The Infinite Hotel Paradox - Jeff Dekofsky
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0:07 - 0:08In the 1920's,
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0:08 - 0:10the German mathematician David Hilbert
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0:10 - 0:12devised a famous thought experiment
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0:12 - 0:14to show us just how hard it is
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0:14 - 0:18to wrap our minds
around the concept of infinity. -
0:18 - 0:22Imagine a hotel with an infinite
number of rooms -
0:22 - 0:24and a very hardworking night manager.
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0:25 - 0:28One night, the Infinite Hotel
is completely full, -
0:28 - 0:31totally booked up
with an infinite number of guests. -
0:31 - 0:34A man walks into the hotel
and asks for a room. -
0:34 - 0:35Rather than turn him down,
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0:35 - 0:38the night manager decides
to make room for him. -
0:38 - 0:39How?
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0:39 - 0:42Easy, he asks the guest in room number 1
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0:42 - 0:44to move to room 2,
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0:44 - 0:46the guest in room 2 to move to room 3,
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0:46 - 0:47and so on.
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0:47 - 0:50Every guest moves from room number "n"
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0:50 - 0:52to room number "n+1".
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0:53 - 0:55Since there are an infinite
number of rooms, -
0:55 - 0:57there is a new room
for each existing guest. -
0:57 - 1:00This leaves room 1 open
for the new customer. -
1:00 - 1:01The process can be repeated
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1:01 - 1:04for any finite number of new guests.
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1:04 - 1:08If, say, a tour bus unloads
40 new people looking for rooms, -
1:08 - 1:10then every existing guest just moves
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1:10 - 1:11from room number "n"
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1:11 - 1:14to room number "n+40",
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1:14 - 1:16thus, opening up the first 40 rooms.
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1:17 - 1:19But now an infinitely large bus
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1:19 - 1:22with a countedly infinite
number of passengers -
1:22 - 1:24pulls up to rent rooms.
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1:24 - 1:26Countedly infinite is the key.
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1:26 - 1:29Now, the infinite bus
of infinite passengers -
1:29 - 1:31perplexes the night manager at first,
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1:31 - 1:32but he realizes there's a way
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1:32 - 1:33to place each new person.
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1:33 - 1:36He asks the guest in room 1
to move to room 2. -
1:36 - 1:39He then asks the guest in room 2
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1:39 - 1:40to move to room 4,
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1:40 - 1:43the guest in room 3 to move to room 6,
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1:43 - 1:44and so on.
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1:44 - 1:47Each current guest moves
from room number "n" -
1:47 - 1:49to room number "2n" --
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1:51 - 1:54filling up only the infinite
even-numbered rooms. -
1:54 - 1:56By doing this, he has now emptied
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1:56 - 1:59all of the infinitely many
odd-numbered rooms, -
1:59 - 2:03which are then taken by the people
filing off the infinite bus. -
2:03 - 2:07Everyone's happy and the hotel's business
is booming more than ever. -
2:07 - 2:10Well, actually, it is booming
exactly the same amount as ever, -
2:10 - 2:13banking an infinite number
of dollars a night. -
2:14 - 2:16Word spreads about this incredible hotel.
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2:16 - 2:19People pour in from far and wide.
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2:19 - 2:21One night, the unthinkable happens.
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2:21 - 2:23The night manager looks outside
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2:23 - 2:28and sees an infinite line
of infinitely large buses, -
2:28 - 2:30each with a countedly infinite
number of passengers. -
2:30 - 2:31What can he do?
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2:31 - 2:34If he cannot find rooms for them,
the hotel will lose out -
2:34 - 2:36on an infinite amount of money,
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2:36 - 2:38and he will surely lose his job.
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2:38 - 2:42Luckily, he remembers
that around the year 300 B.C.E., -
2:42 - 2:45Euclid proved that there
is an infinite quantity -
2:45 - 2:47of prime numbers.
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2:47 - 2:50So, to accomplish this
seemingly impossible task -
2:50 - 2:52of finding infinite beds
for infinite buses -
2:52 - 2:54of infinite weary travelers,
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2:54 - 2:57the night manager assigns
every current guest -
2:57 - 2:59to the first prime number, 2,
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2:59 - 3:02raised to the power
of their current room number. -
3:02 - 3:05So, the current occupant of room number 7
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3:05 - 3:08goes to room number 2^7,
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3:08 - 3:09which is room 128.
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3:10 - 3:14The night manager then takes the people
on the first of the infinite buses -
3:14 - 3:16and assigns them to the room number
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3:16 - 3:18of the next prime, 3,
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3:18 - 3:22raised to the power of their seat
number on the bus. -
3:22 - 3:25So, the person in seat
number 7 on the first bus -
3:25 - 3:28goes to room number 3^7
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3:28 - 3:32or room number 2,187.
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3:32 - 3:34This continues for all of the first bus.
