Collections of mathematical objects (numbers) that can be operated on by some or all of the standard operations of arithmetic: addition, multiplication, subtraction, and division. Such systems have a variety of technical names (e.g., group, ring, field) that are not employed here. This article shall, however, indicate which operations are applicable in the main systems of interest. These main number systems are:

a. The natural numbers N. These numbers are the positive (and zero) whole numbers 0, 1, 2, 3, 4, 5, …. If two such numbers are added or multiplied, the result is again a natural number.

b. The integers Z. These numbers are the positive and negative whole numbers …, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …. If two such numbers are added, subtracted, or multiplied, the result is again an integer.c. The rational numbers Q. These numbers are the positive and negative fractions p/q where p and q are integers and q ? 0. If two such numbers are added, subtracted, multiplied, or divided (except by 0), the result is again a rational number.

d. The real numbers R. These numbers are the positive and negative infinite decimals (including terminating decimals that can be considered as having an infinite sequence of zeros on the end). If two such numbers are added, subtracted, multiplied, or divided (except by 0), the result is again a real number.

e. The complex numbers C. These numbers are of the form x + iy where x and y are real numbers and i = v(-1) . (For further explanation, see the section Complex analysis.) If two such numbers are added, subtracted, multiplied, or divided (except by 0), the result is again a complex number.