Fun with Infinity

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Fun with Infinity
« Reply #30 on: March 28, 2006, 03:09:54 PM »
Quote from: "the grim squeaker"
nice
but theres one floor if all the rooms are full and no one can leave then poeple will be perpetually move rooms and die of exuation

We are not taking these things into consideration.  The only reason a hotel is being used is to make it a little less mundane.  Don't take the convenience or well-being of the occupants into account.
nd that, my liege, is how we know the Earth to be banana shaped.

Fun with Infinity
« Reply #31 on: March 28, 2006, 03:13:21 PM »
Okay, you've gotten your new tenant into the hotel.  Unfortunately, a tourbus pulls into the driveway.  The tour guide gets out and tells you he has an infinite number of people on the bus.  They are tired and cranky and need someplace to sleep.  He won't take no for an answer.  How do you fit these people into your hotel?
nd that, my liege, is how we know the Earth to be banana shaped.

Fun with Infinity
« Reply #32 on: March 29, 2006, 07:48:31 AM »
Quote from: "Erasmus"
Quote from: "the grim squeaker"
nice
but theres one floor if all the rooms are full and no one can leave then poeple will be perpetually move rooms and die of exuation


No individual person will perpetually move; each guest only has to walk the distance from his door to the next door, which is presumably finite.  An infinite number of people move, yes, and it takes infinite time for them all to move (since each one starts a finite time after the previous one), unless you can broadcast the message to all rooms at once.

If you can broadcast, each leaves his room at the same time, and each arrives at his new room at the same time.  Finite-time broadcast to infinite distances violates information theory, but whatever, so do infinite hotels :)
(but what happens is an infinite number of new guests arrive at the doors and only 200 guests leave?-grim)
Anyway every guest is guaranteed to be able to move into the next room, so nobody dies of exhaustion.(unless there room is the constant-daylight-desrt-no-indoor-plumbing-and-no-way-out-room-hahahahaha-take-that-you-bastard.-grim)

-Erasmus

Fun with Infinity
« Reply #33 on: March 29, 2006, 07:49:24 AM »
Quote from: "Pesto"
Okay, you've gotten your new tenant into the hotel.  Unfortunately, a tourbus pulls into the driveway.  The tour guide gets out and tells you he has an infinite number of people on the bus.  They are tired and cranky and need someplace to sleep.  He won't take no for an answer.  How do you fit these people into your hotel?

shoots some guests?

Fun with Infinity
« Reply #34 on: March 29, 2006, 01:44:05 PM »
No.
nd that, my liege, is how we know the Earth to be banana shaped.

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Erasmus

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Fun with Infinity
« Reply #35 on: March 29, 2006, 06:22:33 PM »
Okay, I've given it a day or so.  You tell the person in room N to move to room 2*N; now you have infinitely many vacant rooms, but people die from exhaustion on their way to them.  Problem solved!
Why did the chicken cross the Möbius strip?

Fun with Infinity
« Reply #36 on: March 29, 2006, 10:44:26 PM »
Quote from: "Erasmus"
Okay, I've given it a day or so.  You tell the person in room N to move to room 2*N; now you have infinitely many vacant rooms, but people die from exhaustion on their way to them.  Problem solved!

Thats what i said early, as there is an infinte gap between rooms so unless you had a device that would instantanitusly transport people, if not then there would be no way the get to the outliying rooms.-grim

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Erasmus

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Fun with Infinity
« Reply #37 on: March 30, 2006, 12:54:50 AM »
Quote from: "the grim squeaker"
Thats what i said early, as there is an infinte gap between rooms so unless you had a device that would instantanitusly transport people, if not then there would be no way the get to the outliying rooms.-grim


Er no, there would never be an infinite gap.  There would be arbitrarily large gaps, but each would be finite, and so would take only finite time to get from the starting point to the finishing point.  Not a problem, really.  Just cryogenically freeze the people for whom it will take, say, 50 years or more for them to reach their room.

But again, as Pesto mentioned, that sort of difficulty is not really the crux of the problem.  What we're really doing is demonstrating an interesting feature of infinite sets: you can take a proper subset of them -- that is, you can remove elements -- and the size of the set will not have changed.  You do thing by putting the elements in a one-to-one correspondence.  In the case of the hotel, for example, the one-to-one correspondence is N<-->2N.  The fact that you can do that shows that there's the same number of even integers as there are positive integers, which is kinda counterintuitive.

Another interesting result, along the same lines, is that there are the same number of real numbers between 0 and 1 as there are real numbers in total.  I'll let you try to put these numbers in a one-to-one correspondence with all real numbers.  The next one to try after that is that there are the same number of rational numbers (numbers of the form a/b, where a and b are integers and b is not 0) as there are postive integers.  Next, and hardest, is that there are more real numbers than there are positive integers: you do that by showing that if you put positive integers and real numbers in a one-to-one correspondence, you'll have some real numbers left over, always.  The reason that's cool is it shows that there is more than one infinity, and that some infinities are larger than others.  Mind-blowing.

