# 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 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 that$0$ 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 $ab \in {\mathbb{R}^ + }$.