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History on complex numbers. Between algebra and geometry. (Histoire des nombres complexes. Entre algèbre et géométrie.) (French) Zbl 1089.01001

CNRS Histoire des Sciences. Paris: CNRS Editions (ISBN 2-271-06128-8/pbk). 501 p. (2003).
The occurrence of complex numbers and their gradual introduction into mathematics was a very momentous process in the development of mathematics. The author presents a detailed profound history of this process which took place in the last four centuries. The book is divided into four chapters. The first chapter titled “Les nombres imaginaires comme problème” describes on the one hand the occurrence of the “impossible” numbers in the work of Cardano, Bombelli, Viète, Harriot, Girard, and Descartes during the 16th and 17th century and on the other hand problems like the formula of Moivre as well as the logarithms of complex numbers and the decomposition of rational functions into partial fractions. The second chapter presents the attempts to the representation of complex numbers by Wessel, Buée, Argand, Warren, and Mourrey. However the author explains not only the contributions of those scientists and tries to clarify the winding path of progress but he put this topic in the broader context of the relation between algebra and geometry and its changes. The third chapter encloses the brilliant contributions of Gauss and Cauchy to the foundation of the concept of complex number as well as the beginnings of the theory of function of these numbers, furthermore the Gaussian plane and from Cauchy the symbolic expressions, the algebraic analysis, and the algebraisation of the imaginaries. The fourth chapter is devoted to “William Rowan Hamilton et la généralisation algébraique” containing the English algebraical school, Hamilton’s theory of algebra as the science of pure time, the theory of algebraic couples, triplets and finally of the quaternions which the author characterized as the first flash of the modern algebra.
The author follows the development of the various concepts of complex number by the stages sketched above. He includes in his analysis also the contributions of non-mathematicians and reflects the impact of linguistic, philosophical, metaphysical, as well as socio-economic aspects. That leads to the fact that the reader does not get lost in the details but always keeps the broader context in mind. Since the occurrence and foundation of complex numbers was not a history of a linear progress, the author always endeavours to show the place of the various contributions in the interrelations of algebra and geometry, in symbolism and mathematical language. Many quotations allow the reader to reconsider if he agrees with the argumentation of the author and give him a lively picture of the concrete historical context. The book is completed by an extensive list of references, a name index and an index of important symbols. The latter one gives a translation of the original historical symbols ordered by their occurrence in the book into modern mathematical language.

MSC:

01-02 Research exposition (monographs, survey articles) pertaining to history and biography
01A40 History of mathematics in the 15th and 16th centuries, Renaissance
01A45 History of mathematics in the 17th century
01A50 History of mathematics in the 18th century
01A55 History of mathematics in the 19th century

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