Infinite points on circle

Question

Can we say that circle has infinite points? What if then I took one point out. Does it matter. And then if we took half the infinite points of the circle out of it and still can it be called circle?

Answer 1

In the easiest setup, a circle is all points in the plane $\mathbb R^2$, with distance 1 from the origin $(0,0)$. There are infinitely many solutions to this condition, so YES there are infinitely many points in a circle.

YET removing a single point changes its structure (f.e. a circle missing a point is no longer closed) and the result is no longer a circle. But this shouldn't come as a huge surprise. Removing a single point from an infinite set often changes it. E.g. if you have then set $S=\{0, 1, 2, ...\}$ and remove $0$, then suddenly you achieve the property, that all elements of $S$ are positive.

Answer 2

1. Yes, the circle contains infinitely many points. The usual unit circle consists of all points of the form $$(\cos \theta, \sin\theta)$$ for $0\leq \theta < 2\pi.$

2. No, if you delete even a single point from a circle, it's not a circle anymore. In more advanced mathematics you will study different properties of sets and be able to prove that the circle and circle-with-point-removed are very different, topologically and algebraically. Consider for instance that on the circle, any two points can be connected by two different curves that stay inside the circle; if you take out a point there is only one curve connected two points. That said, if you squint you might argue that deleting a point makes the circle still "look like" a circle (and in some cases the point doesn't matter, for example an integral will have the same value if you integrate it up over the circle or "punctured" circle). There are ways of making precise that the circle with one point removed is "almost" a circle: the full circle is the closure of the punctured circle, and the two sets differ by "a set of measure zero" (which in this case means the part you deleted has zero length compared to the full circle).

3. What is half of infinity? There are different ways of removing "half the points": certainly a semicircle is very different from the full circle. You could delete all points with irrational $\theta$ in the formula above: this deletes far more than half of the points, but the resulting set will still "look like" a circle.