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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?

user237118
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2 Answers2

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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.

Misha Schmalian
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  • So geometrically it is no longer a circle? – user237118 Jul 21 '19 at 15:59
  • If we start from a point on circle and took out half the infinite points on circle, do we end up with semi circle? – user237118 Jul 21 '19 at 16:00
  • @user237118 "So geometrically it is no longer a circle?" Yes, but it still has infinitely many points. (Incidentally, you should say "infinitely many points" rather than "infinite points" - the latter makes it sound like the points themselves are infinite objects, which can actually be a bit confusing since "point at infinity" is a real technical term. This is a totally minor issue but it is technical more correct.) – Noah Schweber Jul 21 '19 at 16:01
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    @user237118 "If we start from a point on circle and took out half the infinite points on circle, do we end up with semi circle?" Well, depends which half we remove ... – Noah Schweber Jul 21 '19 at 16:01
  • The consecutive ones, just next to each other – user237118 Jul 21 '19 at 16:03
  • "took out half the infinite points on circle, do we end up with semi circle?" isn't really a well-defined concepts. Infinite sets are where mathematical formality is first necessary and we can easily get contradictions otherwise.

    The first statement reaches its limits as SIZE isn't defined that way on infinite sets. One might think ${2, 4, 6, ...}$ is "half of" ${1, 2, 3, 4, ...}$. But we can map them one-to-one onto each-other with the function $x\mapsto 2*x$. So they are de-facto the same.

     – Misha Schmalian Jul 21 '19 at 16:09
  • Also "consecutive ones, just next to each other" isn't really clear, because between any two points on the circle there is another. More importantly you can't enumerate (have a one-to-one correspondence with integers) the points on a circle.

    Infinite sets are at first weird to get your head around. I'd recommend just reading some real analysis notes, to get some intuition.

     – Misha Schmalian Jul 21 '19 at 16:12
  • You are comparing a continuous set (points on a circle) with a countable set (the non-negative integers.) I have to ask, is that a valid comparison? And if so, then why is it a valid comparison? I don't know much about the mathematics of infinite sets, but I know enough to know that there is no 1:1 correspondence between those two sets. – Solomon Slow Jul 22 '19 at 00:32
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  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.
user7530
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