Does point have length?

Question

It is often described in books that a point contains neither volume, area, length, nor any other higher dimensional analogue.

But generally a point can be measured in very small prefixes like nm.

Generally the straight line refers to infinite number of points present

Hence if a single point does not have length then how can the straight line be formed and measured

Hence we can conclude that every point has length and it is a straight line

What do you think?

Answer 1

A point is a 1 dimensional object with no length.

A line is a distance between two points.

You can consider a line segment as made up an infinite number of points of infinitesimal size.

As you have noted, there is a bit of a paradox hidden in there.  How can you have points with no length making up a line segment?  Similarly, if I have to travel from point A to point B, how can I ever get there if I have to travel through an infinite number of waypoints?

You might want to read up on Zeno's paradox...

Answer 2

That is mind boggling :O

But I see where you're coming from.

If a point contains neither volume, area, length, nor any other higher dimensional analogue, then it mustn't exist? But it does, because you can see it. That means you can measure it, no? So I agree with you.

What is a mathematical point anyway? In what what does it exist in?

Answer 3

You are confusing a point in mathematics with a picture of a point.

Definitions in mathematics need to be useful for descriptions.  It is useful to say that a point has no dimension, and a line has length but not width, and a plane has length and width but no thickness, etc.  This means that points, lines, planes, etc. are not actual physical objects.  They are useful mathematical ideas.

Think of a Point as referring to a location.  You could put your foot on a location, but it would cover more than that location (Point).  If you put your finger on the location, it would still cover more points than the specific location.  Even your pencil would cover more than the specific location.  So a "point" made by a pencil is not the same as a Point.

It is somewhat similar to the color "blue."  When we speak of a blue sky, we are not referring to some particular, actual blue.  If you wanted to paint a blue sky, you would need some specific blue paint, which is different from "blue."

Answer 4

Such sophistry may be tolerated in philosophy, but has no place in mathematics.

Answer 5

No; a mathematical point, by definition, has no length.  That is precisely why we can say that a line segment (of any length) contains an infinite number of points.

It is not true that points "can be measured" at nm scales. That would not be a mathematical point.

Answer 6

Nope. When you measure any length of a line, no matter how small, you measure a distance from point A to point B on that line. You cannot measure a single point.

Any "length" or "distance" is defined as distance between two points. As soon as you can measure it, there will be start and end point, and an infinite number of points in between.

Now what confuses you is how can a sum of zeros amount to a non zero? Which is a sum of lengths of single points, adds up to a line.

The simple (actually, oversimplified) answer is that although each point has zero length, you have an infinite number of points.

Slightly more correct answer would be that this shouldn't be calculated as a simple sum of point lengths but as a limit of the sum of point lengths as their number approaches infinite. Look up some info about limits of a function on Wikipedia or in a good Math book.