Investigating Logic Behind Line Segment Lengths

[gb7nash]

https://www.physicsforums.com/showthread.php?t=495791".

"What is the biggest number you can divide by 10 exactly?".

I like this one better.

 

[mireazma]

My mathematically profane answer: The points of the second line are bigger
If you say for any point of short(?) line there [STRIKE]exist[/STRIKE] correspond a point on the long(?) line, it means that the points of the latter are the projections of the shorter line's points. He he.

 

[DaveC426913]

My mathematically profane answer: The points of the second line are bigger
If you say for any point of short(?) line there [STRIKE]exist[/STRIKE] correspond a point on the long(?) line, it means that the points of the latter are the projections of the shorter line's points. He he.

Haha. No. Points are points.

The folly in the argument is the part where he equates one infinity with another. You can't treat infinities arithmetically.

 

[Jamma]

Yep, as soon as you realize that the number of points has nothing to do with the length of the line, it's all very obvious.

I can see where the confusion comes from though; in real life, even if we have billions upon billions of points, say all arranged in a little line, like a line of atoms, then we can still stretch them, but the distances will actually increase between the atoms, and if we increase far enough, then we will eventually "see" the gaps. So no matter how close you approximate the infinity of points, even if you use trillions of them all bunched up, it still never has the same properties of what goes on when you have infinitely many.