Infinite Wonders

By Ivars Peterson

Muse, February 2004, p. 27.

What's the biggest number you can think of? A billion? A trillion? A googol? (That's 1 followed by a hundred zeroes.) Whatever number you come up with, there's always a larger one. You could write down 1 and keep adding zeroes after it until you hand gets tired, and you still wouldn't get to the "last" number. There's always another number right after whatever you've written down. Just add 1 and you'll get a bigger number.

Because there's no imaginable "last" number, mathematicians say there are infinitely many whole numbers. When you start thinking about infinity and numbers, you also run into all sorts of weird mind-boggling puzzles and paradoxes. Here's one in which hotel rooms stand in for numbers.

Suppose you walk into an ordinary hotel. If the desk clerk tells you there are no vacancies, there's really nothing you can do. Every room is occupied.

It's very different at the Infinity Hotel, which has an infinite number of rooms. Even if all the rooms are occupied when you arrive, the clerk can still find you a bed. She simply sends a housekeeper to awaken the guests and move them to new rooms. To make sure everyone has a place, the guest in room 1 goes to room 2, the guest in room 2 goes to room 3, the guest in room 3 goes to room 4, and so on, until each occupant has moved one room over. There's an extra room at the end of the line because infinity plus 1 still equals infinity. And with room 1 vacant, you can unpack and settle down for the night.

In fact, you could make room for an infinite number of guests. Move the guest in room 1 to room 2, the guest in room 2 to room 4, the guest in room 3 to room 6, the guest in room 4 to room 8, and so on. This frees all the odd-numbered rooms, and there is an infinite number of them. The new arrivals can move into the vacant rooms, because adding infinity to infinity still gives you infinity.

That's the strangeness of infinity, where a part can be as large as the whole and there's always room for one more.

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