-Erasmus
Why did the chicken cross the Möbius strip?

Fun with Infinity
« Reply #38 on: March 30, 2006, 06:36:20 AM »
Quote from: "Erasmus"
Quote from: "the grim squeaker"
Thats what i said early, as there is an infinte gap between rooms so unless you had a device that would instantanitusly transport people, if not then there would be no way the get to the outliying rooms.-grim


Er no, there would never be an infinite gap.  There would be arbitrarily large gaps, but each would be finite, and so would take only finite time to get from the starting point to the finishing point.  Not a problem, really.  Just cryogenically freeze the people for whom it will take, say, 50 years or more for them to reach their room.

But again, as Pesto mentioned, that sort of difficulty is not really the crux of the problem.  What we're really doing is demonstrating an interesting feature of infinite sets: you can take a proper subset of them -- that is, you can remove elements -- and the size of the set will not have changed.  You do thing by putting the elements in a one-to-one correspondence.  In the case of the hotel, for example, the one-to-one correspondence is N<-->2N.  The fact that you can do that shows that there's the same number of even integers as there are positive integers, which is kinda counterintuitive.

Another interesting result, along the same lines, is that there are the same number of real numbers between 0 and 1 as there are real numbers in total.  I'll let you try to put these numbers in a one-to-one correspondence with all real numbers.  The next one to try after that is that there are the same number of rational numbers (numbers of the form a/b, where a and b are integers and b is not 0) as there are postive integers.  Next, and hardest, is that there are more real numbers than there are positive integers: you do that by showing that if you put positive integers and real numbers in a one-to-one correspondence, you'll have some real numbers left over, always.  The reason that's cool is it shows that there is more than one infinity, and that some infinities are larger than others.  Mind-blowing.

-Erasmus

the probelm with cryogenics is when you unfreeze them the water will expand thus making the person go pop-grim

Fun with Infinity
« Reply #39 on: March 30, 2006, 07:43:22 AM »
Quote from: "Erasmus"
Okay, I've given it a day or so.  You tell the person in room N to move to room 2*N; now you have infinitely many vacant rooms, but people die from exhaustion on their way to them.  Problem solved!

Like I said, we're not worrying about exhaustion, or any of those other pesky inconveniences.

Well, you were able to accomodate the bus, but just as luck would have it, another bus with an infinite number of people pulls into the driveway.  You think, "No problem."  As it turns out, there is a convention in town, and this isn't the only bus.  In fact, there is an infinite number of busses, each with an infinite number of people.  How will you get all these people into your hotel?
nd that, my liege, is how we know the Earth to be banana shaped.

Fun with Infinity
« Reply #40 on: March 30, 2006, 08:45:20 AM »
Quote from: "Pesto"
Quote from: "Erasmus"
Okay, I've given it a day or so.  You tell the person in room N to move to room 2*N; now you have infinitely many vacant rooms, but people die from exhaustion on their way to them.  Problem solved!

Like I said, we're not worrying about exhaustion, or any of those other pesky inconveniences.

Well, you were able to accomodate the bus, but just as luck would have it, another bus with an infinite number of people pulls into the driveway.  You think, "No problem."  As it turns out, there is a convention in town, and this isn't the only bus.  In fact, there is an infinite number of busses, each with an infinite number of people.  How will you get all these people into your hotel?

so tell them to got to the top floor(a hotel with an infinte number of rooms) probelem sloved of course make them pay up front, as they would die of old age on the 100000000th floor assuming they were born on the 20th or higher floor

Fun with Infinity
« Reply #41 on: March 30, 2006, 09:54:48 AM »
No.
nd that, my liege, is how we know the Earth to be banana shaped.

Fun with Infinity
« Reply #42 on: March 30, 2006, 10:35:01 AM »
Quote from: "Pesto"
No.

Okay now you're just being difficult. -grim

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Erasmus

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Fun with Infinity
« Reply #43 on: March 30, 2006, 11:41:19 AM »
Quote from: "Pesto"
Well, you were able to accomodate the bus, but just as luck would have it, another bus with an infinite number of people pulls into the driveway.  You think, "No problem."  As it turns out, there is a convention in town, and this isn't the only bus.  In fact, there is an infinite number of busses, each with an infinite number of people.  How will you get all these people into your hotel?


