Investigating Logic Behind Line Segment Lengths

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In summary, the conversation discusses the misconception that two lines with an equal number of points must also have equal lengths. The flaw in this reasoning is that infinite sets behave differently from finite sets and counting their members is counterintuitive. The conversation also mentions the Hilbert Hotel paradox as an example of this concept. Additionally, the conversation explores the idea of limits and how they do not contradict the fact that two lines can have an equal number of points but different lengths.
  • #1
limitkiller
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I am confused ,whats wrong with the reasoning below ?
Untitled.png

assume that the black lines are parallel and red Line segments are defined between the black lines.
for any point on one of the red line segments there exist only one point on the other one (the points that are touched by the lines between the black lines and parallel to them).
so the Line segments have equal lengths.
 
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  • #2
limitkiller said:
I am confused ,whats wrong with the reasoning below ?
Untitled.png

assume that the black lines are parallel and red Line segments are defined between the black lines.
for any point on one of the red line segments there exist only one point on the other one (the points that are touched by the lines between the black lines and parallel to them).
so the Line segments have equal lengths.

Are you saying that the two red lines are equal? That is not true. They are only equal if they both make the same angle with the black lines.

An easy way (besides actually measuring them) is to use the pythagorean theorem on some simple triangles.
 
  • #3
chiro said:
Are you saying that the two red lines are equal? That is not true. They are only equal if they both make the same angle with the black lines.

An easy way (besides actually measuring them) is to use the pythagorean theorem on some simple triangles.

of course i am sure that they are not equal,but i proved(didnt I?) they have same count of points...
doesnt that mean they are equal?
please tell your reasons...:biggrin:
 
  • #4
limitkiller said:
of course i am sure that they are not equal,but i proved(didnt I?) they have same count of points...
No. That proves nothing.
There are an infinite number of points on each line. But you cannot apply real number arithmetic to infinities (such as infinity = infinity).
 
  • #5
What's wrong is that you don't understand what "infinte" means.

Google for the "Hilbert Hotel paradox" for some simple examples of why you idea doesn't work.

Note, the Hilbert "paradox" is only about a countable but infinite number of objects, but your line segments have an uncountable number of points, so it isn't the whole story - but it will get you started thinking the right way about "infinity".
 
  • #6
Every interval on the Real number line has the same "number" of points. This has nothing to do with the length of the interval. For that you need to pick a metric, under the taxi cab metric then I believe you are correct the lines are the same length.
 
  • #7
i don't get you...
if its true :

DaveC426913 said:
No. That proves nothing.
There are an infinite number of points on each line. But you cannot apply real number arithmetic to infinities (such as infinity = infinity).
the Cavalieri's principle would be also wrong...
and lim (x/x) ,x→infinity =1 and lim (2x/x) ,x→infinity =2 would be wrong for the same reason.
I am not saying such thing as :infinity=infinity ,I just say: be cause for every point on one of the lines there exists only one point on the other so they have equal length...
also "Hilbert Hotel paradox" was pretty interesting but i don't thin its the same thing.
 
  • #8
limitkiller,
Yours isn't a stupid question. It's actually very interesting. As others in this thread have pointed out, infinite sets behave in ways very different from finite sets. If you have two finite sets and you can find a one-to-one mapping between the two, the sets are equal in size. For example A = {1, 2, 3} and B = {one, two, three}. There's an obvious mapping between these two sets, and the cardinality of each set is 3.

Infinite sets are different, and counting their members seems counterintuitive. For example, it would seem that the set of positive integers (N = {1, 2, 3, ... }) is larger than the set of positive even integers (E = {2, 4, 6, ...}) since the first set contains all of the numbers in the second set. However, there is a one-to-one map between the two sets, so the cardinality of both sets is the same.

The mapping is
1 --> 2
2 --> 4
3 --> 6
and so on.

Using this mapping, any number in the first set can be seen to have a counterpart in the second set; conversely, given any number in the second set, its counterpart in the first set can be found.
 
  • #9
limitkiller said:
i don't get you...
if its true :


the Cavalieri's principle would be also wrong...
and lim (x/x) ,x→infinity =1 and lim (2x/x) ,x→infinity =2 would be wrong for the same reason.
No, this is different. What these limits are saying is that no matter how large an x you pick, the fractions x/x and 2x/x evaluate to 1 and 2, respectively.
limitkiller said:
I am not saying such thing as :infinity=infinity ,I just say: be cause for every point on one of the lines there exists only one point on the other so they have equal length...
No, what this says is that both lines have the same number of points. This is NOT the same thing as saying that both lines have the same length.

