Number of circles within a circle

[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 ?

 

[radou]

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 ?

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. Non-formally speaking, you could be talking about a shaded disk of radius R, I guess.

 

[berkeman]

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?

 

[Rade]

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. Non-formally speaking, you could be talking about a shaded disk of radius R, I guess.

I can see the attachment fine--are you still having problem ?

 

[Rade]

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?

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.

 

[berkeman]

I can see the attachment fine--are you still having problem ?

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.

And about the number line thing -- are you familiar with how limits work in a finite section of the number line? A related topic would be Zeno's (sp?) paradox, for example.

BTW, is this a homework problem? We should move it to the homework forums if it is.

 

[Gib Z]

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.

 

[Rade]

BTW, is this a homework problem

No it is not.

 

[Rade]

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.

Neither of these constraints apply--see the figure in post #1--perimeter has width, not a shaded disk.

 

[CRGreathouse]

Neither of these constraints apply--see the figure in post #1--perimeter has width, not 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.

 

[Gib Z]

Neither of these constraints apply--see the figure in post #1--perimeter has width, not a shaded disk.

Well yes, that's why there isn't an infinite number of circles...

CRGreatHouse- The circles I specify for an infinite amount are not existent, just mathematical with the area between the circles zero.

 

[Rade]

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.

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 circle--correct ?

 

[CRGreathouse]

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 circle--correct ?

Yes, certainly. There's a circle for every real number between 0 and the outer radius.

 

[bomba923]

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 circle--correct ?

A disk of some radius R≥0 can be thought of as {(x,y) : 0≤x²+y²≤R²}

A circle with a radius r≥0 can be thought of as {(x,y) : x²+y²=r²}

As you can see,
$$\left\{ {\left( {x,y} \right):x^2 + y^2 = r^2 } \right\} \subset \left\{ {\left( {x,y} \right):0 \leqslant x^2 + y^2 \leqslant R^2 } \right\}{\text{ if }}0 \leqslant r \leqslant R$$

If R>0, there is an infinite quantity of distinct r that satisfy 0≤r≤R;
thus, there is an infinite quantity of distinct circles within a disk of radius R>0.

 

[Rade]

Bomba923--thank you very much for the time and clarity of your answer--what I would expect from the mathematics section of this forum.