Geometry and the measurement of infinity

I've agreed with that, but I still think it is wrong to use infinity in an evaluation of which shape becomes the longest. Infinity in this case doesn't provide any conclusions as infinity should be used for checking limits rather than count comparisons. I think my grudge is that people are using the misconception that infinity is reachable and you can perform operations on it such as subtraction. :c
 
Stratadon said:
I actually strongly disagree with this. You are a bit mistaken. In theory, you could always make the width smaller by adding a 0 after the decimal point, thus making the height larger with a set area.

Remember, Area of Rectangle = Length x Width

therefore, Area of Rectangle / Width = Length

The Length is not infinity when the width is 0. The length would be undefined.

Now, as I was saying, the width can always get smaller forever approaching 0 but never making it there. This is what infinity really is. The concept that you forever approach 0 by adding more 0's after the decimal point. So in fact what I said is 100% correct through infinity and beyond.

ToyStory3Wallpaper800.jpg
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My argument is that you cannot have a rectangle with a width of 0. That would defy one of the fundamental principles of the rectangle itself. Therefore, you can never reach the infinite height, because in order for something to have a set area, it has to have multiple dimensions. Compressing the width ad infinitum will continue to never give you an infinite height, and therefore the rectangle that began with the greater area will continue to have the greater area, and thus the greater height, no matter how much you compress the rectangles.

I think we're on the same page here, I was just adding a bit of clarity to your post, stating that neither will ever reach infinite height.
 
TheGurw said:
My argument is that you cannot have a rectangle with a width of 0. That would defy one of the fundamental principles of the rectangle itself. Therefore, you can never reach the infinite height, because in order for something to have a set area, it has to have multiple dimensions. Compressing the width ad infinitum will continue to never give you an infinite height, and therefore the rectangle that began with the greater area will continue to have the greater area, and thus the greater height, no matter how much you compress the rectangles.

I think we're on the same page here, I was just adding a bit of clarity to your post, stating that neither will ever reach infinite height.
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What is "infinite height"? I will re-read your post only once you have defined this for me, because the way you are using it is quite wrong as t7 has already pointed out.

Because in my opinion, a width approaching 0 ad infinitum is the exact definition of the rectangle reaching an infinite height.
 
t7seven7t said:
I've agreed with that, but I still think it is wrong to use infinity in an evaluation of which shape becomes the longest. Infinity in this case doesn't provide any conclusions as infinity should be used for checking limits rather than count comparisons. I think my grudge is that people are using the misconception that infinity is reachable and you can perform operations on it such as subtraction. :c
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I'm not going to argue that. I was trying to show that infinite height can never be reached in a way that was probably more lazy than I should have been. I claim a sincere lack of sleep for that momentary lapse where I tried to subtract from infinity.
 
Stratadon said:
What is "infinite height"? I will re-read your post only once you have defined this for me, because the way you are using it is quite wrong as t7 has already pointed out.
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Um, no. t7 was pointing out a completely different mistake I made, where I tried to dumb down my point by subtracting from infinity - which I've already addressed as my utter lack of sleep. Unless you can show me exactly how I'm using that phrase incorrectly, I think you actually do need to read the post again.

"Infinite height" as I'm using it here is the impossible height achieved when you compress a rectangle (while retaining total area) until the vertical lines have a null gap. Since you can't do that and still call it a rectangle (because a null gap between two parallel lines creates a single line - and a rectangle requires two parallel vertical lines in order to be called a rectangle), nor can it retain an area (since to have an area you need a second dimension - length x 0 = 0, meaning it doesn't exist), you can never reach that infinite height. In other words, the question is flawed, as I've been saying since my first post.
 
/throws coffee at gurw

edit: I think you would argue that if you threw 2x2 pixels together it would form a square, despite having no pixel in the center area. In the same way its possible to argue that a rectangle can still be a rectangle if formed by two lines sitting exactly side by side with no space in between. Or perhaps you would just call that two parallel lines.
 
Let us say that when the two sides of the rectangle do touch. The height would be a defined number, which is probably the perimeter divided by 2, if I am not mistaken. So, taking this information, the height of the rectangle that is supposedly approaching infinity is actually approaching this perimeter/2 number.
 
