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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