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A line may have infinite points becauase it may be expanded.But in case of a line segment it has 2 distinct points which are not movable.The distance between the end points in finite and known.

But still why do people(in my school) say that there are infinite points in a line segment.When I ask the teacher,she says you will learn at higher levels(what a genius way to get rid of question).

Rahul Shah
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Let's assume that your end points are A(0,0) and B(1,0).

Try to find a point in between them. Ok you found C(0.4,0).

Now try finding another point. Will you ever NOT be able to find another point? No, because there are infinite numbers between 0 and 1 (just think of decimals).

In the same way there are infinite points in a line segment.

Thanos Darkadakis
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Any segment with at least two points has infinitely many points, because, intuitively, given any two distinct points, there is a third one, distinct from both of them, say, the middle point.

If there were a finite number of points, there will be no point between say point 6 and point 7 (etc)

rewritten
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The concept of infinity is used for many different things, and one should not confuse them. A line, be it closed, open, straight, curved, finite in length, or infinite in length always consists of infinitely many individual points (at least for any reasonable notion of 'line'). This can be proven rigorously, and it's not hard at all. Basically, between any two distinct points on a line there is a third point between these two points (between should not necessarily mean the mid-point on a straight path connecting the points, and this may get a bit tricky, but not too tricky). The length of the line is a different issue. It may be finite in length or infinite in length. The totality of the points comprising the line is in any case infinite. In fact it can be shown (quite easily) that the cardinality of points of any line is always the same, so in a sense all lines have the same "amount" of points in them, though the way these points are arranged may give totally different geometric qualities to the line.

Ittay Weiss
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The idea that a line segment is “made up” of an infinite number of points is very common, but there is a conflict between the every day conception and what mathematicians mean, in their technical usage.

Why?

Because the word “point” varies between the common, everyday usage and the specialist mathematical usage.

There is a big difference between the physical parts that make up a line, and the points we can mark on that line.

Simply stated, a “point” means either:

  • common usage: a part of a thing; a concrete thing with physical extension.
  • mathematical usage: marks an exact location in space; an abstraction without physical extension.

The two concepts are very different, but mathematicians often forget that the common notion of “point” conflicts with their precise definition.

And thus, the mathematical explanation of points and line segments can seem confused and contradictory to the non-mathematician, who has a very different notion of what a “point” is.

Conflating the two throws forth a horde of confused notions, like the idea that all line segments (of different lengths) have the same number of parts.

How can any line segment be any different than any other, if they all have the same number of parts!?

Yes, you can mark an infinite number of points on a meter stick, in theory at least: 0.8… 0.87… 0.874… 0.875…

All these points could be marked on a given meter stick—in principle, if not practically.

But, these hypothetical marks do not constitute the meter stick itself.

The marks we make (on the meter stick) are not the “stuff” that make up a meter stick. They are a secondary thing, without substance, an artifact of measurement.

The “stuff” of the meter stick relates to what we call “geometric magnitude”, which means the “space” taken up by something.

A thing is “made up” of its parts, and a larger stick has a greater magnitude than a shorter one, since it has more parts (of the same size) than the smaller one.

That we can make an infinite number of marks on either stick is irrelevant to the number of parts they each contain.

When we mark a line, the marks we make describe the line—they do not “make up” the line.

Sorry for the late, late reply.

You asked an excellent question, but were dismissed by your teacher with a non-response.

You can read more on this topic here: https://reduct.blog/articles/infinite-angels-dancing-on-a-pinpoint/

(I haven't written anything on the use of infinitesimals on my blog yet, but intend to over the next few months—they have some incredible uses)

John Jordan
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