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 \mathbb{R} based on “ > ” are:

  1. Ifa,b \in \mathbb{R}, then one and only one of the following in true a > b,{\text{ }}a = b,{\text{ }}b > a.
  2. If a,b,c \in \mathbb{R} and a > b,{\text{ }}b > c, then a > c.
  3. If a,b,c \in \mathbb{R} and a > b, then a + c > b + c.
  4. If a,b,c \in \mathbb{R} and a > b,{\text{ }}c > 0, then ac > bc.

In view of the axioms above the field of real numbers \mathbb{R} is said to be ordered and \mathbb{R} is said to be an ordered field. The set of rational numbers \mathbb{Q} is also an ordered field.

The above axioms can easily be expressed in terms of the less than relation “ < ” for a > b \Leftrightarrow b < a.

When we write a \geqslant b it means that either a > b,{\text{ or }}a = b. Similar meaning holds for a \leqslant b.

Positive and Negative Real Numbers: A real number a is said to be positive or negative according as a > 0,{\text{ or }}a < 0.

We shall denote sets of positive real numbers by {\mathbb{R}^ + } and {\mathbb{R}^ - } respectively. Thus \mathbb{R} = {\mathbb{R}^ - } \cup \left\{ 0 \right\} \cup {\mathbb{R}^ + }. Similarly symbols {\mathbb{Q}^ + } and {\mathbb{Q}^ - } 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 {\mathbb{R}^ + }, or {\mathbb{R}^ - }. They are said to be of opposite signs if one belongs to {\mathbb{R}^ + } and the other to {\mathbb{R}^ - }.

Keeping with the usual convention, if a > b and b > c we shall simply write a > b > c and say that b lies between a and c. If x \in {\mathbb{R}^ + },y \in {\mathbb{R}^ - }, then x > 0 and y < 0, i.e. 0 > y. Hence x < 0 < y implies that0 a lie between every two real numbers of opposite signs and every positive number is greater than every negative number.

Example: If a,b \in {\mathbb{R}^ + }, then\left( {a + b} \right),{\text{ }}ab \in {\mathbb{R}^ + }, and if a,b \in {\mathbb{R}^ - }, then \left( {a + b} \right) \in {\mathbb{R}^ - }, and ab \in {\mathbb{R}^ + }.