MrFrog90 said:
Let us say that when the two sides of the rectangle do touch. The height would be a defined number, which is probably the perimeter divided by 2, if I am not mistaken. So, taking this information, the height of the rectangle that is supposedly approaching infinity is actually approaching this perimeter/2 number.
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They can never touch. You're forgetting that the lines on a graph don't have a width - when lines on a graph touch, they merge and become one line. This removes the second dimension and 3 of the lines of the rectangle, meaning you no longer have either an area or a perimeter.
 
TheGurw said:
Um, no. t7 was pointing out a completely different mistake I made, where I tried to dumb down my point by subtracting from infinity - which I've already addressed as my utter lack of sleep. Unless you can show me exactly how I'm using that phrase incorrectly, I think you actually do need to read the post again.

"Infinite height" as I'm using it here is the impossible height achieved when you compress a rectangle (while retaining total area) until the vertical lines have a null gap. Since you can't do that and still call it a rectangle (because a null gap between two parallel lines creates a single line - and a rectangle requires two parallel vertical lines in order to be called a rectangle), nor can it retain an area (since to have an area you need a second dimension - length x 0 = 0, meaning it doesn't exist), you can never reach that infinite height. In other words, the question is flawed, as I've been saying since my first post.
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You seem to have it stuck in your head that a rectangle can only have an 'infinite height' when the lines are touching, or the width is 0. You are using 'infinite height' almost as if it is a set number that you reach at some point and it kinda sounds like a 12 year old kid. "I reached infinite height guise, yeah!&^&@" You seem to understand that when the width is 0, the rectangle is undefined because it has has no area. This scenario is outside of the parameters of the problem so we should not even be thinking about it.

Infinity, as defined by wikipedia, is 'something without any limit' .

The width becoming smaller and approaching 0 ad infinitum but never reaching 0 is the definition of infinity as I understand it. That number has no limit because you can always add another 0 after the decimal point and before a number. (.01 -> .001 -> .0001 -> etc..) Therefore, the height of both rectangles with a set area of 4 or 5 is still infinity because it has no limit. The height could always be larger by making the decimal of the width smaller. With the same width, however, no matter how small, the rectangle with an area of 5 will always be taller than the rectangle with an area of 4 at the same width.
 
TheGurw said:
Their limit is actually 1 unit (whatever unit we're using, however small) less than infinity. If the limit were actually infinity, the lines could merge. However, they can't. So the rectangle with the area of 5 will have a much greater height, and will only widen the gap that much more as you approach 0 width.
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One less than limitless is still limitless, just as limitless plus one is limitless. And whatever height you find for rectangle 5 can be achieved with rectangle 4.
A=width*height
height=A/width
y=A/x ---> y=5/x and y=4/x

Take any height in the rectangle 5 equation, and you can find a width that gives the same height in the rectangle 4 equation.
 
Stratadon said:
You seem to have it stuck in your head that a rectangle can only have an 'infinite height' when the lines are touching, or the width is 0. You are using 'infinite height' almost as if it is a set number that you reach at some point and it kinda sounds like a 12 year old kid. "I reached infinite height guise, yeah!&^&@" You seem to understand that when the width is 0, the rectangle is undefined because it has has no area. This scenario is outside of the parameters of the problem so we should not even be thinking about it.

Infinity, as defined by wikipedia, is 'something without any limit' .

The width becoming smaller and approaching 0 ad infinitum but never reaching 0 is the definition of infinity as I understand it. That number has no limit because you can always add another 0 after the decimal point and before a number. (.01 -> .001 -> .0001 -> etc..) Therefore, the height of both rectangles with a set area of 4 or 5 is still infinity because it has no limit. The height could always be larger by making the decimal of the width smaller. With the same width, however, no matter how small, the rectangle with an area of 5 will always be taller than the rectangle with an area of 4 at the same width.
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If you're going to resort to insults on top of obviously not reading my posts (or the original posed question for that matter), you aren't deserving of further replies. Good day, sir.
MiniCacti said:
One less than limitless is still limitless, just as limitless plus one is limitless. And whatever height you find for rectangle 5 can be achieved with rectangle 4.
A=width*height
height=A/width
y=A/x ---> y=5/x and y=4/x

Take any height in the rectangle 5 equation, and you can find a width that gives the same height in the rectangle 4 equation.
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First part: I've already addressed that mistake.

Second part: The widths of the rectangles are identical at the end of the original question. You can't change that. In other words:
(y1) = 4/x
(y2) = 5/x
 
TheGurw said:
If you're going to resort to insults on top of obviously not reading my posts (or the original posed question for that matter), you aren't deserving of further replies. Good day, sir.

