Infinity: Bizarre examples
Created | Updated Jan 28, 2002
The idea of infinity is a concept that many people feel they understand. Unfortunately, this theoretical number has some very strange properties. Quite apart from the fact that mathematics using infinite numbers gets quickly boring (infinity + 1 = infinity, infinity + 100 = infinity, and so on) there are some more interesting logic problems that mathematicians with time on their hands have invented. Here are two examples:
Question 1:
Question: A man owns a hotel with an infinite number of rooms in which he has an infinite number of guests staying. All at once an infinite number of further guests arrives. How can the hotel owner accommodate this infinite number of new guests?Answer: He takes the guest in room 1 and moves him into room 2. The displaced guest in room 2 is moved into room four. The person staying in room 3 is moved into room 6. This process can be repeated until all of the guests have been moved into the even numbered rooms. This leaves an infinite number of odd numbered rooms into which the new guests can be moved.
Explanation: The answer above seems not to make sense, but mathematics of this sort rarely does. The reason that this works is because there are the same number of even numbers as there are numbers, i.e. infinity.
NOTE - You can see a more detailed version of this in entry A414523
Question 2:
Question: A woman has a bag and an infinite number of beads that are numbered from one to infinity. At 12:00 (twelve noon) she takes the beads numbered 1 to 10 and places them in the bag, and then takes out the bead numbered 1. At 12:30 she puts in the beads numbered 11 to 20 and takes out the bead numbered 2. At 12:45 she places beads 21 to 30 into the bag and takes out the bead numbered 3. At 12:52½ she puts in 31 to 40 and takes out the bead numbered 4. This process is repeated as the time to 1 o'clock halves. At 1 o'clock how many beads are there in the bag?Answer: Zero.
Explanation: Logically this would seem to be impossible, as you would expect there to be far more beads in the bag than she ever takes out. The reason this is true is because there is no number bead where you cannot calculate the time at which it was taken out, meaning that at one o'clock the bag must be empty.
I told you infinity was weird!
If you have another question about infinty I'll be happy to add it.
