Comparing the Lengths of Two Infinite Lines

[Acala]

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 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?

 

[SW VandeCarr]

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 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?

Just a thought. If you require the areas to be preserved, the sides of the rectangles can never be brought into coincidence. The rectangles must remain rectangles no matter how "long" they get. The lengths in the x direction can be arbitrarily large, but not infinite.

 

[chiro]

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.

 

[SW VandeCarr]

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.

 

[Acala]

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?

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.

I do not understand this deduction. Are you saying that because of their continuity, their infinite values must be equal? I was under the impression that their continuity essentially made them geometric analogues of uncountable infinite sets, and I was under the impression that uncountable sets do not necessarily have the same size as one another.

 

[SW VandeCarr]

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?

To bring the sides into coincidence, you have to give up preservation of area. At any point in the deformation process where you do this, the length of the resulting line at the completion of the process will be half the perimeter of the rectangle at the time you suspended the conservation of area requirement. Do you see why? The only property that will be preserved will be the length of the long diagonal at the moment the process ends. The long diagonal becomes the line that results and is equal to half the former perimeter.

I do not understand this deduction. Are you saying that because of their continuity, their infinite values must be equal? I was under the impression that their continuity essentially made them geometric analogues of uncountable infinite sets, and I was under the impression that uncountable sets do not necessarily have the same size as one another.

This is the most interesting part. The cardinality of the continuum C applies to all uncountable infinite sets afaik. That means there is a one to one mapping (bijection) between such sets. The shortest imaginable line segment has the same number of points (as a point set) as an infinite line. And, of course, all infinite lines have the same measure. The measure is not the number of points, but some external metric.

 

[Acala]

This is the most interesting part. The cardinality of the continuum C applies to all uncountable infinite sets afaik. That means there is a one to one mapping (bijection) between such sets. The shortest imaginable line segment has the same number of points (as a point set) as an infinite line. And, of course, all infinite lines have the same measure. The measure is not the number of points, but some external metric.

Excellent, this makes a lot of sense. Could the same reasoning be applied if we were to add a dimension? (i.e. do rectangles of different areas all contain the same number of lines?)

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?

 

[SW VandeCarr]

Excellent, this makes a lot of sense. Could the same reasoning be applied if we were to add a dimension? (i.e. do rectangles of different areas all contain the same number of lines?)

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?

Well, I never heard of a line set, so I don't know. But there are "curves" which can "pave" a 2D space and "fill" spaces of higher dimension such that every point in the space is on the curve. This reduces the problem to point sets.

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.

 

[SteveL27]

This is the most interesting part. The cardinality of the continuum C applies to all uncountable infinite sets afaik.

No, that's not true. If C is the cardinality of the reals, we know that P(R), the power set of the reals, has strictly larger cardinality. There are lots and lots of different uncountable cardinalities.

 

[SW VandeCarr]

No, that's not true. If C is the cardinality of the reals, we know that P(R), the power set of the reals, has strictly larger cardinality. There are lots and lots of different uncountable cardinalities.

Thanks. Point taken. I think in this case we are talking about the cardinality of the reals however. Is this not the case? In what cases, when talking about a continuum, are we not talking about the cardinality of the reals?

 

[SteveL27]

Thanks. Point taken. I think in this case we are talking about the cardinality of the reals however. Is this not the case? In what cases, when talking about a continuum, are we not talking about the cardinality of the reals?

Perhaps I misunderstood your post. I thought you said all uncountable sets had the same cardinality. That's not true. If you were only talking about uncountable sets of reals having the same cardinality, now we're back to the Continuum Hypothesis that we were discussing in another thread.

 

[SW VandeCarr]

Perhaps I misunderstood your post. I thought you said all uncountable sets had the same cardinality.

I did say that. I should have specified sets of real numbers.