# Order Axioms for Real Numbers

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:

- If , then one and only one of the following is true .
- If and , then .
- If and , then .
- 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 . A similar meaning holds for .

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

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 the same sign if both of them either belongs 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 lies 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 .