Real numbers possess an ordering relation. This relation we denote by the symbol “” which is read as “greater than”. The axioms of order in  based on “” are:

1. If, then one and only one of the following in true .
2. If  and , then .
3. If  and , then .
4. If  and , then .

In view of the axioms above the field of real numbers is said to be ordered and is said to be an ordered field. The set of rational numbers is also an ordered field.

            The above axioms can easily be expressed in terms of the less than relation “” for .
            When we write  it means that either. Similar meaning holds for .

Positive and Negative Real Numbers: A real number is said to be positive or negative according as.

            We shall denote sets of positive real numbers by  and  respectively. Thus . Similarly symbols  and  shall be used to denote the sets of positive and negative rational numbers respectively. Two real numbers are said to be same sign if both of them either belong to, or. They are said to be of opposite signs if one belongs to  and the other to.
            Keeping with the usual convention, if  and  we shall simply write  and say that  lies between and. If, then  and, i.e. . Hence  implies that a lie between every two real numbers of opposite signs and every positive number is greater than every negative number.

Example: If, then, and if, then, and.