A complex number is a number which can be put in the form a + bi, where a and b are real numbers and i is called the imaginary unit, where i² = −1. In this expression, a is called the real part and b the imaginary part of the complex number. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number Failed to parse (Missing texvc executable; please see math/README to configure.): a+bi

Complex numbers are used in many scientific fields, including engineering, electromagnetism, quantum physics, and applied mathematics, such as chaos theory. Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers. He called them "fictitious", during his attempts to find solutions to cubic equations in the 16th century.

Complex numbers allow for solutions to certain equations that have no real solution: the equation

has no real solution, since the square of a real number is either 0 or positive. Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the imaginary unit i where Failed to parse (Missing texvc executable; please see math/README to configure.): i^2=-1 , so that solutions to equations like the preceding one can be found. In this case the solutions are −1 ± 3i. In fact not only quadratic equations, but all polynomial equations in a single variable can be solved using complex numbers.