Question

State whether the following statement is true or false:
A line segment is made of an infinite (uncountable) number of points.
(a) True
(b) False

Answer 1

Hint: In the above problem, we have to comment upon the statement “A line segment is made of an infinite (uncountable) number of points”. We know that a line segment is of a finite length but let's say the one end of the line segment is at 0 and the other end of the line segment is at 4. Then we can find a number 1 between them and between 0 and 1 we can find a lot of decimal numbers. By taking into consideration these explanations we can comment on the statement.

Complete step-by-step answer:
We know that a line segment is a segment of an infinite length line. Segmenting of a line means we cut a definite length from that infinite line.

As you can see that the infinite line has no boundaries whereas the line segment AB has boundaries which are point A and point B.
Now, let us suppose that this line segment lies on the x axis and the point A is at (0, 0) and point B at (4, 0).

In the above diagram, as you can see that between A and B there is a point (2, 0) lies which we have marked in the below diagram as C.

If we zoom the above graph, then we will find that between A and C there lies a point D which we have shown below:

If you see the above figure, you will find that point D (1, 0) lies between A and C. Similarly, you can find a point between A and D also.

In the above figure, you can see a point E (0.5, 0) that lies between the points A and D.
If we zoom in further between A and E then we again find some decimal number so there are an infinite number of points possible between the points A and B or any line segment.
Hence, the correct option is true.

Note: You might have thought that a line segment is of a finite length so it is made up of a finite number of points which is wrong. We can draw any equation of line from two points but that doesn’t mean we can find all the points on the line. Similarly, if we know the end points of a line segment we cannot tell the total number of points contained in that line segment.