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.