Rade
There are an infinite number of radii within any given circle, are there also an infinite number of circles within a circle as shown in the attached image ?
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Given a radius R, there is an infinite number of 0 < r < R, so the answer is yes. I can't see the attachment though, but there is no way to display that. Nonformally speaking, you could be talking about a shaded disk of radius R, I guess.There are an infinite number of radii within any given circle, are there also an infinite number of circles within a circle as shown in the attached image ?
Seems like an analogous question to "Are there an infinite number of discrete points between 0 and 1 on the number line?" Is there something special about the circle aspect of this question?There are an infinite number of radii within any given circle, are there also an infinite number of circles within a circle as shown in the attached image ?
I can see the attachment fineare you still having problem ?Given a radius R, there is an infinite number of 0 < r < R, so the answer is yes. I can't see the attachment though, but there is no way to display that. Nonformally speaking, you could be talking about a shaded disk of radius R, I guess.
Don't know, that is why I ask. Consider that 0 and 1 that serve as limits in your example are similar limits in that they are both integers, but in the circle example it is not clear to me that both limits are similar.Seems like an analogous question to "Are there an infinite number of discrete points between 0 and 1 on the number line?" Is there something special about the circle aspect of this question?
It takes an average of a few minutes for the Staff and Mentors to see that there is an attachment awaiting approval. Once I saw it and approved it, everybody can see it. radou just viewed your post before I approved it.I can see the attachment fineare you still having problem ?
No it is not.BTW, is this a homework problem
Neither of these constraints applysee the figure in post #1perimeter has width, not a shaded disk.There are an infinite number of circles if the perimeter has "no" width and the distances between the radii are zero. ie A Shaded Disk.
I'm not entirely sure what you're saying, but if each of the 'circles' is actually an object with nonzero area (say, the set of all points within 0.001 units of a circle) then only finitely many can fit into the (large) circle without overlapping, since the area of a circle is finite.Neither of these constraints applysee the figure in post #1perimeter has width, not a shaded disk.
Well yes, thats why there isn't an infinite number of circles...Neither of these constraints applysee the figure in post #1perimeter has width, not a shaded disk.
OK, but there would be an infinite number if the perimeter of added circles "has no width"in the same way that there are an infinite number of radii (without width) in a circlecorrect ?I'm not entirely sure what you're saying, but if each of the 'circles' is actually an object with nonzero area (say, the set of all points within 0.001 units of a circle) then only finitely many can fit into the (large) circle without overlapping, since the area of a circle is finite.
Yes, certainly. There's a circle for every real number between 0 and the outer radius.OK, but there would be an infinite number if the perimeter of added circles "has no width"in the same way that there are an infinite number of radii (without width) in a circlecorrect ?
A disk of some radius R≥0 can be thought of as {(x,y) : 0≤x^{2}+y^{2}≤R^{2}}OK, but there would be an infinite number if the perimeter of added circles "has no width"in the same way that there are an infinite number of radii (without width) in a circlecorrect ?