Axioms for Real Numbers

Axioms for Real Numbers:

The axioms for real numbers are classified as 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 that 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 operation 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 exist a real number b such that a + b = b + a = 0
Additive inverses are most commonly known as negative. The real numbers b above is called the negative of a and written as - a. Since0 + 0 = 0, therefore 0 is the negative of itself, i.e. 0 = - 0.

Commutative Law: Addition operation 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 operation. This means that the multiply of any two real numbers, i.e. a,b \in \mathbb{R} \Rightarrow ab \in \mathbb{R}.

Associative Law: Multiplication operation 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 exist a real numberb such that ab = ba = 1
Multiplicative inverses are most commonly known as inverses. The real numbers b above is called the inverse or reciprocal of a and written as1/a or{a^{  - 1}}etc. Since1 \cdot 1 = 1, therefore 1is the negative of itself, i.e. {1^{ - 1}} = 1.

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

Distributive Laws: It states that multiplication is distributive over 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 the set of real numbers \mathbb{R} is called a Field. The set of rational numbers \mathbb{Q} is also a field.

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