Complex Numbers and Functions

  • Nakhlé H. Asmar
  • Loukas GrafakosEmail author
Part of the Undergraduate Texts in Mathematics book series (UTM)


This chapter starts with the early discovery of complex numbers and their role in solving algebraic equations. Complex numbers have the algebraic form \(x+i\, y\), where xy are real numbers, but they can also be geometrically represented as vectors (xy) in the plane. Both representations have important advantages; the first one is well-suited for algebraic manipulations while the second provides significant geometric intuition. There is also a natural notion of distance between complex numbers that satisfies the familiar triangle inequality. Complex numbers also have a polar form \((r,\,\theta )\) based on their distance r to the origin and angle \(\theta \) from the positive real semi-axis. This alternative representation provides additional insight, both algebraic and geometric, and this is explicitly manifested even in simple operations, such as multiplication and division.

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA

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