114
D. Rocchesso: Sound Processing
ring
commutative ring
complex numbers
imaginary unity
The two operations of sum and product are jointly characterized by the dis-
tributive properties:
D1, Distributive Property : u(v + w) = uv + uw
D2, Distributive Property : (v + w)u = vu + wu
The existence of the opposite and the reciprocal implies the existence of two
other operations, namely, the difference u - v = u + (-v) and the quotient
u/v = u(v
-1
).
Given the properties of a field, we can say that the natural numbers N =
0, 1, . . . do not form a field since, for instance, they do not have an opposite.
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.
The real numbers R are all those numbers that can be expressed in decimal
notation as x.y, where the number of digits of y is not necessarily bounded. Real
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.
A.1.2
Rings
A set of numbers provided with sum and product, and such that the properties
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.
Whenever we want to indicate the sets of ordered couples or triples of el-
ements belonging to a field (or a ring) F we will use the notation F
2
or F
3
,
respectively.
A.1.3
Complex Numbers
The classes of numbers introduced so far are instrumental to a hierarchical
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.
In order to define the complex numbers, we have to define the imaginary
unity i as that number that multiplied by itself (i.e., squared), gives -1. There-
fore,
i
2
= ii = -1 .
(1)
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