field

zero

opposite

unity

inverse

Appendix A

Mathematical

Fundamentals

A.1

Classes of Numbers

A.1.1

Fields

Given a set F of numbers, two operations called sum and product over these

numbers, and some algebraic properties that we are going to enumerate, F is

called a field. The sum of two elements of the field u, v F is still an element

of the field and has the following properties:

S1, Associative Property : (u + v) + w = u + (v + w)

S2, Commutative Property : u + v = v + u

S3, Existence of the Zero : There exists one and only element in F, called

the zero, that is the neutral element for the sum, i.e., u + 0 = u , for all

u F

S4, Existence of the Opposite : For each u F there exists one and only

element in F, called the opposite of u, and written as -u, such that

u + (-u) = 0.

The product of two elements of the field u, v F is still an element of the field

and has the following properties:

P1, Associative Property : (uv)w = u(vw)

P2, Commutative Property : uv = vu

P3, Existence of the Unity : There exists one and only element in F, called

the unity, that is the neutral element for the product, i.e., u1 = u , for all

u F

P4, Existence of the Inverse : For each u F different from zero, there

exists one and only element in F, called the inverse of u, and written as

u

-1

, such that uu

-1

= 1.

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