Topics Mathematics

Question I have one question concerning about lines in mathematics. My teacher told me that two lines of different lengths are made up of the same number of points. he told me that if we placed one above the other and join its end points and extend it they will meet at a point (for eg.) R. he told me that we can prove that by joining one point of the longer line to the shorter line and then to the point R and by continuing doing the same. If we do so we will feel that it is made up of the same number of points. But in my view if we place one line above the other and join its end points then both the line would be slanting towards each other (because one is longer than the other). If we remove those points and the line that we joined then equals will be left because we are removing the same number of points. If we continue doing this by drawing parallel lines then both of them will meet at a point on the centre of the shorter line and if we stii continue drawing then the lines will meet at a point such that it does not lie on the shorter line. This shows that the longer line has more number of lines than the shorter one. Please clarify my doubt.

Accepted:
August 31, 2010

Comments

On the standard account, given two finitely long lines, even of different lengths, their pointscan indeed be matched up one-to-one, e.g. by the kind of projection theteacher indicated. And the possibility of that kind of one-to-one matchingis just what we mean when we say the two lines "contain the same number ofpoints".

What makes this possible, despite the different lengths? In part, the fact that there are an infinite number of points along a finite line (the issue we are dealing with here is one of those initially puzzling matters which arises when we deal with the non-finite: intuitions tutored by finite examples can lead us astray).

And there being an infinite number of points along a finite line is related to the fact that the points on a line are dense -- that is to say, between any two points, however close together, there is another point.

Now, consider a line with end points. Between the left hand end-point a and any other point on the line there is a further point. So it follows that there is no 'next' point, immediately to the right after a. (Suppose that b is a candidate 'next' point: then by denseness, there will in fact be another point c between a and b. So b isn't 'next' after all. Nor is c, because by the same reasoning there is another point d between a and c. And so it goes.)

This means that if you remove the left-hand end-point a what remains is an 'open' line with no left-most point. (It can't be that point b is leftmost since c remains to the left of it; and it can't be point c as d remains to the left of it; and so it goes.) There are points arbitrarily close to the position that a has vacated, but no left-most one.

Hence the imagined construction where we remove the end point of a line and then remove the new end point etc. in fact makes no good sense. After we chip off the left-most point a of a line with endpoints we are left with an 'open' line with no left-most point, and the described construction falters at the very first step after the initial one!