 The set of integers is not closed with respect to division. 1 divided by 2 is not in the set of integers. To include these, and make the set close with respect to division, we need to add all the rational numbers. The numbers that can be expressed as a ratio of integers, or the denominator is not zero. To map these numbers to the number line, we simply divide each segment into the number of sub-segments indicated by the denominator. For example, here's 9 fourths. The line that contains all integers and rational numbers and zero is known as the rational number line. It has all the properties of the integer number line and is closed for all the basic operations. It not only contains all rational numbers, it provides for ordering. One number is greater than another if its coordinate on the line is to the right. It is less than another if its coordinate is to the left and it is equal to the other if its coordinate is at the same location. With the rational number line in hand, we can now define distance on the line. If p and q are rational points on the line with coordinates x and y respectively, such that x is less than or equal to y, then the distance between p and q is y minus x. It is straightforward to find the midpoint between any two rational points p and q. We'll call it m with a coordinate equal to t. The distance to t would be the distance to p plus half the distance between p and q. That would be y minus x divided by 2. We see that the coordinate of the midpoint of two points on the number line is half the sum of the given points. Consequently, there is always a rational point between any two rational points on the line because we can divide by 2 without limit. It follows that there are infinitely many rational numbers between any two given rational numbers no matter how close together they are. We say the rational number line is dense. An important consequence of the set of rational numbers being dense is that the length of any segment can be approximated to any degree of accuracy by a rational number. And there is one more important point to make about this line. If we shift the origin, every coordinate on the line will change. For example, if we shift it to the right two units, the coordinates on the new line will be different from the originals by two units. But there is one thing that does not change when we change the coordinates. And that is the length of any line segment. It is said to be invariant with respect to coordinate transformations. You can see how the shift in the values of the coordinates cancel out in the length calculation. Before we leave the number line, there are two more considerations we need to examine. One, there are missing points on the rational number line. And two, the number zero has some unique and relevant characteristics.