We say that a set is countably infinite if we can pair the elements with set of counting numbers 1, 2, 3, and so on. Believe it or not, the number of positive integers and the number of integers (both negative and positive including 0) have the same number of elements. It is because we can pair them in a one-to-one correspondence such as shown in the below.
As shown on the table, if we continue indefinitely, we know that we can pair each counting number with an integer in a one-to-one correspondence without missing any element.
Using this concept, we show intuitively that the number of points on two line segments is equal even if they have different lengths. We can do this by showing that for each point on segment , there is a corresponding point on segment .
Given two line segments with different lengths, let us label the longer segment and the shorter segment . We place them in a horizontal position such that they do not overlap.
Next, connect the end points of the segment and extend them on the part of shorter segment to form a triangle. We call the point of intersection .
Notice that each ray drawn from that passes through segment will also pass through segment . That is, if ray passes through segment at , then it also passes through at . If a ray passes through at , then it also passes through at . Clearly, this is true for all rays that can be drawn.
Now since each point on has a corresponding point on , there is a one-to-one correspondence between the points in and the points on . As a result, the number of points on the two segments is equal.