ABSTRACT
The real number system consists of both the rational numbers (numbers with terminating or repeating decimal expansions) and the irrational numbers (numbers with infinite, non-repeating decimal expansions). Historically the most important fact about the complex numbers is that they provide negative numbers with square roots. More generally, the complex numbers provide any polynomial equation with roots. Addition of complex numbers corresponds exactly to addition of vectors in the plane. Complex multiplication does not correspond to any standard vector operation. Indeed it cannot. For the standard vector, dot product has no concept of multiplicative inverse; and the standard vector cross product has no concept of multiplicative inverse. But one of the main points of the complex number operations is that they turn the number system into a field: every nonzero number does indeed have a multiplicative inverse. This is a very special property of two-dimensional space.