To make things simple, move everybody currently in the hotel hotel into a bus (we already know how to do this).  Now, have the busses all park side-by-side at the hotel's curb, so you effectively have a big grid of bus seats.  Again, just to make things simple, assume that each bus has one long column of seats (people are seated in single-file).  Now you can just assign people to rooms by walking a zigzag path among the busses.  Every time you get to the "top" of the grid, you walk right; every time you get to the left side, you walk down.  As you go, give each rider a room number one higher than the number you gave to the previous rider (put the first rider of the first bus in room 1).

To illustrate:



You just walk along the red path, handing out numbers.  Alternatively, you could have the riders file off the busses by following the red arrow, and fill the rooms in the order in which they arrive in the hotel.

-Erasmus
Why did the chicken cross the Möbius strip?

Fun with Infinity
« Reply #44 on: March 30, 2006, 12:05:59 PM »
or you could send them to an alternate universe with an infinite number of empty identical hotels

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Erasmus

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« Reply #45 on: March 30, 2006, 12:53:51 PM »
Quote from: "the grim squeaker"
or you could send them to an alternate universe with an infinite number of empty identical hotels


You could, but you don't need to -- there's enough room in just the one hotel for everybody.
Why did the chicken cross the Möbius strip?

Fun with Infinity
« Reply #46 on: March 30, 2006, 12:57:57 PM »
that is a point its a hotel with and infinite number of rooms they cant all be full

Fun with Infinity
« Reply #47 on: March 30, 2006, 03:05:17 PM »
They can be if there is an infinite number of people.
nd that, my liege, is how we know the Earth to be banana shaped.

Fun with Infinity
« Reply #48 on: March 30, 2006, 03:09:25 PM »
Quote from: "Erasmus"
Quote from: "Pesto"
Well, you were able to accomodate the bus, but just as luck would have it, another bus with an infinite number of people pulls into the driveway.  You think, "No problem."  As it turns out, there is a convention in town, and this isn't the only bus.  In fact, there is an infinite number of busses, each with an infinite number of people.  How will you get all these people into your hotel?


To make things simple, move everybody currently in the hotel hotel into a bus (we already know how to do this).  Now, have the busses all park side-by-side at the hotel's curb, so you effectively have a big grid of bus seats.  Again, just to make things simple, assume that each bus has one long column of seats (people are seated in single-file).  Now you can just assign people to rooms by walking a zigzag path among the busses.  Every time you get to the "top" of the grid, you walk right; every time you get to the left side, you walk down.  As you go, give each rider a room number one higher than the number you gave to the previous rider (put the first rider of the first bus in room 1).

To illustrate:



You just walk along the red path, handing out numbers.  Alternatively, you could have the riders file off the busses by following the red arrow, and fill the rooms in the order in which they arrive in the hotel.

-Erasmus

The pic is broken, but I know what you're talking about.  You're basically using the mapping of the rational numbers onto the natural numbers.  That's not the solution I first heard, but it will work.

Here's the solution I first heard.  Move all the people in the hotel into the prime numbered rooms.  Assign each bus a prime number.  Give each bus the rooms numbered p^n, for n >= 2.  Then, not only have you put all the people from busses in rooms, but you have an infinite number of empty rooms left over.
nd that, my liege, is how we know the Earth to be banana shaped.

?

Erasmus

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Fun with Infinity
« Reply #49 on: March 30, 2006, 06:30:06 PM »
Quote from: "Pesto"
The pic is broken, but I know what you're talking about.


If by broken you mean, "has a flaw", could you tell me what it is?  If you mean, "Won't download," er, works fine from this computer, which isn't the one I created/uploaded the figure from.

Quote
You're basically using the mapping of the rational numbers onto the natural numbers.


Yep.

Quote
Here's the solution I first heard.  Move all the people in the hotel into the prime numbered rooms.  Assign each bus a prime number.  Give each bus the rooms numbered p^n, for n >= 2.  Then, not only have you put all the people from busses in rooms, but you have an infinite number of empty rooms left over.


Pretty swank.

-Erasmus
Why did the chicken cross the Möbius strip?

Fun with Infinity
« Reply #50 on: March 31, 2006, 06:45:51 AM »
the word infinite means never ending so there is a never ending number of rooms for a never ending stream of people meaning that you can just keep going forever
-grim

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joffenz

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Fun with Infinity
« Reply #51 on: March 31, 2006, 02:32:37 PM »
What happens if you take someone out a room? Is the hotel full?

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Erasmus

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« Reply #52 on: March 31, 2006, 03:05:30 PM »
Quote from: "cheesejoff"
What happens if you take someone out a room? Is the hotel full?


Depends on how you define full.

If "full" = "# occuppied rooms = # rooms", then yes, it's still full.

If "full" = "# of unoccuppied rooms = 0", then no, it's not full.

If "full" = "You can't put any more people," then no, it's not full.
Why did the chicken cross the Möbius strip?

