do Ceará, Fortaleza, Ceará, Brazil

It is an obvious fact that the set of reals is too small to provide a complete description of the set of roots of polynomial functions; for instance, the function x↦x 2 + 1, with , does not have any real root. Historically, the search for such roots strongly motivated the birth of complex numbers and the flowering of complex function theory. In this respect, a major first crowning was the proof, by Gauss, of the famous Fundamental Theorem of Algebra, which asserts that every polynomial function with complex coefficients has a complex root.

In this chapter, we concentrate ourselves in the construction and discussion of the elementary properties of the set of complex numbers, postponing to Sect. 15.​1 the presentation of one simple proof of the aforementioned theorem of Gauss.

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