## Complex Numbers

Especially in algebra, but also in the theory of analytic functions and differential equations, many algebraic theorems have much simpler statements if one extends the real number system R to a larger field C of "complex" numbers. This chapter shows that it is what one gets from the real field if one desires to make every polynomial equation have a root. There is a fundamental one-one mapping of the complex numbers onto the points of a Cartesian plane. Namely, each complex number z = x + iy is mapped onto the point P = (x, y) with the real component x of z as abscissa and the imaginary component y as ordinate. Polar coordinates may be used in this plane. Conjugate complex numbers are very useful in mathematics and physics (especially in wave mechanics).