By your reasoning, every triangle would have to be isosceles (two equal sides). Consider a right triangle ABC, with AB perpendicular to AC, and with altitude AB and base AC. If you construct a line that is parallel to the base running from AB (altitude) to BC (hypotenuse), each point on the altitude corresponds to a point on the hypotenuse, so by your reasoning, the altitude has the same number of points as the hypotenuse (true), and hence, has the same length. Obviously this conclusion can't be true, which means there is a flaw in your reasoning about sets with an equal number of points implying that the sets have equal length.
limitkiller said:
also "Hilbert Hotel paradox" was pretty interesting but i don't thin its the same thing.
 
  • #10
Mark44 said:
No, this is different. What these limits are saying is that no matter how large an x you pick, the fractions x/x and 2x/x evaluate to 1 and 2, respectively.
No, what this says is that both lines have the same number of points. This is NOT the same thing as saying that both lines have the same length.
yeah you are right,the limit thing is a little different but what about cavalieri's principle ??
Mark44 said:
No, what this says is that both lines have the same number of points. This is NOT the same thing as saying that both lines have the same length.

By your reasoning, every triangle would have to be isosceles (two equal sides). Consider a right triangle ABC, with AB perpendicular to AC, and with altitude AB and base AC. If you construct a line that is parallel to the base running from AB (altitude) to BC (hypotenuse), each point on the altitude corresponds to a point on the hypotenuse, so by your reasoning, the altitude has the same number of points as the hypotenuse (true), and hence, has the same length. Obviously this conclusion can't be true, which means there is a flaw in your reasoning about sets with an equal number of points implying that the sets have equal length.
I know my reasoning is wrong(actually I don't know how do I know:biggrin:) if it was true all the lines segments were equal...
but seriously it (my reasoning) is the same thing with Cavalieri's principle isn't it?
 
  • #11
Yes, limitkiller, you are absolutely correct. A similar problem WILL arise in Cavalieri's principle. But the point is that Cavalieri was not correct, he himself realized this.

Cavalieri stated his principle around 1630, this is a time where mathematics is not rigourous and often wrong. Many of the papers of his time would be considered seriously flawed these days. So saying that a problem will arise in Cavalieri's principle is nothing new, this is already known.

Luckily, Cavalieri's method can be saved (in a lot of cases). It was Fubini and Tonelli who made the method rigourous and correct.
 
  • #12
Mark44 said:
limitkiller,
Yours isn't a stupid question. It's actually very interesting. As others in this thread have pointed out, infinite sets behave in ways very different from finite sets. If you have two finite sets and you can find a one-to-one mapping between the two, the sets are equal in size. For example A = {1, 2, 3} and B = {one, two, three}. There's an obvious mapping between these two sets, and the cardinality of each set is 3.

Infinite sets are different, and counting their members seems counterintuitive. For example, it would seem that the set of positive integers (N = {1, 2, 3, ... }) is larger than the set of positive even integers (E = {2, 4, 6, ...}) since the first set contains all of the numbers in the second set. However, there is a one-to-one map between the two sets, so the cardinality of both sets is the same.

The mapping is
1 --> 2
2 --> 4
3 --> 6
and so on.

Using this mapping, any number in the first set can be seen to have a counterpart in the second set; conversely, given any number in the second set, its counterpart in the first set can be found.
than you
wow, two sets have one to one relation and in the same time they have one to two :biggrin: relation
1,2 --> 2
3,4 --> 4
5,6 --> 6
though I am not able to understand it...
Mark44 said:
Yes, limitkiller, you are absolutely correct. A similar problem WILL arise in Cavalieri's principle. But the point is that Cavalieri was not correct, he himself realized this.

Cavalieri stated his principle around 1630, this is a time where mathematics is not rigourous and often wrong. Many of the papers of his time would be considered seriously flawed these days. So saying that a problem will arise in Cavalieri's principle is nothing new, this is already known.

Luckily, Cavalieri's method can be saved (in a lot of cases). It was Fubini and Tonelli who made the method rigourous and correct.
this means i should study more than i do now...:grumpy:
and now i understand what does Hilbert Hotel paradox have to do with this.
 
  • #13
I think the Hilbert Hotel was simply brought up to show that infinity is a very strange thing! You shouldn't reason with infinity as you reason with ordinary numbers, you need to be very, very, very careful with infinity!

Your OP also shows this. The question posed in the OP is only a paradox if you reason with infinity like you do with normal numbers. So you can't really do this!
 
  • #14
Also limitkiller I would have a look at micromass' blog which contains a description of the Banach-Tarski paradox. This has the same kind of behavior which you are describing and might be informative for you.
 
