Complex Numbers

A complex number is a number consisting of a real and imaginary part. It can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit with the property i^ 2 = −1 The complex numbers contain the ordinary real numbers, but extend them by adding in extra numbers and correspondingly expanding the understanding of addition and multiplication.

Complex numbers were first conceived and defined by the Italian mathematician Gerolamo Cardano, who called them "fictitious", during his attempts to find solutions to cubic equations.The solution of a general cubic equation in radicals (without trigonometric functions) 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, a solution exists to every polynomial equation of degree one or higher. Complex numbers thus form an algebraically closed field, where any polynomial equation has a root.

A complex number can be visually represented as a pair of numbers forming a vector on a diagram called an Argand diagram, representing the complex plane.

Operations

Complex numbers are added, subtracted, multiplied, and divided by formally applying the associative, commutative and distributive laws of algebra, together with the equation i^ 2 = −1:

Addition:

Subtraction:

Multiplication:


Division:

where c and d are not both zero. This is obtained by multiplying both the numerator and the denominator by the conjugate of the denominator c + di, which is (c − di).

Reference:
http://en.wikipedia.org/wiki/Complex_number