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3:34 - 3:36The passengers on the second bus
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3:36 - 3:39are assigned powers of the next prime, 5.
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3:39 - 3:41The following bus, powers of 7.
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3:42 - 3:43Each bus follows:
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3:43 - 3:45powers of 11, powers of 13,
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3:45 - 3:47powers of 17, etc.
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3:47 - 3:49Since each of these numbers
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3:49 - 3:51only has 1 and the natural number powers
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3:51 - 3:53of their prime number base as factors,
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3:53 - 3:55there are no overlapping room numbers.
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3:55 - 3:58All the buses' passengers
fan out into rooms -
3:58 - 4:01using unique room-assignment schemes
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4:01 - 4:03based on unique prime numbers.
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4:04 - 4:06In this way, the night
manager can accommodate -
4:06 - 4:08every passenger on every bus.
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4:08 - 4:11Although, there will be
many rooms that go unfilled, -
4:11 - 4:12like room 6,
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4:12 - 4:15since 6 is not a power
of any prime number. -
4:15 - 4:18Luckily, his bosses
weren't very good in math, -
4:18 - 4:19so his job is safe.
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4:20 - 4:22The night manager's strategies
are only possible -
4:22 - 4:27because while the Infinite Hotel
is certainly a logistical nightmare, -
4:27 - 4:30it only deals with the lowest
level of infinity, -
4:30 - 4:34mainly, the countable infinity
of the natural numbers, -
4:34 - 4:371, 2, 3, 4, and so on.
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4:37 - 4:41Georg Cantor called this level
of infinity aleph-zero. -
4:41 - 4:43We use natural numbers
for the room numbers -
4:43 - 4:45as well as the seat numbers on the buses.
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4:46 - 4:48If we were dealing
with higher orders of infinity, -
4:48 - 4:50such as that of the real numbers,
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4:50 - 4:53these structured strategies
would no longer be possible -
4:53 - 4:56as we have no way
to systematically include every number. -
4:57 - 4:59The Real Number Infinite Hotel
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4:59 - 5:01has negative number rooms in the basement,
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5:01 - 5:02fractional rooms,
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5:02 - 5:04so the guy in room 1/2 always suspects
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5:05 - 5:07he has less room than the guy in room 1.
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5:07 - 5:10Square root rooms, like room radical 2,
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5:10 - 5:11and room pi,
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5:11 - 5:14where the guests expect free dessert.
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5:14 - 5:17What self-respecting night manager
would ever want to work there -
5:17 - 5:19even for an infinite salary?
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5:19 - 5:21But over at Hilbert's Infinite Hotel,
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5:21 - 5:22where there's never any vacancy
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5:22 - 5:24and always room for more,
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5:24 - 5:27the scenarios faced by the ever-diligent
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5:27 - 5:29and maybe too hospitable night manager
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5:29 - 5:31serve to remind us of just how hard it is
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5:32 - 5:34for our relatively finite minds
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5:34 - 5:37to grasp a concept as large as infinity.
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5:37 - 5:39Maybe you can help tackle these problems
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5:39 - 5:40after a good night's sleep.
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5:40 - 5:42But honestly, we might need you
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5:42 - 5:45to change rooms at 2 a.m.
- Title:
- The Infinite Hotel Paradox - Jeff Dekofsky
- Speaker:
- Jeff Dekofsky
- Description:
-
View full lesson: http://ed.ted.com/lessons/the-infinite-hotel-paradox-jeff-dekofsky
The Infinite Hotel, a thought experiment created by German mathematician David Hilbert, is a hotel with an infinite number of rooms. Easy to comprehend, right? Wrong. What if it's completely booked but one person wants to check in? What about 40? Or an infinitely full bus of people? Jeff Dekofsky solves these heady lodging issues using Hilbert's paradox.
Lesson by Jeff Dekofsky, animation by The Moving Company Animation Studio.
- Video Language:
- English
- Team:
- closed TED
- Project:
- TED-Ed
- Duration:
- 06:00
Helene Batt edited English subtitles for The Infinite Hotel Paradox | ||
Retired user commented on English subtitles for The Infinite Hotel Paradox | ||
Retired user commented on English subtitles for The Infinite Hotel Paradox | ||
Krystian Aparta edited English subtitles for The Infinite Hotel Paradox | ||
Krystian Aparta edited English subtitles for The Infinite Hotel Paradox | ||
Krystian Aparta commented on English subtitles for The Infinite Hotel Paradox | ||
TED edited English subtitles for The Infinite Hotel Paradox | ||
TED edited English subtitles for The Infinite Hotel Paradox |
Krystian Aparta
The English transcript was updated on 3/23/2015.
Retired user
Please note a typo in 1:19 countedly infinite => countebly infinite
Retired user
Amendment to the previous comment:
in 1:19, 1:23, and 2:28 should be countably infinite instead of countedly infinite. :)