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WTF

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Fun with Infinity
« Reply #53 on: March 31, 2006, 06:42:25 PM »
The only reason people get hung up being unable to believe that 0.9999...=1 (exactly) and .33333...=1/3 (exactly) is because we use a base 10 number system.

There isn't any special property about 0.333333...it's a concrete value just like a single digit.  But because of the properties of using a decimal number system, certain values (such as 1/3) can't be represented exactly without using an infinite number of digits.  If we used a a base 9 system for example, "one third" would be 0.3.

0.3+0.3+0.3 = 1.  No one would argue that it isn't exactly one-third if we were all used to base9.  Yet if we transition over to a base 10 system, it gets represented differently.  0.3 becomes 0.3333333....  But they are the exact same values.

0.9999...=1 and 0.3333...= 1/3.  Exactly.

Fun with Infinity
« Reply #54 on: April 23, 2006, 04:15:04 AM »
Theres an infinite number of busses, with an infinite number of people, so it's infinity multiplied by infinity and that is equal to infinity, so I just cut the workload by infinity!

Now, theres only infinite people, which is easier than an infinite amount of busses holding infinite people each, to solve it, you just need some logic.

i x i = i
i = hotel space
i already in hotel
i + i = i
so theres over all i
theres i in the hotel, so everybody must already be in!

So everybody got into the hotel within a nanosecond, job well done!

?

MAL

Fun with Infinity
« Reply #55 on: May 02, 2006, 01:43:49 PM »
Quote from: "the grim squeaker"
Quote from: "Erasmus"
Quote from: "the grim squeaker"
Thats what i said early, as there is an infinte gap between rooms so unless you had a device that would instantanitusly transport people, if not then there would be no way the get to the outliying rooms.-grim


Er no, there would never be an infinite gap.  There would be arbitrarily large gaps, but each would be finite, and so would take only finite time to get from the starting point to the finishing point.  Not a problem, really.  Just cryogenically freeze the people for whom it will take, say, 50 years or more for them to reach their room.

But again, as Pesto mentioned, that sort of difficulty is not really the crux of the problem.  What we're really doing is demonstrating an interesting feature of infinite sets: you can take a proper subset of them -- that is, you can remove elements -- and the size of the set will not have changed.  You do thing by putting the elements in a one-to-one correspondence.  In the case of the hotel, for example, the one-to-one correspondence is N<-->2N.  The fact that you can do that shows that there's the same number of even integers as there are positive integers, which is kinda counterintuitive.

Another interesting result, along the same lines, is that there are the same number of real numbers between 0 and 1 as there are real numbers in total.  I'll let you try to put these numbers in a one-to-one correspondence with all real numbers.  The next one to try after that is that there are the same number of rational numbers (numbers of the form a/b, where a and b are integers and b is not 0) as there are postive integers.  Next, and hardest, is that there are more real numbers than there are positive integers: you do that by showing that if you put positive integers and real numbers in a one-to-one correspondence, you'll have some real numbers left over, always.  The reason that's cool is it shows that there is more than one infinity, and that some infinities are larger than others.  Mind-blowing.

-Erasmus

the probelm with cryogenics is when you unfreeze them the water will expand thus making the person go pop-grim


The problem with that, is that water expands when it freezes, not when it unfrezees - It actually is the only (? - or one of the very few) materials that do so.

So the people would explode during the cryogenisation and not during the unfreezing process

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MAL

Fun with Infinity
« Reply #56 on: May 02, 2006, 01:46:46 PM »
Quote from: "tigerhawkvok"
Besides, math is correct by definition.  Its an analytical truth.


I agree with the fact that 0,9999.... = 1, but just let me redirect you towards Gödels' works, which logically proved that math isn't actually 'correct by definition'

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Erasmus

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« Reply #57 on: May 02, 2006, 04:41:11 PM »
Quote from: "MAL"
... let me redirect you towards Gödels' works, which logically proved that math isn't actually 'correct by definition'


Care to explain that?  Oh, and, you can pretend I'm at least vaguely familiar with Gödel's theorems.

-Erasmus
Why did the chicken cross the Möbius strip?

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MAL

Fun with Infinity
« Reply #58 on: May 02, 2006, 08:26:13 PM »
Just saying that Gödel said that no logical system can be both complete and consistent.  My understanding of it is that you can't base yourself on math to say that math is right, and, that said, you can't just say that math is absolutely correct, whatever the context.

I only posted that 'cause I didn't like the "math is correct, no matter what - so I'm absolutely right too" argument  :wink:

Fun with Infinity
« Reply #59 on: May 03, 2006, 08:00:24 AM »
i'm pretty sure that 0.333... = 1/3 is an approximation and not exact.

While 0.999... might as well be considered 1 or 9/9 in some cases, it's still an approximation and not exact.