We started off with a wish list for calculating with an extension of the real numbers, in which the equation #z^2=-1# has a solution. The requirements consisted of preserving as many properties that we know about the real numbers.

Subsequently, we constructed the complex numbers by starting from the points in the flat plane. The sum is the familiar vector addition, but the multiplication was new. We saw that the multiplication can be described conveniently in terms of polar coordinates: the angles with the #x#-axis of the two complex numbers which are to be multiplied, are added, and the lengths are multiplied.

All kinds of rules of the real numbers were found to be working for complex numbers as well, and strong new facts came up, like: Every polynomial has a zero, and each complex number is a zero of a real quadratic polynomial.

The construction of complex numbers can be understood as a special case of a construction of algebraic systems, which is dealt with in abstract algebra courses. The formulation of the complex numbers using pairs of real numbers as presented in this chapter dates from 1833 and goes back to W.R. Hamilton (1805-1865). Hamilton knew to generalize this construction to an algebraic system with elements of the form #a+b\cdot{\rm i} +c\cdot{\rm j} +d\cdot{\rm k}# (with #a,b,c,d\in \mathbb{R}# ), in which #{\rm i}^2=-1#, #{\rm j}^2 =-1#, #{\rm k}^2=-1#, #{\ii }\cdot{\rm j}={\rm k}=-{\rm j}\cdot{\rm i}#, #{\rm j}\cdot{\rm k} =\ii=-{\rm k}\cdot {\rm j}#, #{\rm k}\cdot{\rm i}={\rm j}=-{\rm i}\cdot{\rm k}#; this is the algebraic system of the so-called quaternions.

For linear algebra we can use the complex numbers to solve all kinds of polynomial equations (which will appear in the following chapters). Calculating with polynomials will be discussed in more depth in the chapter Polynomial and rational functions.

The Fundamental theorem of algebra goes back to C.F. Gauss (1777-1855); There are several proofs of this theorem, some algebraic, some analytical.

It requires more algebraic techniques than we deal with here to prove that no explicit formulas exist to solve polynomial equations of degree 5 and higher.

The analysis of functions #f:\mathbb{C} \rightarrow \mathbb{C}# (that is to say, issues like differentiability and integrability) is part of the subject of Complex functions and Fourier theory.

The complex numbers are widely used in electrical engineering and mathematical physics.