Complex numbers became more prominent in the 16th century, when closed formulas for the roots of cubic and quartic polynomials were discovered by Italian mathematicians. Imaginary numbers were defined in 1572 by Rafael Bombelli, an Italian mathematician who wrote an influential algebra text and made free use of both negative numbers and complex numbers. Rafael Bombelli lived from 1526 - 1572; he was born in Bologna, died in Rome, Italy. Bombelli used a method related to continued fractions to calculate square roots, and the lunar crater Bombelli is named after him. Bombelli solved equations, and he introduced +i and -i and described how they both worked in Algebra, thus defining imaginary numbers.
At the time, such numbers were thought not to exist, much as zero and the negative numbers were regarded by some as fictitious or useless. Many other mathematicians were slow to believe in imaginary numbers at first, including Descartes who wrote about them in his La Géométrie, where the term was meant to be derogatory. Although Descartes originally used the term imaginary number to mean what is currently meant by the term complex number, the term imaginary number today usually means a complex number with a real part equal to 0. Zero is the only number that is both real and imaginary.
Until Algebra 2, it's impossible to take the square root of a negative number. A new number was invented around the time of the Reformation that can be used to solve the above problem. Since no "real world" use can be applied to this number, other than solving certain complex equations. This number is called "i", and it's called imaginary since it wasn't real. "i" is stated as the square root of -1, so "i" is equal to negative 1 in mathematical terms. Since i is not considered to exist, a set of numbers called complex numbers is sought about that included the use of i.
Complex numbers are the extension of the real numbers obtained by adjoining an imaginary unit, i. Every complex number can be written in the form x + iy, where x and y are real numbers called the real part and the imaginary part of the complex number, respectively. Complex numbers have addition, subtraction, multiplication, and division operations which extend the corresponding operations on real numbers, although with a number of additional elegant and useful properties. Notably, negative real numbers can be obtained by squaring complex numbers. Complex numbers were first discovered by Cardan, who called them "fictitious", during his attempts to find solutions to cubic equations. The solution of a general cubic equation may require intermediate calculations containing the square roots of negative numbers, even when the final solutions are real numbers, a situation known as casus irreducibilis. This ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, it is always possible to find solutions to polynomial equations of degree one or higher.
Complex numbers are used in many different fields including applications in engineering, electromagnetism, quantum physics, applied mathematics, and chaos theory. When the underlying field of numbers for a mathematical construct is the field of complex numbers, the name usually reflects that fact; examples of the above are complex analysis, complex matrix, and complex polynomial.
