Conceptual error in approximating a straight line as a circle

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.

• "If we consider a circle as an infinite sided polygon" $\leftarrow$ That is already an error, everything based on this assumption is too. – Zev Chonoles Dec 26 '16 at 7:50
• Why is it wrong to assume that a circle is an infinite sided polygon? – user401830 Dec 26 '16 at 7:55
• 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) – Zev Chonoles Dec 26 '16 at 7:59
• No line is part of a"circle of infinite radius" because there is no such thing as a circle of infinite radius. – littleO Dec 26 '16 at 8:33
• Every circle has a radius which is a positive real number. And "infinity" is not a real number. – littleO Dec 26 '16 at 9:07

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.

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)

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.