First part: I've already addressed that mistake.

Second part: The widths of the rectangles are identical at the end of the original question. You can't change that. In other words:
(y1) = 4/x
(y2) = 5/x
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I apologize if that sounded like an insult. That was in no way my intent. That was merely the image I got in my head as I was reading your post. I felt the need to state that because you keep repeating the same stuff over and over again without really understanding what I was saying, and you once again dodged my logic with your reply.

I have read the original post. I know it contradicts itself when it says "until the two vertical lines join." I thought we both came to an agreement and established that this scenario is not possible, so my response was then focusing on the concept of infinity relating to the problem while disregarding the part of the original problem that we debunked.

You can either defend your argument that the height of the rectangles will never 'reach infinity' or try to counter my logic. If you feel that I am too blunt in my responses, feel free to read the disclaimers under the information tab on my profile page. If there is ever any part of my logic that is unclear or needs further explanation, I would be happy to clarify.
 
Stratadon said:
I apologize if that sounded like an insult. That was in no way my intent. That was merely the image I got in my head as I was reading your post. I felt the need to state that because you keep repeating the same stuff over and over again without really understanding what I was saying, and you once again dodged my logic with your reply.

I have read the original post. I know it contradicts itself when it says "until the two vertical lines join." I thought we both came to an agreement and established that this scenario is not possible, so my response was then focusing on the concept of infinity relating to the problem while disregarding the part of the original problem that we debunked.

You can either defend your argument that the height of the rectangles will never 'reach infinity' or try to counter my logic. If you feel that I am too blunt in my responses, feel free to read the disclaimers under the information tab on my profile page. If there is ever any part of my logic that is unclear or needs further explanation, I would be happy to clarify.
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OK. Now I understand where you're going.

If I'm understanding your clarification correctly, you're stating that the height of the rectangles will continue to increase infinitely as you continue to compress the width infinitely. Neither rectangle will ever have an infinite height nor a width of zero - which was the whole basis of my posts. We never disagreed to begin with - we were arguing over two different parts of the same paragraph.
 
MiniCacti said:
I am having trouble seeing one as taller than the other. To me, this is like comparing two dead bodies; one may have died first, but the other is right there with it now. Like you said, the 5-area may approach its limit more quickly, but the 4-area is approaching the exact same thing. I have heard bits and pieces about other types of limitless-ness (and love the Numberphile video on it. /me <3's Numberphile.) but I don't see the difference here. When comparing solely where two things are, "I got here first" doesn't seem relevant.
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Careful. Infinity cannot be thought of as a "destination." The heights of both rectangles do not have upper limits and are not going anywhere, per se. They just keep getting bigger and bigger as the widths become smaller and smaller. If the two lines do meet, then the height is undefined because we have a "division by zero" case, so number mathematics does not apply.

But I am curious if there is a different kind of mathematics where this question does have a solid answer

MiniCacti said:
Take any height in the rectangle 5 equation, and you can find a width that gives the same height in the rectangle 4 equation.
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Yes, but only within the real number scale, which infinity is not a part of

But, now that I have thought about this infinity sizes topic a little more, it looks to me that the set of possible heights between the two rectangles are both the same size infinity since the range of both is between 0 and infinity, exclusive.
 
lacar1601 said:
Careful. Infinity cannot be thought of as a "destination." The heights of both rectangles do not have upper limits and are not going anywhere, per se. They just keep getting bigger and bigger as the widths become smaller and smaller. If the two lines do meet, then the height is undefined because we have a "division by zero" case, so number mathematics does not apply.

But I am curious if there is a different kind of mathematics where this question does have a solid answer



Yes, but only within the real number scale, which infinity is not a part of

But, now that I have thought about this infinity sizes topic a little more, it looks to me that the set of possible heights between the two rectangles are both the same size infinity since the range of both is between 0 and infinity, exclusive.
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Hold up. I can't think of limitless as an upper limit? Could you elaborate on this, it is not sitting right and really breaks my simplified wording of the problem:
Out of these two functions, which upper limit is larger?
They are the same says I! Both upper limits are limitless!
But Cacti, neither has an upper limit...
:confused:

Yes, the height is undefined, but the whole point of this thread is to squable about which undefined height is larger (if at all, as I stand) - get the brain working and learn some math along the way.
 
So, for the last two pages we've basically been arguing over this:

biggerinfinity.png



In other words, we've been arguing about which infinity is bigger? :confused:
 
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