ABSTRACT
The complex numbers appear at the end of a hierarchy (Sections 1.1 and 1.2) formed by the positive integers, integers, rationals, irrationals, and real numbers; the complex numbers are the simplest for which all three direct (sum, product, power) and inverse (subtraction, division, root) operations are closed, that is, when applied to a complex number the result is also a complex number. A complex number is an ordered pair of real numbers, which can be represented as a point on the plane (Sections 1.3-1.6); this provides a graphical illustration of some properties of elementary real and complex functions (Sections 1.7 and 1.8). A quaternion (Section 1.9) is a generalization of complex number in four dimensions for which the product is noncommutative. Further generalizations of the concept of number (transfinite, hypercomplex) may have less properties. The operations like sum, subtraction, product, division, power, and root may be applied not only to numbers but also to other entities (functions, multiplicities, sets, rings) as long as they retain similar properties.
