chapter  5
17 Pages

Complex Numbers

WithGarrett Birkhoff, Saunders Mac Lane

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).