To infinity - and beyond

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sexy maths .

To infinity - and beyond
  To infinity - and beyond
             he credit crunch has caused            whole numbers: 1,2, 3 etc. Hotel Uncoun-        choose some infinite decimal numbers at
             some big numbers to be cited           table has rooms numbered using all the          random), suppose that room 1 is paired
             over the past months: millions         infinite decimal numbers - those that con-      with room 0.342565 ... , room 2 with room
             lost in the financial meltdown         tinue for ever after the decimal point -        0.578027466 , room 3 with room
  •••        in Iceland, billions promised to                           ...
                                                    such as IT = 3.14159 and '1/2= 1.41421   ....   0.55472882 and so on. Hotel Uncount-
  save British banks, trillions wiped off the       Both have infinitely many rooms but Can-        able's owner can always come up with a
- markets in one day of trading. But, as all        tor showed why Hotel Uncountable's infin-       room that has been missed. To do this, she
  children know, it's easy to beat any of           ity is bigger than Hotel Infinity's. If two     cooks up an infinite decimal number so
  those numbers. You lost a trillion dollars?       hotels have a finite number of rooms, the       that the first decimal differs from the first
  Well, I lost a trillion plus one. But the play-   way to tell which has the most is to pair up    decimal of the room paired with room 1;in
  ground clever-clogs may venture a                 the rooms; the hotel with rooms left over       this case change the 3 to a 4. The second
  number to trump all others: infinity. It          is the bigger one. This is how Cantor real-     decimal is chosen to differ from the
  can' so easily beaten by adding one.          ised that you should compare infinities.        second one of room 2; for example, change
     Suppose an hotel has an infinite                  So, imagine that Hotel Infinity's owner      the 7 to an 8. Keep going, arranging each
  number of rooms, and each one is occu-            thinks that he's found a way to pair up all     time that, for example, the 100th decimal
  pied. If a new guest arrives, Hotel Infini-       his rooms so that they match all the rooms      is different from the 100th decimal of the
  ty's proprietor can still accommodate the         in Hotel Uncountable; for example (to           room paired with room 100. So, from these
  new guest. By shifting each guest one                                                             examples, the new number starts 0.485 ....
  room along, room 1is freed up for the new                                                            Why has the room with this door
  guest. So no one is left without a room,                                                          number not been counted? Suppose that
  because there is always another one.                                                              Hotel Infinity's owner claims that this
     You might try to beat it by adding infin-                                                      room was paired with a later number,
  ity to infinity. But Hotel Infinity can still                                                     room 412,say. Hotel Uncountable's owner
  soak up infinitely many new guests. Ask                                                           can show that it wasn't. Look at the 412th
  the existing guests to move to the room                                                           decimal place of this new number:
  number double that of their present one,                                                          because of the way we constructed the
  ie, the person in room 5 moves to room 10.                                                        number, it must be different- from the
  Now all the odd-number rooms are empty                                                            412th decimal place of the room you
  and can take the new guests.                                                                      paired with room 412. So it's not the
     So perhaps infinity is the biggest                                                             number paired with room 412.
  number? One exciting moment in mathe-                                                                Hotel Infinity's owner missed this room
  matical history was the realisation at the                                                        number. But even if he tried to shift all the
  end of the 19th century that infinity isn't                                                       rooms along one and add this new r-oomto
  the biggest number: there are infinite infin-      THE CONUNDRUM                                  the count, Hotel Uncountable's proprietor
  ities, each bigger than the previous one. It       If infinitely many buses arrive at Hotel       can play the same trick, producing another
  was Georg Cantor, a German mathemati-              Infinity with infinitely many guests in        missing room number. Hence the infinity
  cian, who came up with a beautiful argu-           each bus, can the proprietor   fit them        of rooms in Hotel Uncountable is bigger
  ment for why there is more than one. So            all in?                                        than the infinity of Hotel Infinity.•
  hold on to your mathematical hats as I                                                            Marcus du Sautoy
  take you to infinity - and beyond.                 Answer on page 21                              The author appears in Zero to Infinity at the
     Hotel Infinity's rooms are numbered by                                                         Dana Centre on November 20 at 7pm.

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