Abstract
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 \(x\in \mathbb R\), 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.
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Notes
- 1.
The names complex and imaginary are rooted in the historical development of complex numbers. More precisely, when mathematicians started using complex numbers, even without having a precise definition of what they should be, they called them complex or imaginary, in allusion to the oddness of guessing the existence of “numbers” whose squares could be negative.
- 2.
Also called the Argand-Gauss plane, in honor of the amateur Swiss mathematician of the eighteenth century Jean-Robert Argand, and of the great J. C. F. Gauss.
- 3.
After William R. Hamilton , Irish astronomer, mathematician and physicist of the nineteenth century. Quaternions have many important applications in Mathematics and Physics, but unfortunately they lie far beyond the scope of these notes.
- 4.
Recall that this means we rotate u through an angle of α radians, in the counterclockwise sense if α > 0 and in the clockwise sense if α < 0.
- 5.
Cf. Klarner and Göbel [24]. Nevertheless, the proof we present here is different from the original one.
References
A. Caminha, An Excursion Through Elementary Mathematics I - Real Numbers and Functions (Springer, New York, 2017)
A. Caminha, An Excursion Through Elementary Mathematics II - Euclidean Geometry (Springer, New York, 2018)
D.A. Klarner, F. Göbel, Packing boxes with congruent figures. Indag. Math. 31, 465–472 (1969)
J.H. Van Lint, R.M. Wilson, Combinatorics (Cambridge University Press, Cambridge, 2001)
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Caminha Muniz Neto, A. (2018). Complex Numbers. In: An Excursion through Elementary Mathematics, Volume III. Problem Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-77977-5_13
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