Question about points on a circle ?

[cragar]

Lets say i draw a circle, and then from the center i draw a line to the outer edge and let's say i do this for every point on the line. So I've gone completely around the circle. I should have an infinite amount of lines . And now let's say i draw a bigger circle around that and then extend the lines from the inner circle to the outer one. Wont those lines diverge and there will be gaps on the bigger circle. So if there is an infinite amount of lines why are there gaps. Unless I'm missing something. I remember seeing something from a you tube video that talked about it. But hey just brought it up and didn't really talk about it .

 

[Jimmy Snyder]

This is a really good question and exposes the nature of the concept of infinity. Actually, there is no gap and here is the proof.

Suppose that there is a gap and consider a point A in the gap. That means that none of the radial lines intersects A. Draw a line from A to the common center of the two circles. It will intersect the smaller circle at a point B. Consider the line segment from the center to B and extend it to the larger circle. It will intersect A. This contradiction proves that there is no gap.

 

[cragar]

ok you i see what your saying .

 

[cragar]

can we say that the larger circle has more points on it

 

[Jimmy Snyder]

can we say that the larger circle has more points on it

No. You can label the points of the smaller and larger circles by the angle formed by the positive x-axis and the line from the center of the circle to the point on the circle. There are just enough labels to go around (ha ha) in both cases.

 

[cragar]

so you are saying that those 2 sets of points are equal.

 

[Jimmy Snyder]

so you are saying that those 2 sets of points are equal.

The number of points in both sets are equal, not the sets themselves.

 

[alt]

Stuff like this can make your head spin (ha ha).

A question; are the radial lines assumed to have any width ? If not, how do they fill the spaces - all spaces.

If they do, what happened to that width of each of them and collectively, when they converge back to the one dimensional centre point (projecting back inwards) ?

 

[cragar]

The number of points in both sets are equal, not the sets themselves.

ok . Can we say that their are more real numbers than natural numbers .

 

[Jimmy Snyder]

A question; are the radial lines assumed to have any width ?

No.

If not, how do they fill the spaces - all spaces.

In the same way that points, which also have no width, fill the spaces to make a line. The way to see that the radial lines fill all spaces is to suppose there is a point A that is not 'filled'. Draw a line from A to the center and extend the line from A so it intersects the circle at point B. The line from the center to B, 'fills' A. This is related to the fact that the circle itself is 'filled' by its points.

 

[cragar]

ok I am having second thoughts about the lines diverging on the 2nd outer circle , If i take two lines let's say at 1 o'clock and 2 o'clock on the smaller circle and then extend them to the bigger circle their endpoints will now be farther apart . So let's say we now take 2 lines right next to each other line x1 and then next line x2 and extend those lines won't they now be farther apart and we will have a gap . I can kind of see where the flaw in my argument is that there is no next number on the continuum. but why can't there be a next line there hast to be? Or at least i think

 

[Jimmy Snyder]

ok I am having second thoughts about the lines diverging on the 2nd outer circle , If i take two lines let's say at 1 o'clock and 2 o'clock on the smaller circle and then extend them to the bigger circle their endpoints will now be farther apart . So let's say we now take 2 lines right next to each other line x1 and then next line x2 and extend those lines won't they now be farther apart and we will have a gap . I can kind of see where the flaw in my argument is that there is no next number on the continuum. but why can't there be a next line there hast to be? Or at least i think

The problem with this reasoning is that there is no such thing as two lines "right next to each other". Just are there are no two points right next to each other. Between any two points that are not the same, there is a third point halfway between them. This is true no matter how small the difference between the two points. I like to call this the "a miss is as good as a mile" principle. This is also true for the points on a circle. If there were two lines right next to each other, they would intersect two points on the circle right next to each other. But since there are no two points right next to each other, there are no two lines right next to each other.

 

[cragar]

ok i got it now .

 

[Lacy33]

Excellent! Thank you for asking.

 

[jobyts]

I remember seeing something from a you tube video that talked about it. But hey just brought it up and didn't really talk about it .

I remember the video that you are talking about, but cannot remember the name either. This video was posted in npr/bbc/cnn/PF on a mathematician who tried to do arithmetic using infinity. All his ideas and the lifelong works were entirely rejected by the Mathematics community.