Ustulo
Well-Known Member
Ти си кебаб. Морам те уклонити.
Geometry and the measurement of infinity
- Thread starter lacar1601
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JHawk
Active Member
I feel I could have enjoyed the hell that was calculus if they had included more geometry in our curriculum.
Cacher
Well-Known Member
Can I have in phonetic spelling? Because that makes no sense.
They both have either 2 points or infinite points depending on which type of geometry you want to apply.
You can say that both line segments contain an infinite number of points but each of them only HAS two points.
They height of both will approach infinity but not reach it, as the requirement of a rectangle and the concept of area fail as the edges come together and the rectangle becomes a line segment.
lacar1601
Well-Known Member
By definition of a mathematical limit (which the two of us have discussed before), it is the value that our function of interest converges on as the domain becomes sufficiently close to, though not necessarily reaching, the location of interest.
In this case, I observe the height of the two rectangles as a function of its width. As the width of both rectangles becomes smaller and smaller, the 5-area rectangle's height approaches infinity faster than the 4-area rectangle. Hence, I arrive at the conclusion I draw in my previous post. Granted, the limit of the heights two rectangles are both infinity as their widths go to zero. Infinity is not a real number, so, as Gurw said, I can only draw a qualitative answer for this.
TheGurw
Well-Known Member
We aren't, however, attempting to list the possible points, just determining which has the greater infinity. I believe my argument still stands - that the line with a greater length has a greater amount of points (or a lesser distance to the next point if the distance is greater than the length of the line). My reasoning here is that the basics of math still apply: no matter the distance between points, the line with the greater length will have more points. Extrapolating that to an infinitely small distance between points, the line with the longer length would continue to have a greater amount of points, even though they both have an infinite number of points.
MiniCacti
Well-Known Member
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
'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.
TheGurw
Well-Known Member
The point being, neither can truly reach infinity - since to remain a rectangle, the lines can never truly merge. They may be extremely tall - but the closer the lines get, the greater the difference between the heights of the two rectangles.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
MiniCacti
Well-Known Member
But their limit is infinity. Whatever height rectangle 5 reaches, the other can reach with a shorter width.
EDIT: Another way of looking at it is that they ALWAYS have the same height, and rectangle 4 simply gets "skinnier" more quickly.
TheGurw
Well-Known Member
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.
Pasaria
Well-Known Member
Not quite, but close enough. It's Serbian.
Stratadon
Well-Known Member
Here is my take on the rectangles. When you are comparing the height or length of any two objects, you have to measure them by the same unit.
Let's say we are measuring two humans. Would it be fair to measure human 1 in inches and human 2 in feet? The human measured in inches would have a much larger number associated with his height, but he may actually be shorter than the other human.
When comparing the heights of the rectangles, you must set their width's equal to each other. Through this process, we can say definitively that the rectangle with an area of 5 will always be taller than the rectangle with an area of 4 at any given width. Therefore, no matter what number you choose through infinity, the rectangle with an area of 5 will always be taller given an equal unit of measurement between the two.
Let's say we are measuring two humans. Would it be fair to measure human 1 in inches and human 2 in feet? The human measured in inches would have a much larger number associated with his height, but he may actually be shorter than the other human.
When comparing the heights of the rectangles, you must set their width's equal to each other. Through this process, we can say definitively that the rectangle with an area of 5 will always be taller than the rectangle with an area of 4 at any given width. Therefore, no matter what number you choose through infinity, the rectangle with an area of 5 will always be taller given an equal unit of measurement between the two.
TheGurw
Well-Known Member
*with the exception of infinity itself - since that would mean the rectangles had become single lines -
t7seven7t
Active Member
Infinity has no limit, infinity is not a number. The concept of infinity is that it never ends, and if it never ends you can never perform ordinary mathematical operations on it.
The widths of both rectangles get endlessly closer to - but never actually reach - zero, so both rectangles have an immeasurable height, hence why we term that they are infinitely long. If we were to take the problem into a real world universe rather than the fictitious one where points and lines consume no measurable space, we would be able to describe the problem with real numbers and find that the rectangle with the larger surface area would finish with a greater height when the walls of the rectangles rub up against one another.
The widths of both rectangles get endlessly closer to - but never actually reach - zero, so both rectangles have an immeasurable height, hence why we term that they are infinitely long. If we were to take the problem into a real world universe rather than the fictitious one where points and lines consume no measurable space, we would be able to describe the problem with real numbers and find that the rectangle with the larger surface area would finish with a greater height when the walls of the rectangles rub up against one another.
TheGurw
Well-Known Member
I never said infinity has a limit. I said it exists as the goal which can never be achieved.
For a rectangle to be a rectangle, it must continue to exist as four sides with 90 degree corners. If the rectangles are compressed so far that they have infinite height, that means they've been compressed into a single line and therefore no longer satisfy the requirements of being a rectangle. Therefore the height can NEVER reach infinity and still remain within the bounds of the question. Therefore the rectangle that began with the greater area will continue to have the greater area, and since the width of the rectangles is the same, the height is the determining factor in which rectangle has the greater area. As I bring my case to a close, I'll point out that the rectangle that has the greater area now also has the greater height by the fundamental laws of mathematics, and I challenge you to find a single flaw in this argument; regardless of how tired I am.
Stratadon
Well-Known Member
*with the exception of infinity itself - since that would mean the rectangles had become single lines -
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.
