The Complex Numbers

  • Joseph Bak
  • Donald J. Newman
Part of the Undergraduate Texts in Mathematics book series (UTM)


Numbers of the form \( a + b\sqrt { - 1} \), where a and b are real numbers—what we call complex numbers.appeared as early as the 16th century. Cardan (1501–1576) worked with complex numbers in solving quadratic and cubic equations. In the 18th century, functions involving complex numberswere found by Euler to yield solutions to differential equations. As more manipulations involving complex numbers were tried, it became apparent that many problems in the theory of real-valued functions could be most easily solved using complex numbers and functions. For all their utility, however, complex numbers enjoyed a poor reputation and were not generally considered legitimate numbers until the middle of the 19th century. Descartes, for example, rejected complex roots of equations and coined the term “gimaginary” for such roots. Euler, too, felt that complex numbers “exist only in the imagination” and considered complex roots of an equation useful only in showing that the equation actually has no solutions.


Complex Number Complex Plane Cauchy Sequence Real Solution Stereographic Projection 
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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsCity College of New YorkNew YorkUSA

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