I have this thought that circle in 'real' is not a closed figure. We all know that 'pi' is irrational.And integers are nodes in a 'monstrous' line of real numbers. Irrational numbers are non-terminating i.e. to whatever lower scales you go and try to measure, there is still something left (so are repeating decimals).

Lets say in a Cartesian plane we start at a point (x0 ,y0), x0 = 0.0000000.... and y0 = 0.000000... and draw a circle ( let the center be (2.000000....., 0.00000000...) ). If its perimeter itself is non-terminating how are we closing circle at (x0 ,y0) again. i.e. even though how much you draw shouldn't there be even more to draw to close that circle?

(Perimeter = 2*pi*radius, and 'pi' is irrational => perimeter is irrational or non-terminating)

The same problem occurs also when

a) repeating decimals are present,

b) In isosceles right angled triangle, where hypotenuse has Sqrt(2),

Is this problem related anyhow to Achilles-tortoise problem? ( I don't see any-- the reason is that, here a infinite series is not converging 'miraculously' to a finite value instead we have a irrational value to realize)

Am I going wrong somewhere?


  • $\begingroup$ What do you mean by "the perimeter itself is non-terminating"? $\endgroup$ – fixedp Aug 26 '14 at 11:37
  • $\begingroup$ Yes this seems to be a rehashing of en.wikipedia.org/wiki/Zeno's_paradoxes - we can sum infinite series and obtain finite values so it's not a problem. $\endgroup$ – lemon Aug 26 '14 at 11:41
  • $\begingroup$ Take an infinitely long spring, compress it to a 2D figure: Infinite volume turns to finite area. Now, with respect to the 2D plane, the resultant is closed. But in 3D space, an airplane could fly through it. Is it not rational to define a closed shape as being one in which King Kong could not escape from if enclosed in the shape and corresponding plane? $\endgroup$ – Nick Aug 26 '14 at 12:04
  • $\begingroup$ See Completeness of the real numbers : without this axioms (or the equivalent theorem) we cannot assert that the Cartesian plane contains the point (SQRT(2), SQRT(2)). $\endgroup$ – Mauro ALLEGRANZA Aug 26 '14 at 13:22

This question may be reduced to that how can the real line has no hole? A general common misconception is to think of a (mathematical) line as a physical, material line. So one proceeds with dividing a line and argue thereon.

Instead, in mathematics what actually concerns is how to cover a given line with numbers. As you know, using integers is not sufficient. Rationals are also insufficient, as $\sqrt{2}$ was found. But it can be shown that real numbers suffice. And it is in this sense we understand the real line and speak of continuum.

Poincare's illustration may also be inspirational here. In what sense we understand a straight line on a plane in mathematics? A straight line on a plane is simply the limit of the procedure that we pinch any two points on a given circle joined by a given diameter of the circle and keep stretching this circle.

  • $\begingroup$ But how you define 'a line', which you later try to 'cover with numbers'? I understand 'a line' must be defined without using numbers, otherwise it would already be covered... $\endgroup$ – CiaPan Aug 26 '14 at 13:02
  • $\begingroup$ @CiaPan: I am trying to separate mathematical line from physical line, introducing the notion of continuum, so that mathematical thought can be emphasized. As Poincare pointed out, a straight line is the limit of the aforementioned procedure :) $\endgroup$ – Megadeth Aug 26 '14 at 13:06

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