Is a Circle with Radius 0 Real?

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Homework Help Overview

The discussion revolves around the concept of a circle with a radius of zero, exploring its implications in geometry and mathematics. Participants are examining whether such a circle can be considered real or if it simply represents a point, as well as the definitions and properties associated with circles in various contexts.

Discussion Character

  • Conceptual clarification, Assumption checking, Mixed

Approaches and Questions Raised

  • Participants are questioning the definition of a circle when the radius is zero, considering it as a degenerate case or a point. There is also discussion about the implications of negative radii and their relation to the concept of distance in metric spaces.

Discussion Status

The conversation is ongoing, with various interpretations being explored. Some participants suggest that a circle with a radius of zero can be viewed as a point, while others emphasize the necessity of a positive radius for a circle to exist meaningfully. There is no explicit consensus, but multiple perspectives are being examined.

Contextual Notes

Participants are navigating definitions and properties of circles, particularly in relation to real and complex spaces. The discussion includes considerations of how these definitions apply in different mathematical contexts, such as metric spaces.

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if it is 0 ...

hey all ...
if we have the circle equation and we found the radius ... if it is positive the circle would off course be real ... but if it was 0, do we say that we don't have a circle or the circle is a dot ?
cheers
abc
 
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The graph is a single point. Technically, that's NOT a circle but you can consider it a "degenerate" circle.

(and if r< 0, your graph is the empty set- some people consider that a REALLY degenerate circle!)
 
"Technically",the equation would be (for one with the center in the origin)
[tex]x^{2}+y^{2}=R^{2}[/tex] and the radius being negative would not mean anything wrong.The pairs (x,y) satisfying it would be the same.As for "zero",well,that should be seen as the "degenerate" case...

The wrong part with negative radius would come from the definition of the circle,which uses the word "distance",and the definition of distance for a metric space (in this case [itex]R^{2}[/itex]) which is an application to [itex]R_{+}\cup \{0\}[/itex]..

Daniel.
 
One could argue that the circle in question exists in complex space where either or both of x and y are complex.

Then again, if R^2 is negative the radius is also complex. does this invalidate the nature of a circle or is it reasonable?
 
No,it does not invalidate the "nature" of the circle,it just pushes our power of understanding...The abstractization is sometimes unappropriate.The circle MUST have a real positive nonzero radius.And it's preferable to have as a one-dimensional object...

Daniel.
 

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