# All line segments equal?

Hello,

If someone takes two segments, one longer than the other, it is naturally assumed that the longer one is most definitely longer than the other. However, if the longer segment is put at an angle with the other to make a triangle, suddenly someone can draw a one to one correspondence between all the points on the original two segments. So the line was not really longer. How can we resolve this paradox?

There is no paradox. Bijections do not, in general, preserve length.

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I agree it looks weird and suspicious. But the correct resolution is to stop seeing it as a paradox and more as a fact of nature/math.

However, if the longer segment is put at an angle with the other to make a triangle, suddenly someone can draw a one to one correspondence between all the points on the original two segments.

Couldn't you do that even without putting the longer line at an angle with the other?

Is this what you have in mind? Here I'm showing two points on each line, with their distance exaggerated for clarity. Although there's a one to one correspondence, the distance between the two points on the longer line is greater than the distance between the two points on the shorter line.

Mark44
Mentor
If someone takes two segments, one longer than the other, it is naturally assumed that the longer one is most definitely longer than the other.
The longer segment has a length that is larger, but both segments contain the same number of points. Another way to say this is that the cardinality of both segments is the same.