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.