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. If $$a,b \in \mathbb{R}$$, then one and only one of the following is 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 $$a > b,{\text{ or }}a = b$$. A similar meaning holds for $$a \leqslant b$$.

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

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 the same sign if both of them either belongs 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 that $$0$$ lies 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 $$ab \in {\mathbb{R}^ + }$$.