For English-speaking people, here is a mailI wrote and sent to explain to an Australian guy he was wrong to think that the series 1,2,3,4,... was twice as numerous as the series 2,4,6,8,... :
I also took the opportunity to make a litle fun about French and English people :)
The series 1,2,3,4,... (whole numbers) is an infinite series, true.
The series 2,4,6,8,... (even numbers) is also an infinite series, which is the exact same size (or, cardinal) than the first one, or the the odd numbers 1,3,5,7,...
To see that, during one of your sleepless nights, you can think about the example of the Infinite Hotel of Hilbert (famous French mathematician). If you have some courage, you can try and read my long paragraph below :
Here it is :
If a hotel in London has an infinite number of rooms, numbered 1,2,3,4,5,... , and one night the hotel is completely full with French tourists (all rooms are occupied by a French client, I know that is a lot of clients (if not, a lot of French)). What happens if a new client comes along to sleep ? Will the hotel receptionist say 'No, sorry, no room left' ?
Not at all, he will just say to every French client : 'Please, can you please move to the room that has one number higher than yours, please ? ' (all the 'please' because this hotel is in England, of course)
And, problem solved, since room number 1 will be free for the newcomer (the former French client of room number 1 is now in room 2, then the former French of room 2 is in room 3, etc).
But, but .... what if an infinite bus comes along full up with an infinite number of English clients ? They all come down one after the other in a big infinite rush, to try and have a room in the Infinite Hotel (they don't know it is full (if they knew it was full with French, they would go elsewhere I'm sure), and they really want to sleep. Have you already seen a big rush of an infinite number of English tourists ? It is very strange because it is all very organized for a rush, nobody tries to go over in the queue, they just speed up to the reception but then wait for their turn ^^)
Well, so the receptionist is very puzzled, he is about to refuse all the other clients but the first one (with the same trick as before he can fit one customer, then another one, then another one, etc, but he can't keep asking his French clients to move up a room, every time a new English client comes along, they would keep moving for infinite time if they want to fit the infinite bus of English ! And the French all want to sleep, so, no good).
Suddenly, a brilliant idea comes up his mind (and pay attention here, because it explains why there is the same amount of whole numbers than of even numbers) : he asks all his French clients to go to the room with the even number that is double of his own number : for example, client in room 1 goes to room 2, client in room 2 goes to room 4, client in room 3 goes to room 6, client in room 4 goes to room 8, etc ...
Everyone can agree that :
1) All the French clients know exactly where they are supposed to go (but will they manage in no mess ? After all, they are French)
2) They have only one choice in where to go to spend their night
3) They can't find a room already occupied, since for instance the client in room 1 goes to room 2, but room 2 would be empty since the client in room 2 would go in room 4, etc
4) The infinity of French clients have accommodation, AND they leave an infinite number of empty rooms (all the odd ones : 1, 3, 5, 7, etc, will be free) where the receptionist can now fit in the whole infinity of English tourists from the bus.
One might wonder, why such a brilliant man with such clever ideas is still "only" a receptionist in a hotel ? Maybe because it is Hilbert's infinite hotel, so everyday is a challenge ...
Mathematical corner (for non-allergic) :
This in maths is called a bijection, showing that 2 sets of numbers (or other stuff also) have the same "cardinal" by finding a way of relating one and only one element from the first set, to one and only one element of the second set, so that all elements are related in a one-to-one way.
We say that to element "n" of the whole numbers, we relate element "2n" (his double) of the even numbers, it shows that there is the same cardinal in both the set of whole numbers, and the set of even numbers.
Like I said before :
1 <-> 2
2 <-> 4
3 <-> 6
4 <-> 8
5 <-> 10