This is the cue for imaginary numbers to make their entrance. This topic is one that we shall take up again in the final chapter;for the time being, we willjust make some introductory comments.
This first struck home in the 16th century when Italian mathematicians learnt how to solve cubic and fourth-degree polynomial equations in a fashion that extended that used to solve quadratic equations. The Cardano method, as it came to be known, would of ten involve square roots of negatives even though the solutions to the equations eventually turned out to be positive integers. By stages from this point, the use of complex numbers,which are those of the form a+bi, where a and b are ordinary real numbers, was shown to facilitate a variety of mathematical calculations. For example, in the 18th century Euler revealed and exploited the stunning little equation eiπ=-1, which cannot fail to surprise anyone on their first encounter.