Linear Algebra, 5th Edition

Appendix D Complex Numbers

For the purposes of algebra, the field of real numbers is not sufficient, for there are polynomials of nonzero degree with real coefficients that have no zeros in the field of real numbers (for example, $x^2+1$). It is often desirable to have a field in which any polynomial of nonzero degree with coefficients from that field has a zero in that field. It is possible to “enlarge” the field of real numbers to obtain such a field.

Definitions.

A complex number is an expression of the form $z=a+bi$, where a and b are real numbers called the real part and the imaginary part of z, respectively.

The sum and product of two complex numbers $z=a+bi$ and $w=c+di$ (where a, b, c, and d are real numbers) are defined, respectively, as follows: ...