  • #15
chiro said:
Also limitkiller I would have a look at micromass' blog which contains a description of the Banach-Tarski paradox. This has the same kind of behavior which you are describing and might be informative for you.
okay,thank you



one last thing, in Hilbert Hotel paradox did we really accommodate every one in the hotel?
i mean there was always someone who didnt have a room an we just changed that someone.
 
  • #16
limitkiller said:
one last thing, in Hilbert Hotel paradox did we really accommodate every one in the hotel?
i mean there was always someone who didnt have a room an we just changed that someone.

That's the beauty of it, everybody will have a room! On the other hand, you are correct: we are just changing the person without a room. It's weird because it's a paradox :biggrin:
 
  • #17
micromass said:
That's the beauty of it, everybody will have a room! On the other hand, you are correct: we are just changing the person without a room. It's weird because it's a paradox :biggrin:
:yuck:
then if we have a room and two people we are able give each a different room(by giving ones room to the other and repeating it infinitely) ??
 
  • #18
Yes! Even if you have a room and infinitely many people, you can give them all different rooms!
 
  • #19
micromass said:
Yes! Even if you have a room and infinitely many people, you can give them all different rooms!

heeeeey everybody will have a room!:biggrin:
 
  • #20
micromass said:
Yes! Even if you have a room and infinitely many people, you can give them all different rooms!
so why the hotel had infinitely many rooms? when finite number of rooms would do?
 
  • #21
limitkiller said:
i don't get you...
if its true :


the Cavalieri's principle would be also wrong...
and lim (x/x) ,x→infinity =1 and lim (2x/x) ,x→infinity =2 would be wrong for the same reason.
I am not saying such thing as :infinity=infinity ,I just say: be cause for every point on one of the lines there exists only one point on the other so they have equal length...
also "Hilbert Hotel paradox" was pretty interesting but i don't thin its the same thing.
Cavalieri's principle talks about lengths, areas, and volumes. It says nothing about the "number of points" on a line, area, or region and has no connection with this problem.
 
  • #22
limitkiller said:
so why the hotel had infinitely many rooms? when finite number of rooms would do?
There are at least a couple of parts to this problem. All of the rooms in the Infinite Hotel are assumed to be occupied by an infinite number of guests. In one part of the problem, a traveler comes to the front desk, asking for a room. The desk clerk tells the traveler that there are no empty rooms, but the clerk arrives at a solution by realizing that if he can get each of the current guests to move to the next room (the room with a number that is 1 more than the current room), then room 1 will be empty, and the traveler can take that room. If there are only a finite number of rooms in the hotel, this won't be a solution.

In another part of the problem, a bus with an infinite number of passengers pulls up to the hotel, and each passenger wants a room. After some time, the desk clerk realizes that if he can get each of the current guests to move to a room whose number is twice the number of his/her current room, then all of the odd-numbered rooms will be empty, and the bus passengers can take those rooms. This scheme also wouldn't work if the hotel had only a finite number of rooms.
 
  • #23
I want to say something here that's relevant to the title of the thread. I would help with the question, but that's out of my depth right now. However, I just wanted to say...

This is not a stupid question. There are no stupid questions. If you feel you do not understand something, than going for help is quite simply the best thing you can do. Not stupid at all.

As long as you come with a willingness to learn, that is.
 
  • #24
Mark44 said:
There are at least a couple of parts to this problem. All of the rooms in the Infinite Hotel are assumed to be occupied by an infinite number of guests. In one part of the problem, a traveler comes to the front desk, asking for a room. The desk clerk tells the traveler that there are no empty rooms, but the clerk arrives at a solution by realizing that if he can get each of the current guests to move to the next room (the room with a number that is 1 more than the current room), then room 1 will be empty, and the traveler can take that room. If there are only a finite number of rooms in the hotel, this won't be a solution.

In another part of the problem, a bus with an infinite number of passengers pulls up to the hotel, and each passenger wants a room. After some time, the desk clerk realizes that if he can get each of the current guests to move to a room whose number is twice the number of his/her current room, then all of the odd-numbered rooms will be empty, and the bus passengers can take those rooms. This scheme also wouldn't work if the hotel had only a finite number of rooms.
thank you!



Char. Limit, I make lots of mistakes and my questions are risen from those mistakes . so they are not really questions ,just mistakes...



HallsofIvy,as far as I know a principle is a fundamental truth that can be followed, how do you else follow it?
 
  • #25
The only mistakes that you can be blamed for are the ones you refuse to learn from.
 
  • #26
micromass said:
Yes! Even if you have a room and infinitely many people, you can give them all different rooms!

