# Why point is a circle with radius zero?

I was reading this What is a point circle, a real circle and an imaginary circle? and i get confused with the statement that is written in the accepted answer , i.e

A point "circle" is just a point; it's a circle with a radius of zero

But point itself is a circle and when we zoom it enough there is still a radius $$>0$$ , so how it is a circle of radius zero ???

• "But point itself is a circle and when we zoom it enough there is still a radius >0, so how it is a circle of radius zero ???" - why do you think that?
– JC12
Commented Aug 2, 2020 at 7:37
• You seem to be thinking of a point you draw with like a pencil. We're talking like a "mathematically ideal" point, which has no width, no length -- nothing going in any direction. Commented Aug 2, 2020 at 7:37

A point is defined to have no size. It is simply a location in space. I think the confusion arises because if you put a "point" of ink on a piece of paper and look closely enough, it seems to have a size.

But that dot of ink isn't really a point; it has a size. If you wanted a true mathematical point on paper, you will need a dot of ink that is so tiny, no other dot can be made that is smaller than it.

• Even a single electron is not a point. Commented Aug 3, 2020 at 0:39

To quote from Euclid's Elements, Book I, Definition 1

A point is that which has no part.

In modern sense, part corresponds to dimensions. So, by the definition and tautology, when you zoom a point you shouldn't observe any dimensions.

In the question you have linked to, the OP asks about the meaning/intuition of some of the things, so when the radius is zero (that is when $$g^2 +f^2 -c^2 =0$$) we get a point, that is "something which has no dimensions", we call it a point circle in the context of analytic geometry just to make things a little intuitive, but in rigorous sense a point is not a circle of zero radius, it is not a square of zero length or any other thing like that.

Hope I was clear and it helps you!

In various high school textbooks, when the radius of a circle is zero, it is said that it is a degenerate circle of zero radius. That means we're looking at a circumference of zero radius. Consequently, only one point belongs to the circumference, which is the centre.

• Very nice explanation :). Commented Aug 2, 2020 at 16:22
• @Knight Thank you very very much... Commented Aug 2, 2020 at 18:53

There's a discrepancy between what can be drawn and what should exist in theory. If you "zoom in" into the point you would see a circle. However, if you zoom in more, you would be able to delete the "outer" reaches of the circle and be left with a smaller circle which would still be considered a point. Theoretically a point is a circle with no radius. However there is no way to draw it. It is not a concept you can visualize. Do you understand the logic though?

I am not sure I agree with your statement "point itself is a circle", This may be true according to this definition from the other post; "a circle is merely the set of points that are a fixed distance away from the center. If that fixed distance is zero, then that set of points is simply the center point itself.", or as I would put it, a point circle is a set containing exactly one point.

However, this a mathematical and theoretical construct only. When you talk about "zooming in" you seem to be implying that its a real physical thing that has dimensions. In the physical world ( so far as we know) there is no such thing as a point circle, but in maths, there is.