![\"\"](\"https:\/\/images.theconversation.com\/files\/352759\/original\/file-20200813-18-pqzzju.PNG?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip\")
Illustration of the Infinite Hotel by Julie De Saedeleer.<\/span> Author provided\n<\/span><\/span>Let\u2019s imagine that every room in our infinite hotel is already occupied\u2014the hotel is fully booked! A new guest arrives at the front desk and wants a room. Bad news? No! The properties of infinity will allow us to find one for him. But how do we do that? The front desk clerk has an idea. He knocks on the door of Room 1 and asks the guest to move to Room 2; then he asks the guest in Room 2 to move to Room 3, and so on. Each guest moves from room N to room N+1, and since there are an infinite number of rooms, there\u2019s a new room for every guest. After this lengthy process, the tireless receptionist informs the traveler that the first room is now available for the night; all that\u2019s left is to put fresh sheets on the beds for everyone and say good night.<\/h5>\n
This remarkable hotel can also accommodate an entire busload of travelers even when the hotel is already fully booked. Let\u2019s say a bus carrying 20 new guests arrives. The ingenious front desk clerk uses the same idea and this time asks the guest in Room 1 to move to Room 21, the guest in Room 2 to move to Room 22, and so on. Each guest moves from room N to room N+20, and the problem is solved: the 20 travelers from the bus each settle into one of the first 20 rooms, which are now empty. This process works regardless of the finite number of travelers who arrive. Fantastic, but it\u2019s very likely that you\u2019ll be asked to move several times during the night.<\/p>\n
Now, let\u2019s imagine that an infinite bus, filled with a countably infinite number of passengers (Passenger 1, Passenger 2\u2026), arrives at our hotel, which is fully booked. Our trusty receptionist, initially perplexed, quickly figures out how to accommodate this infinite number of new passengers!<\/p>\n
What about you?<\/p>\n
To do this, an infinite number of rooms must be vacated\u2014for example, all the odd-numbered rooms. So our receptionist asks the guest in Room 1 to move to Room 2, the guest in Room 2 to move to Room 4, the guest in Room 3 to move to Room 6, and so on. This time, the guest in Room N moves to Room 2N. The even-numbered rooms are now all occupied, leaving all the odd-numbered rooms free to accommodate the new guests. Everyone is happy\u2014except perhaps the hotel staff, who definitely never stop changing the sheets.<\/p>\n
Our infinite hotel is building quite a reputation, because there's always room, no matter how many people show up.<\/p>\n
We can take this thought experiment even further by imagining that an infinite countable number of buses, each carrying an infinite countable number of passengers, arrive at our hotel, which is always full. The receptionist can once again rely on mathematics for help, drawing on the infinity of prime numbers. Prime numbers are natural numbers that have exactly two divisors: 1 and themselves. For example: 2, 3, 5\u2026 The countable infinity of prime numbers was proven by Euclid in the 4the<\/sup>\u00a0century before Christ. He uses this result to create an infinite number of rooms capable of accommodating an infinite number of guests arriving on an infinite number of buses.<\/p>\n Hotel guests will move to rooms whose numbers are powers of 2 (the first prime number). For example, the guest in room 5 will move to room 2.5<\/sup> (=32). The guest in Room N is therefore moving to Room 2N<\/sup>. Let's now move on to the first infinite bus, whose passengers will be sent to rooms numbered by the exponents of 3 (the second prime number). The passenger in seat 5 of the first bus will sleep in room 35<\/sup> (=243). And so on for the infinite number of passengers on the infinite number of buses; passenger N on the 2e<\/sup>\u00a0Bus will sleep in Room 5N<\/sup>, Customer N from 3e<\/sup>\u00a0Bus will be staying in Room 7N<\/sup>. Since we use the exponents of prime numbers, there are no duplicate room numbers, and every guest can sleep soundly. Note that there are still an infinite number of empty rooms\u2014those that are not exponents of prime numbers, such as room 6e<\/sup>\u00a0bedroom.<\/p>\n Despite the colossal logistical effort required by this hotel, these experiences are only possible because we work with infinity\u2014and in particular its smallest version, known as \u00abcountable infinity\u00bb\u2014a concept we\u2019ve used throughout when counting rooms, buses, guests\u2026 On the other hand, if a finite hotel\u2014no matter how large it may be\u2014is fully booked, it is impossible to accommodate even one more guest.<\/p>\n Our infinite hotel, often called Hilbert's Hotel, is a thought experiment devised by the mathematician David Hilbert (1862\u20131943) that illustrates the eponymous paradox describing the counterintuitive properties of infinite sets.<\/p>\n Above all, this thought experiment illustrates the difficulties we face in dealing with infinity, forcing us to abandon our habits of counting in finite sets.<\/p>","protected":false},"excerpt":{"rendered":" Soyons fous cette ann\u00e9e pour les vacances, ouvrons un h\u00f4tel infini. Oui, vous avez bien lu, un h\u00f4tel hypoth\u00e9tique avec une infinit\u00e9 d\u00e9nombrable de chambres. Un ensemble infini est dit d\u00e9nombrable, lorsque ses \u00e9l\u00e9ments peuvent \u00eatre list\u00e9s sans omission ni r\u00e9p\u00e9tition dans une suite index\u00e9e par les entiers. D\u00e8s lors on peut num\u00e9roter les chambres\u00a0:<\/p>","protected":false},"author":33,"featured_media":17923,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"_uf_show_specific_survey":0,"_uf_disable_surveys":false,"footnotes":"","rank_math_focus_keyword":"","rank_math_title":"","rank_math_description":"","rank_math_canonical_url":"","rank_math_robots":null,"rank_math_facebook_title":"","rank_math_facebook_description":"","rank_math_twitter_title":"","rank_math_twitter_description":""},"categories":[9],"tags":[],"class_list":["post-17748","post","type-post","status-publish","format-standard","has-post-thumbnail","category-latribune"],"acf":[],"_links":{"self":[{"href":"https:\/\/antilla-martinique.com\/en\/wp-json\/wp\/v2\/posts\/17748","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/antilla-martinique.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/antilla-martinique.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/antilla-martinique.com\/en\/wp-json\/wp\/v2\/users\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/antilla-martinique.com\/en\/wp-json\/wp\/v2\/comments?post=17748"}],"version-history":[{"count":0,"href":"https:\/\/antilla-martinique.com\/en\/wp-json\/wp\/v2\/posts\/17748\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/antilla-martinique.com\/en\/wp-json\/wp\/v2\/media\/17923"}],"wp:attachment":[{"href":"https:\/\/antilla-martinique.com\/en\/wp-json\/wp\/v2\/media?parent=17748"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/antilla-martinique.com\/en\/wp-json\/wp\/v2\/categories?post=17748"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/antilla-martinique.com\/en\/wp-json\/wp\/v2\/tags?post=17748"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"\"](\"https:\/\/images.theconversation.com\/files\/352760\/original\/file-20200813-20-f93o7m.PNG?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip\")