Topics Mathematics

Question This is more like a comment to the question in Mathematics that starts with: "If you have a line, and it goes on forever, and you choose a random point on that line, is that point the center of that line? And if you ..." The answer provided by the panelist, as well as the initial question, assume that one can distinguish between points at infinity. As far as Math goes however, one cannot do that, and this is the reason the limit for cos(phi) does not exist, as phi goes to infinity. Revisiting the argumentation provided by the panelist, the error starts with the 'definition' of the distance between a fixed point and infinity - this distance cannot be defined, and therefore it cannot be compared (at least, as math goes). A somewhat similar problem can be stated, without the pitfalls of the infinity concept, for a point on a circle, or any closed curve.

Accepted:
October 11, 2005

Comments

It seems to me that you are reading things into the original question, and my answer to it, that were not there. I do not see, either in the original question or in my answer, any reference to "points at infinity". The orignal question talks about a line going on forever, and my answer talks about the line extending infinitely far in either direction from some point P on the line. But this just means that for every number x, there are points on the line more than x units away from P in either direction, not that there are points that are infinitely far away from P. I claimed that the parts of the line on either side of P are congruent, and you can see this by observing that if you rotate the line 180 degrees around P, each side gets moved so that it coincides with the other side.

My previous answer was based on a particular definition of "center". There is another, slightly different definition of "center" that could lead to the sorts of worries that you raise. Suppose we define the center point to be the point that is equidistant from the endpoints. This works fine for a finite line segment, and leads to exactly the same center as the definition I originally proposed. But for an infinite line, if you tried to apply this definition then you would, indeed, find yourself looking for endpoints of the line--points at infinity--and you would find yourself trying to compute the distances from those points at infinity to other points. So this definition of "center" would lead to the sorts of worries that you raise, but it is not the definition I was using in my previous post.