__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 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 is closed under addition operation. This means that the sum or addition of any two real numbers, i.e. .

**Associative Law: **Addition operation in is associative. .

**Existence of additive identity: **There is a real number (zero) such that , .

**Existence of additive inverse: **Corresponding to each there exist a real number such that

Additive inverses are most commonly known as negative. The real numbers above is called the negative of and written as. Since, therefore is the negative of itself, i.e. .

**Commutative Law: **Addition operation in is commutative, i.e.

__Multiplication Axioms__:

**Closure Law: **The set is closed under multiplication operation. This means that the multiply of any two real numbers, i.e. .

**Associative Law: **Multiplication operation in is associative. .

**Existence of multiplicative identity: **There is a real number (one) such that , .

**Existence of multiplicative inverse: **Corresponding to each there exist a real number such that

Multiplicative inverses are most commonly known as inverses. The real numbers above is called the inverse or reciprocal of and written as oretc. Since, therefore is the negative of itself, i.e. .

**Commutative Law: **Multiplication operation in is commutative, i.e.

**Distributive Laws: **It states that multiplication is distributive over addition operation,

i.e. (Right Distributive Law)

i.e. (Left Distributive Law)

In view of addition axioms, multiplication axioms and distributive laws the set of real numbers is called a **Field**. The set of rational numbers is also a field.