6  Complex Numbers, Cubic Equations, and SixteenthCentury Italy pp. 3944By Daniel J. Curtin
Mathematical Time Capsules
MAA Notes
(No. 77)
Online ISBN: 9780883859841
Chapter DOI: http://dx.doi.org/10.5948/UPO9780883859841.007
Subjects: Recreational mathematics, History of science and technology 

Image View Text View  Enlarge Image ‹ Previous Chapter ›Next Chapter
Introduction
The complex numbers are important in modern mathematics and science, yet they receive almost no attention in the modern curriculum, which is heavily weighted towards preparation for the Calculus. Most precalculus treatments of the complex numbers give no insight into where they came from. They are mainly seen as supplying a full set of roots for polynomials that do not have all real roots. In fact, they first arose because they were needed to find real roots for cubic equations, precisely in the case where all three roots are real. The material in this article can be used anywhere complex numbers are introduced. It mixes geometric and algebraic ideas, in a way that should be particularly useful in a functional approach to precalculus. Technology can be used or not, as seems appropriate to the instructor.
Historical Background
The idea of complex numbers, at least in the sense of calculations involving square roots of negative numbers, first arose in Italy around 1540, as mathematicians solved cubic and quartic equations. Inmodern courses, complex numbers often first arise to discuss solutions of quadratic equations where ax2 + bx + c = 0 has solutions
and the question of what happens when b2 − 4ac is a negative number arises. Solutions to quadratic problems, analogous to the quadratic formula, have been known since at least 2000 BC, yet no one appears to have shown any interest in the possibility of negatives under the square root. Most likely this is because they only occur when both roots are complex, so from the point of view of real numbers the equation has no solutions.