Circle With Radius of Zero
Date: 12/28/2004 at 15:22:59 From: Jack Subject: A circle with a radius = 0? Is it possible for a circle to have a radius that equals zero? Is it possible for a set of points (e.g., multiple points) to occupy the same location? One textook defines a circle as the set of points that is equidistant from a center point and that the radius is greater than or equal to zero. I dispute this. Am I wrong?
Date: 12/28/2004 at 22:48:10 From: Doctor Peterson Subject: Re: A circle with a radius = 0? Hi, Jack. If the radius is zero, then it isn't really a circle, but might be called a degenerate circle--that is, what you get if you slightly stretch the definition of a circle by using the same equation but taking it to extremes by making the radius zero. The point is (no pun intended!) that many things you can say about a circle will still be true if the radius is zero (making a single point), and they have for some reason chosen to allow that. I wouldn't do so, because there are too many other things that would no longer work in that case. I hope, for example, that in theorems about tangents to a circle they specify that the radius has to be greater than zero. If not, then they are inconsistent in their use of the definition, which is not uncommon in textbooks. By the way, there is nothing wrong in talking about a set of points that consists only of one point; nothing in that wording should be taken to imply that there are multiple points. The problem with this definition lies in the difficulty of writing theorems based on it, not on how many points there are. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 12/29/2004 at 02:14:15 From: Jack Subject: Thank you (A circle with a radius = 0?) Dear Dr. Peterson: Thank you very much for your thoughtful reply. Unfortunately, I lost an argument, but I appreciate knowing the truth even more than being "right." :-) - Jack
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