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 be a straight line extending indefinitely on both sides. Mark two points and on it between , such that is on the right of , i.e. lies between and . We now regard the part on the right of as positive and on the left of 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 be taken to represent the number and the point be taken to represent the number . Since any rational number can be expressed as , we can divide into equal parts and take equal to such parts on the right or left of accordingly as m is positive or negative. In case , point is at itself. The point thus obtained represents the rational number and we say that corresponds to the point , or corresponds to the rational number . Hence, we see that every rational number is represented by some point on the directed straight line with as the unit of the scale.
However, there remain points on whose representation corresponds to no rational number. In particular, consider a point on the right of such that is equal to the side of the equilateral triangle whose circumscribing circle is of radius . Then . Since is not rational, no rational number corresponds to the point . We find that the set of rational numbers is insufficient to provide a complete picture of a straight line.