Complex_numbers

Complex numbers are a natural extension of the real numbers. Firstly, we define the imaginary number i to be a root of the equation x²+1=0. We may then form complex numbers by using the form a + i b, where a and b are real numbers.

Formally we may also define complex numbers as an ordered pair of real numbers (a,b) along with the operations:

(a,b) + (c,d) = (a+c,b+d)
(a,b) * (c,d) = (ac - bd, bc + ad)

So defined, the complex numbers form a FielD.

A most striking result is that all polynomials of degree "n" have exactly "n" complex roots (counting multiple roots according to their multiplicity). This is known as the Fundamental Theorem of Algebra.

The study of functions of complex variables is known as complex analysis and has an enormous practical use in applied mathematics.