Complex Numbers

Abstract

The quadratic equation x ²+ 1 = 0 has no solutions in the field ℝ of real numbers, because every sum of squares r ² + 1 with r ∈ ℝ is positive. A new epoch in the mathematics of modern times was inaugurated by the recognition that this incompleteness of the real number system could be obviated by yet another simple extension of the number domain, the extension of ℝ to the field ℂ of complex numbers.

Ex irrationalibus oriuntur quantitates impossibiles seu imaginariae, quarum mira est natura, et tarnen non contemnenda utilitas (Leibniz).

[From the irrationals are born the impossible or imaginary quantities whose nature is very strange but whose usefulness is not to be despised.]

I am indented to the Volkswagen Foundation for the award of research grant during the academic year 1980/81, as a result of which the work on Chapter3, 4 and 5 of this book was very considerably facilitated.