Comparing the Lengths of Two Infinite Linesby Acala Tags: countable, dimension, infinite sets, infinity, uncountable 

#1
Feb2812, 09:24 PM

P: 19

Hello, everyone.
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 xaxis of a separate graph. Now, I begin to "smash" each rectangle into the xaxis. The area of each rectangle is preserved, but the shapes slowly become longer in the xdirection and shorter in the ydirection until I smash them into the xaxis 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? 



#2
Feb2812, 09:45 PM

P: 2,490





#3
Feb2912, 12:11 AM

P: 4,570

Hey Acala and welcome to the forums.
As mentioned by SW VandeCarr, if you require the areas to be preserved you won't ever stop the rectangle from being 'a rectangle'. If this is the case then the area information will never vanish and the only thing that you will lose is the original information about one of the sides. 



#4
Feb2912, 09:43 AM

P: 2,490

Comparing the Lengths of Two Infinite Lines
To answer the question posed in the title; by two infinite "lines", I assume you mean continuous lines. Obviously, if they are both infinite, they have the same length.




#5
Feb2912, 10:54 AM

P: 19

Thanks for the replies. That is an excellent point about the preservation of the areas; could we get around this, for instance, by letting the rectangles approach a "height" of zero?
Furthermore, is there some sort of property that is conserved as the sides are brought into coincidence? 



#6
Feb2912, 12:29 PM

P: 2,490





#7
Feb2912, 02:08 PM

P: 19

And if so, how exactly do we make calculus work? When we take an integral, are we not adding up an infinite (continuous) number of rectangles, each of zero width (which is, in essence, a line)? Is the information not translatable across dimensions? 



#8
Feb2912, 02:26 PM

P: 2,490

As far as calculus is concerned, we have the concept of a limit. In principle the limit is never realized, but is taken as a value which a function continuously approaches as one of its arguments approaches infinity or zero (usually). Analytically, we take the function's value when its argument is "at" infinity or zero so we can calculate. 



#9
Feb2912, 02:28 PM

P: 799





#10
Feb2912, 03:00 PM

P: 2,490





#11
Feb2912, 04:38 PM

P: 799





#12
Feb2912, 05:37 PM

P: 2,490




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