Axioms for Real Numbers

Axioms for Real Numbers

The axioms for real numbers are classified under:

(1) Extend Axiom
(2) Field Axiom
(3) Order Axiom
(4) Completeness Axiom

Extend Axiom

This axiom states that \mathbb{R} has at least two distinct members. We shall be using this axiom quite frequently without making any specific reference to it.

Field Axiom

Real numbers are combined by means of two fundamental operations which are well known as addition and multiplication. The axioms these operations obey are given below as the laws of computation.

Addition Axioms

Closure Law: The set \mathbb{R} is closed under addition operation. This means the sum or addition of any two real numbers, i.e. a,b \in \mathbb{R} \Rightarrow a + b \in \mathbb{R}.

Associative Law: Addition operations in \mathbb{R} is associative. a,b,c \in \mathbb{R}  \Rightarrow \left( {a + b} \right) + c = a + \left( {b + c} \right).

Existence of additive identity: There is a real number 0(zero) such that a + 0 = 0 + a = a, \forall a \in \mathbb{R}.

Existence of additive inverse:  Corresponding to each a \in \mathbb{R} there exists a real number b such that a + b = b + a = 0.
Additive inverses are most commonly known as negative. The real number b above is called the negative of a and written as - a. Since 0 + 0 = 0, therefore 0 is the negative of itself, i.e. 0 = - 0.

Commutative Law: Addition operations in \mathbb{R} is commutative, i.e. a,b \in \mathbb{R}  \Rightarrow a + b = b + a

Multiplication Axioms

Closure Law: The set \mathbb{R} is closed under multiplication operations. This means that the multiplication of any two real numbers, i.e. a,b \in \mathbb{R} \Rightarrow ab \in \mathbb{R}.

Associative Law: Multiplication operations in \mathbb{R} is associative. a,b,c \in \mathbb{R}  \Rightarrow \left( {ab} \right)c = a\left( {bc} \right).

Existence of multiplicative identity: There is a real number 1(one) such that a \cdot 1 = 1 \cdot a = a, \forall a \in \mathbb{R}.

Existence of multiplicative inverse:  Corresponding to each a \in \mathbb{R} there exists a real number b such that ab = ba = 1.
Multiplicative inverses are most commonly known as inverses. The real number b above is called the inverse or reciprocal of a and written as 1/a or{a^{ - 1}}etc. Since 1 \cdot 1 = 1, therefore 1 is the negative of itself, i.e. {1^{ - 1}} = 1.

Commutative Law: Multiplication operations in \mathbb{R} is commutative, i.e. a,b \in \mathbb{R}  \Rightarrow ab = ba

Distributive Laws: It states that multiplication is distributive over the addition operation,
i.e. a,b,c \in \mathbb{R} \Rightarrow a\left( {b + c} \right) = ab + ac (Right Distributive Law)
i.e. a,b,c \in \mathbb{R} \Rightarrow \left( {b + c} \right)a = ba + ca (Left Distributive Law)

In view of addition axioms, multiplication axioms and distributive laws of the set of real numbers \mathbb{R} is called a field. The set of rational numbers \mathbb{Q} is also a field.