What is the Fraction of One Point Compared to an Infinite Line?

[CrossboneSRB]

So here's a question for you, if on a line we chose one point, then what is the fraction of this point compared to the line?
I think we all agree that between two points there are infinite amount of other points, correct? So what fraction of a line is one point among infinite others? Isn't it 1/∞?

Here's another question, if a perfectly round object is placed on perfectly flat surface, what is the area of contact?

 

[Mark44]

So here's a question for you, if on a line we chose one point, then what is the fraction of this point compared to the line?

0. The "length" of a single point is 0.

I think we all agree that between two points there are infinite amount of other points, correct? So what fraction of a line is one point among infinite others? Isn't it 1/∞?

No, because we don't calculate length on the basis of how many points are in an interval. We calculate length by subtracting the position of the point on the left from the position of the point on the right. The length of an interval [a, b] between two points (assuming a < b) is b - a.

The ratio of the "length" of a point to the length of the line segment [a, b] is 0/(b - a) = 0.

In any case, division by ∞ is not allowed.

Here's another question, if a perfectly round object is placed on perfectly flat surface, what is the area of contact?

Your question isn't very precise. A circular disk is a perfectly round object, so the area of contact would be the area of the disk.

If by "perfectly round object" you mean a sphere, there is only one point of contact, so the area of contact would be zero for reasons similar to what I already gave.

 

[DutchNight]

Infinity cannot be treated as a normal integer or real number. It is not part of one of these sets of numbers. In that sense 1/ꝏ is meaningless.

You can use the infinity concept in limits. Going to infinity means taken larger and larger values (ꝏ is not a number that can ever reach)

The limit of 1/x as x approaches Infinity is: 0For the second question: when a plane "touches" a sphere they have one point in common. The area of contact is therefore 0

 

[A. Bahat]

The concept of division by infinity can also be made rigorous, instead of using limits, by considering complex infinity on the Riemann sphere.