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  • In the 1920's,

  • the German mathematician David Hilbert

  • devised a famous thought experiment

  • to show us just how hard it is

  • to wrap our minds around the concept of infinity.

  • Imagine a hotel with an infinite number of rooms

  • and a very hardworking night manager.

  • One night, the Infinite Hotel is completely full,

  • totally booked up with an infinite number of guests.

  • A man walks into the hotel

  • and asks for a room.

  • Rather than turn him down,

  • the night manager decides to make room for him.

  • How?

  • Easy, he asks the guest in room number 1

  • to move to room 2,

  • the guest in room 2 to move to room 3,

  • and so on.

  • Every guest moves from room number "n"

  • to room number "n+1".

  • Since there are an infinite number of rooms,

  • there is a new room for each existing guest.

  • This leaves room 1 open for the new customer.

  • The process can be repeated

  • for any finite number of new guests.

  • If, say, a tour bus unloads

  • 40 new people looking for rooms,

  • then every existing guest just moves

  • from room number "n"

  • to room number "n+40",

  • thus, opening up the first 40 rooms.

  • But now an infinitely large bus

  • with a countedly infinite number of passengers

  • pulls up to rent rooms.

  • Countedly infinite is the key.

  • Now, the infinite bus of infinite passengers

  • perplexes the night manager at first,

  • but he realizes there's a way

  • to place each new person.

  • He asks the guest in room 1

  • to move to room 2.

  • He then asks the guest in room 2

  • to move to room 4,

  • the guest in room 3

  • to move to room 6,

  • and so one.

  • Each current guest moves from room number "n"

  • to room number "2n",

  • filling up only the infinite even-numbered rooms.

  • By doing this, he has now emptied

  • all of the infinitely many odd-numbered rooms,

  • which are then taken by the people

  • filing off the infinite bus.

  • Everyone's happy and the hotel's business

  • is booming more than ever.

  • Well, actually, it is booming

  • exactly the same amount as ever,

  • banking an infinite number of dollars a night.

  • Word spreads about this incredible hotel.

  • People pour in from far and wide.

  • One night, the unthinkable happens.

  • The night manager looks outside

  • and sees an infinite line

  • of infinitely large buses,

  • each with a countedly infinite number of passengers.

  • What can he do?

  • If he cannot find rooms for them,

  • the hotel will lose out

  • on an infinite amount of money,

  • and he will surely lose his job.

  • Luckily, he remembers

  • that around the year 300 B.C.E.,

  • Euclid proved that there is an infinite quantity

  • of prime numbers.

  • So, to accomplish this seemingly impossible task

  • of finding infinite beds

  • for infinite buses

  • of infinite weary travelers,

  • the night manager assigns every current guest

  • to the first prime number, 2,

  • raised to the power of their current room number.

  • So, the current occupant of room number 7

  • goes to room number 2^7,

  • which is room 128.

  • The night manager then takes the people

  • on the first of the infinite buses

  • and assigns them to the room number

  • of the next prime, 3,

  • raised to the power of their seat number on the bus.

  • So, the person in seat number 7 on the first bus

  • goes to room number 3^7

  • or room number 2,187.

  • This continues for all of the first bus.

  • The passengers on the second bus

  • are assigned powers of the next prime, 5.

  • The following bus, powers of 7.

  • Each bus follows:

  • powers of 11,

  • powers of 13,

  • powers of 17, etc.

  • Since each of these numbers

  • only has 1 and the natural number powers

  • of their prime number base as factors,

  • there are no overlapping room numbers.

  • All the buses' passengers fan out into rooms

  • using unique room assignment schemes

  • based on unique prime numbers.

  • In this way, the night manager can accomodate

  • every passenger on every bus.

  • Although, there will be many rooms that go unfilled,

  • like room 6

  • since 6 is not a power of any prime number.

  • Luckily, his bosses weren't very good in math,

  • so his job is safe.

  • The night manager's strategies are only possible

  • because while the Infinite Hotel

  • is certainly a logistical nightmare,

  • it only deals with the lowest level of infinity,

  • mainly, the countable infinity

  • of the natural numbers,

  • 1, 2, 3, 4, and so on.

  • Georg Cantor called this level of infinity aleph-zero.

  • We use natural numbers for the room numbers

  • as well as the seat numbers on the buses.

  • If we were dealing with higher orders of infinity,

  • such as that of the real numbers,

  • these structured strategies

  • would no longer be possible

  • as we have no way

  • to systematically include every number.

  • The Real Number Infinite Hotel has

  • negative number rooms in the basement,

  • fractional rooms,

  • so the guy in room 1/2 always suspects

  • he has less room than the guy in room 1.

  • Square root rooms, like room radical 2

  • and room pi,

  • where the guests expect free dessert.

  • What self-respecting night manager

  • would ever want to work there

  • even for an infinite salary?

  • But over at Hilbert's Infinite Hotel,

  • where there's never any vacancy

  • and always room for more,

  • the scenarios faced by the ever diligent

  • and maybe too hospitable night manager

  • serve to remind us

  • of just how hard it is

  • for our relatively finite minds

  • to grasp a concept as large as infinity.

  • Maybe you can help tackle these problems

  • after a good night's sleep.

  • But honestly, we might need you

  • to change rooms at 2 a.m.

In the 1920's,

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B2 H-INT TED-Ed infinite room bus hotel guest

【TED-Ed】The Infinite Hotel Paradox - Jeff Dekofsky

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    Sofi posted on 2014/04/09
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