Field Axioms

Let R be the set of real numbers having at least two distinct elements equipped with two fundamental algebraic operations called addition and multiplications and denoted by ‘+’ and ‘.’ respectively.

A.1. The set R is closed with respect to addition i.e., a + b is a unique real number for any two real numbers a and b.

A.2. (a + b) + c =a + (b + c) ∀ a, b, c ∈ R Associativity

A.3. a + b = b + a ∀ a, b ∈ R Commutativity

A.4. There exists an element 0 in R : 0 + a = a ∀ a, b ∈ R

A.5. To each real number a there corresponds a real number, viz., -a, such that

a + (- a) = 0

M1.The set R is closed with respect to multiplication i.e., a.b is a unique real number for any two real numbers a and b.

M2. a.b = b.a ∀ a,b ∈ R (commutativity)

M3. (a.b).c =a.(b.c) ∀ a,b,c ∈ (associativity)

M4.There exists an element namely 1 ≠ 0 in R such that

1.a = a ∀ a ∈ R

M5. To each element a ≠ 0 in R there exists an element 1/a in R such that

(1/a) . a = 1 (1/a = a-1)

AM a(b + c) = ab + ac ∀ a ,b, c ∈ R. ( Distributive law)

Because of the above properties the algebraic structure (R, +, . ) is called a field. As a matter of fact any mathematical system satisfying the above axioms is called a field. Thus we may speak of the field Q of rational numbers or the field C of complex numbers.

Subtraction and Division in R

Def. The difference between two real numbers a and b is defined by a + (-b) and is denoted by a – b.

The operation of finding the difference is called subtraction.

Def. The quotient of a real number a by b real number b (b ≠ 0 ) is defined by a.b-1 , and is denoted by a/b.

The operation of finding quotient is called division.