Depends what size of infinity :P

With the analogy of 1 -> 2
2 -> 4 etc.
there is one a bit closer to your problem.

Think about the line segment of all points (inclusive) from 0 up to 1. Now apply the x2 map to it (i.e. send 1 to 2, 0 to 0, pi/4 to pi/2, etc.) Now the number of points must be the same: to every point in the new line segment, there is precisely 1 point from the old line segment and every point from the old line segment goes somewhere, the x2 map "pairs up" all of the points! But after the x2 map, the new line has twice the length.

Intuitively, we think "where do these new points come from?" because we seem to have more, but the truth is that we don't!
 
  • #29
Jamma said:
Depends what size of infinity :P

With the analogy of 1 -> 2
2 -> 4 etc.
there is one a bit closer to your problem.

Think about the line segment of all points (inclusive) from 0 up to 1. Now apply the x2 map to it (i.e. send 1 to 2, 0 to 0, pi/4 to pi/2, etc.) Now the number of points must be the same: to every point in the new line segment, there is precisely 1 point from the old line segment and every point from the old line segment goes somewhere, the x2 map "pairs up" all of the points! But after the x2 map, the new line has twice the length.

Intuitively, we think "where do these new points come from?" because we seem to have more, but the truth is that we don't!
if there are same number of points ,then why 2/1=2 it should be 1 !
I think what HallsofIvy said might be true,after all its a principle... what do you think?
 
  • #30
HallsofIvy said:
Cavalieri's principle talks about lengths, areas, and volumes. It says nothing about the "number of points" on a line, area, or region and has no connection with this problem.

limitkiller said:
if there are same number of points ,then why 2/1=2 it should be 1 !
I think what HallsofIvy said might be true,after all its a principle... what do you think?
You are still conflating the ideas about length and number of points, even though there is no direct connection between them. The interval [0 , 2] has length 2, and the interval [0, 10] has length 10. Even though these two intervals have different lengths, each one has exactly the same number of points as the other. All that's needed is a one-to-one mapping from one set to the other. Here's that mapping f(x) = 5x, where x is a number in the first interval.
 
  • #32
My mathematically profane answer: The points of the second line are bigger :smile:
If you say for any point of short(?) line there [STRIKE]exist[/STRIKE] correspond a point on the long(?) line, it means that the points of the latter are the projections of the shorter line's points. He he.
 
  • #33
mireazma said:
My mathematically profane answer: The points of the second line are bigger :smile:
If you say for any point of short(?) line there [STRIKE]exist[/STRIKE] correspond a point on the long(?) line, it means that the points of the latter are the projections of the shorter line's points. He he.

Haha. No. Points are points.

The folly in the argument is the part where he equates one infinity with another. You can't treat infinities arithmetically.
 
  • #34
Yep, as soon as you realize that the number of points has nothing to do with the length of the line, it's all very obvious.

I can see where the confusion comes from though; in real life, even if we have billions upon billions of points, say all arranged in a little line, like a line of atoms, then we can still stretch them, but the distances will actually increase between the atoms, and if we increase far enough, then we will eventually "see" the gaps. So no matter how close you approximate the infinity of points, even if you use trillions of them all bunched up, it still never has the same properties of what goes on when you have infinitely many.
 

1. What is the purpose of investigating the logic behind line segment lengths?

The purpose of investigating the logic behind line segment lengths is to understand the relationship between different line segments and the factors that affect their length. This can help in solving geometric problems and making accurate measurements in various fields such as engineering, architecture, and physics.

2. How do you determine the length of a line segment?

The length of a line segment can be determined by using the Pythagorean theorem, which states that the square of the hypotenuse (longest side) of a right triangle is equal to the sum of the squares of the other two sides. Another method is to use a ruler or measuring tape to directly measure the distance between the two endpoints of the line segment.

3. What factors can affect the length of a line segment?

The length of a line segment can be affected by various factors such as the angle at which it is measured, the precision of the measuring tool, and any errors in measurement. Additionally, the length of a line segment can also be affected by the position and orientation of the line in relation to other objects or lines.

4. How can investigating line segment lengths be useful in real-world applications?

Investigating line segment lengths can be useful in various real-world applications such as construction, surveying, and design. Understanding the logic behind line segment lengths can help in accurately measuring and creating geometric shapes and structures, as well as solving problems related to distance and space.

5. What are some common methods used to investigate the logic behind line segment lengths?

Some common methods used to investigate the logic behind line segment lengths include geometric proofs, mathematical equations, and visual representations such as diagrams and graphs. Computer simulations and modeling can also be used to analyze and understand the relationship between different line segments and their lengths.

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