There are an infinite number of points on a line.
No two points are ever next to each other.
Why?
_______________________________________________
Here’s an explanation for curious non-mathematicians (from https://math.stackexchange.com/questions/590677/why-how-are-there-infinite-points-in-a-line-segment):
“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.”
Try it this way:
Take 2 points next to each other on a line...get out a magnifier and zoom in - cut the line in half between the two points - after all there are infinite points...then between where you cut and one of the original points do the same...and on and on and on...thus no matter the number of points you can always divide further .... mathematically...
In a way we can do it physically too but current technology presents limitations on the magnification...and Heisenberg gets in the way with that pesky principal that says if you measure or observe something then it’s different just due to the measurement/observation interaction then it would be without the measurement/observation...
2 points make a line?
No, 2 points make an infinite number of points between the 2 points - with any 2 points IN that line made up of an infinite number of points between these 2, new, points.
Huh...makes sense.
Now, someone help us with determining the slope of an infinite number of points between 2 points on a graph.
So, it is safe to jump off a building because there are an infinite number of halfway points for you to pass before you don’t hit the ground.