commutative ring

complex numbers

imaginary unity

tributive properties:

other operations, namely, the difference u - v = u + (-v) and the quotient

u/v = u(v

Similarly, the integer numbers Z = . . . , -2, -1, 0, 1, . . . do not form a field

because, in general, they do not have an inverse. On the other hand, the rational

numbers Q, which are given by ratios of integers, do satisfy all the properties

of a field.

numbers can be obtained as the union of the set of rational numbers with the

set of transcendental numbers, i.e., those numbers that can not be expressed as

a ratio of integers. An example of transcendental number is , which is the ratio

between the circumference and the diameter of any circle. The real numbers do

form a field, and the rationals are a subfield of the reals.

S14, P1 e D12 are satisfied is called a ring. If P2 is satisfied we have a com-

mutative ring, and if P3 is satisfied the ring has a unity. For instance, the set Z

of integer numbers forms a commutative ring with a unity.

system, where the natural numbers are contained in the integers, which are

part of the rationals, and this latter class in contained in the real numbers.

This hierarchy is resemblant of the temporal evolution of the classes of numbers

since the antiquity to the XVI century. The extension of the hierarchy was always

motivated by the ease with which practical and formal problems could be solved

by manipulation of numerical symbols. The same kind of motivation led to the

introduction of the class of complex numbers. As we will see in sec. A.3), they

come into play when one wants to represent the solutions of a second-order

equation.

fore,