Real Number Line

Points on a straight line can be used to represent real numbers. This geometrical representation of real numbers as points along a straight line is sometimes very helpful. It provides a preliminary aid to understanding relationships among real numbers involved in certain cases of analysis. It readily suggests various ideas which may be followed for the necessary solution to a problem. In particular, in cases involving the consideration of real numbers very close to each other, it is of great help to a beginner.

Let $x’x$ be a straight line extending indefinitely on both sides. Mark two points $O$ and $A$ on it between $x’x$, such that $A$ is on the right of $O$, i.e. $A$ lies between $O$ and $x$. We now regard the part $Ox$ on the right of $O$ as positive and $Ox’$ on the left of $O$ as negative. Such a straight line for which positive and negative sides are fixed is called a directed straight line. As shown above, let the point $A$ be taken to represent the number $1$ and the point $O$ be taken to represent the number $0$. Since any rational number can be expressed as $m/n{\text{ }}(n \in \mathbb{N},m \in \mathbb{Z})$, we can divide $OA$ into $n$ equal parts and take $OP$ equal to $m$ such parts on the right or left of $O$ accordingly as m is positive or negative. In case $m = 0$, point $P$ is at $O$ itself.  The point $P$ thus obtained represents the rational number $m/n$ and we say that $m/n$ corresponds to the point $P$, or $P$ corresponds to the rational number $m/n$. Hence, we see that every rational number is represented by some point $P$ on the directed straight line $x’x$ with $OA$ as the unit of the scale.

However, there remain points on $x’x$ whose representation corresponds to no rational number. In particular, consider a point $Q$ on the right of $O$ such that $OQ$ is equal to the side of the equilateral triangle whose circumscribing circle is of radius $OA$. Then $OQ = OA\sqrt 3$. Since $\sqrt 3$ is not rational, no rational number corresponds to the point $Q$. We find that the set of rational numbers $\mathbb{Q}$ is insufficient to provide a complete picture of a straight line.