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Let's say I have two rectangles, j and k, having areas J and K, respectively, such that area J is less than area K. For clarity's sake, I will set each rectangle on the x-axis of a separate graph.

Now, I begin to "smash" each rectangle into the x-axis. The area of each rectangle is preserved, but the shapes slowly become longer in the x-direction and shorter in the y-direction until I smash them into the x-axis entirely.

My question is: how do the "lengths" of these "lines" compare (assuming that they can even be considered to be lines or to have lengths)? I am not sure whether the lines form something comparable to countably infinite sets, so that the information is lost entirely upon losing a dimension, or whether the infinite length of k is "longer" than the infinite length of j. Is there any information stored in the nature of these lines that could be recovered in order to find the original areas of the rectangles, or even to find which rectangle had greater area?

Any thoughts?

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# Comparing the Lengths of Two Infinite Lines

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