The credit crunch has caused some big numbers to be cited over the past months: millions lost in the financial meltdown in Iceland, billions promised to save British banks, trillions wiped off the markets in one day of trading. But, as all children know, it's easy to beat any of those numbers. You lost a trillion dollars? Well, I lost a trillion plus one. But the playground clever-clogs may venture a number to trump all others: infinity. It can't be so easily beaten by adding one.

Suppose an hotel has an infinite number of rooms, and each one is occupied. If a new guest arrives, Hotel Infinity's proprietor can still accommodate the new guest. By shifting each guest one room along, room 1 is freed up for the new guest. Thus no one is left without a room, because there is always another one.

You might try to beat it by adding infinity to infinity. But Hotel Infinity can still soak up infinitely many new guests. Ask the existing guests to move to the room number double that of their present one, ie, the person in room 5 moves to room 10. Now all the odd-number rooms are empty and can take the new guests.

So perhaps infinity is the biggest number? One exciting moment in mathematical history was the realisation at the end of the 19th century that infinity isn't the biggest number: there are infinite infinities, each bigger than the previous one. It was Georg Cantor, a German mathematician, who came up with a beautiful argument for why there is more than one. So hold on to your mathematical hats as I take you to infinity - and beyond.

Hotel Infinity's rooms are numbered by whole numbers: 1, 2, 3 etc. Hotel Uncountable has rooms numbered using all the infinite decimal numbers - those that continue for ever after the decimal point - such as Pi = 3.14159... and the square root of 2 = 1.41421.... Both have infinitely many rooms but Cantor showed why Hotel Uncountable's infinity is bigger than Hotel Infinity's. If two hotels have a finite number of rooms, the way to tell which has the most is to pair up the rooms; the hotel with rooms left over is the bigger one. This is how Cantor realised that you should compare infinities.

So, imagine that Hotel Infinity's owner thinks that he's found a way to pair up all his rooms so that they match all the rooms in Hotel Uncountable; for example (to choose some infinite decimal numbers at random), suppose that room 1 is paired with room 0.342565..., room 2 with room 0.578027466..., room 3 with room 0.55472882... and so on. Hotel Uncountable's owner can always come up with a room that has been missed. To do this, she cooks up an infinite decimal number so that the first decimal differs from the first decimal of the room paired with room 1; in this case change the 3 to a 4. The second decimal is chosen to differ from the second one of room 2; for example, change the 7 to an 8. Keep going, arranging each time that, for example, the 100th decimal is different from the 100th decimal of the room paired with room 100. So, from these examples, the new number starts 0.485....

Why has the room with this door number not been counted? Suppose that Hotel Infinity's owner claims that this room was paired with a later number, room 412, say. Hotel Uncountable's owner can show that it wasn't. Look at the 412th decimal place of this new number: because of the way we constructed the number, it must be different from the 412th decimal place of the room you paired with room 412. So it's not the number paired with room 412.

Hotel Infinity's owner missed this room number. But even if he tried to shift all the rooms along one and add this new room to the count, Hotel Uncountable's proprietor can play the same trick, producing another missing room number. Hence the infinity of rooms in Hotel Uncountable is bigger than the infinity of Hotel Infinity.

Marcus du Sautoy

The author appears in Zero to Infinity at the Dana Centre on November 20 at 7pm.

Explore Tech & Web

• Personal Tech

• The Web

• Gadgets & Gaming