Complex Numbers

  • Mariano Giaquinta
  • Giuseppe Modica


As already stated, the process of formation of numerical systems has been very slow. For instance, while Heron of Alexandria (IAD) and Archimedes of Syracuse (287BC–212BC) essentially accepted irrational numbers, working with their approximations, Diophantus of Alexandria (200–284) thought that equations with no integer solutions were not solvable; and only in the fifteenth century were negative numbers accepted as solutions of algebraic equations.1 In the sixteenth century complex numbers enter the scene, with Girolamo Cardano (1501–1576) and Rafael Bombelli (1526–1573), in the resolution of algebraic equations as surdes numbers, that is numbers which are convenient to use in order to achieve correct real number solutions. But René Descartes (1596–1650) rejected complex roots and coined the term imaginary for these numbers. Despite the fact that complex numbers were fruitfully used by Jacob Bernoulli (1654–1705) and Leonhard Euler (1707–1783) to integrate rational functions and that several complex functions had been introduced, such as the complex logarithm by Leonhard Euler (1707–1783), complex numbers were accepted only after Carl Friedrich Gauss (1777–1855) gave a convincing geometric interpretation of them and proved the fundamental theorem of algebra (following previous researches by Leonhard Euler (1707–1783), Jean d’Alembert (1717–1783) and Joseph-Louis Lagrange (1736–1813)). Finally, in 1837 William R. Hamilton (1805–1865) introduced a formal definition of the system of complex numbers, which is essentially the one in use, giving up the mysterious imaginary unit \(\sqrt { - 1} \). Meanwhile complex functions reveal their importance in treating the equations of hydrodynamics and electromagnetism, and, in the eighteenth century develop into the theory of functions of complex variables with Augustin-Louis Cauchy (1789–1857), Karl Weierstrass (1815–1897) and G. F. Bernhard Riemann (1826–1866).


Complex Number Equilateral Triangle Complex Exponential Complex Notation Elementary Application 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Mariano Giaquinta
    • 1
  • Giuseppe Modica
    • 2
  1. 1.Dipartimento di MatematicaScuola Normale SuperiorePisaItaly
  2. 2.Dipartimento di Matematica ApplicataUniversità degli Studi di FirenzeFirenzeItaly

Personalised recommendations