Friday, 7 December 2012

My (current) favourite infinity paradox

I remember one of my undergraduate tutors telling me about this paradox, of which I have just been reminded by Littlewood's Miscellany.

At one minute to noon the numbers 1 to 10 are put into a box, and the number 1 is removed.

At 1/2 minute to noon the numbers 11 to 20 are added to the box, and 2 is removed.

At 1/3 minute to noon the numbers 21 to 30 are added to the box, and 3 is removed.

And so on.

How many numbers are in the box at noon?  The answer is obviously 9+9+9+9+... which looks as if it should be infinite.  But in fact there are no numbers in the box, because if you suggest that number n might be there, I point out that it was taken out at 1/n of a minute before noon.  The box is empty: nine times infinity is zero.

One has to be careful in dealing with infinity!

No comments:

Post a comment