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Please see if there is any error in the statements given below.

If we consider a circle as an infinite sided polygon,then if the radius of the circle is made infinite ,then each of it's sides will also become infinitely long . Therefore one cannot prove that any given line is not a part of a circle of infinite radius,because the line (the side of infinite sided polygon) like any other line will have no ending and hence it's properties are indistinguishable from other lines in reality.

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    $\begingroup$ "If we consider a circle as an infinite sided polygon" $\leftarrow$ That is already an error, everything based on this assumption is too. $\endgroup$ – Zev Chonoles Dec 26 '16 at 7:50
  • $\begingroup$ Why is it wrong to assume that a circle is an infinite sided polygon? $\endgroup$ – user401830 Dec 26 '16 at 7:55
  • $\begingroup$ The definition of "polygon" from Wikipedia: In elementary geometry, a polygon is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit. (emphasis mine) $\endgroup$ – Zev Chonoles Dec 26 '16 at 7:59
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    $\begingroup$ No line is part of a"circle of infinite radius" because there is no such thing as a circle of infinite radius. $\endgroup$ – littleO Dec 26 '16 at 8:33
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    $\begingroup$ Every circle has a radius which is a positive real number. And "infinity" is not a real number. $\endgroup$ – littleO Dec 26 '16 at 9:07
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At times it can be useful to think of a line as an infinite radius circle. Specifically, in complex analysis there's a concept called a linear fractional transformation, that has the property that it sends circles/lines to circles/lines in a "nice" way. A common way to think of this is that it sends circles to circles, but a line is just a circle with a point at infinity.

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It is not correct to say that a circle is "an infinite sided polygon" – or more precisely, it would take more clarification about what you mean by "polygon" to even make it mean anything to say that. The definition of "polygon" from Wikipedia:

In elementary geometry, a polygon is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit.

(emphasis mine)

Therefore your reasoning based on this statement is already flawed.

Putting that aside, you'd also have to put a lot more work into making precise what it means for the radius of the circle to be "made infinite". Infinity is not a single concept and most versions of it cannot be treated as a number, so you can't just say $r=\infty$ without a lot of clarification.

Mark's answer is correct, that in complex analysis (and other parts of math) it is sometimes useful to treat circles and lines similarly, but I would say that it is not useful in the context of this discussion.

Putting that aside, you'd have to explain what it would mean for the sides of a polygon to be "infinitely long", and actually prove that "making the circle radius infinite" also makes the "sides" "become infinitely long".

That's just the first sentence, but I hope you see that the general problem is that you need to be very clear and rigorous about the mathematical words you use, and particularly infinity is a tricky concept and prone to causing mistaken reasoning without proper understanding and precision.

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  • $\begingroup$ Can the above statements be applicable in classical physics(since they are used in complex analysis) if the space is assumed to be Euclidean ? $\endgroup$ – user401830 Dec 26 '16 at 9:06
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    $\begingroup$ I don't think that question makes any sense, unfortunately. $\endgroup$ – Zev Chonoles Dec 26 '16 at 9:21
  • $\begingroup$ There is a scientific topic called Möbius Geometry which uses 4-dimenstional vectors to describe both lines and circles. In this setup again a line is a special case of circle, and performing any radius computation will lead to infinity.So , can we apply this result in classical mechanics? $\endgroup$ – user401830 Dec 26 '16 at 12:53

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