Tired and grumpy after your journey to the metropolis of Gangaipudupettai, you’re nevertheless ready for my thought experiment. Right?
So: You arrive at the snootiest establishment in town, 24 Lodge. “May I have a room?” you ask politely at reception. Burly woman with a moustache snaps, “We’re full!”
“What, you mean all 24 rooms?” you ask, for you’ve been here before and the name is a giveaway. She nods her head.
“Yes, all full. But check that place,” and wrinkling her nose, she points across the street: “Googol Lodge.”
“That place”, you note, is named for what inspired the more famous moniker “Google”. Googol is 1 followed by 100 zeros, a number so large that there is pretty much no way to count one googol things, not even runs scored by Sachin T. Then again, the Googol Lodge is named thus because it actually has that many rooms. (This is still a thought experiment.) Surely at least one will be available? “May I have a room?” you ask at reception. A bubbly teenager without a moustache warbles, “We’re full!”
“What, you mean all 10,000…000,000 rooms?” you ask, enunciating dots and commas and each of the hundred zeros with precision.
“Yes, all full. But check that place,” and with a contemptuous sniff, he points down the road: “Infinity Lodge.”
This third place, you note, is named for still another number, because it actually has an infinite number of rooms. (Yes, still a thought experiment.) Surely at least one will be available? “May I have a room?” you ask at the reception. A winsome lass says, “We’re full!”
But just as despondency and grumpiness start to overwhelm you, just as you start asking, “What, you mean all…”, she interrupts to say, “Yes, we are full, but don’t worry, I can give you a room!” Which she does.
She picks up a microphone. Her dulcet charms working overtime, she asks each resident of the Infinity Lodge to move to the room whose number is one more than the room they are in. Thus resident #1 goes to Room #2, resident #2 to Room #3, on and on. Which exercise leaves Room #1 free. You thank the winsome lass, walk over to Room #1 and fall into bed, asleep before your head hits the pillow.
What happened here? Gangaipudupettai has three Lodges, all chock-a-block full when you arrive. But while two have no space for you—when they’re chock-a-block, they’re chock-a-block—Infinity Lodge actually found you a room easily. How?
Well, it happened because infinity is a fundamentally different number from every other in Gangaipudupettai, or in fact on this planet. Different in this way: with every other number, you can start at 1 and count up, confident that in some definite stretch of time—a few seconds, an hour, a year—you’ll reach the number and stop counting.
Example: 24. Try it: 1, 2, 3, 4 … 23, 24. How long did it take you to get there?
But with infinity, you won’t get there. Ever.
And what happens if you start counting from 2? You’ll still get to 24, of course, but you’ll take a little less time than before, because you have only 23 numbers to count.
But with infinity, you still won’t get there. Ever.
So put it this way with the Infinity Lodge: there’s always room for one more guest. Because you need only start counting from 2.
Actually, put it this way: there’s always room for any number more. If you came to Gangaipudupettai with a friend, or with 1,783 friends, or with a googol friends, Infinity Lodge will still give every one of you a room. Just start counting from one more than the number you have in your party.
And in fact, Infinity Lodge would be able to oblige even if you turned up with an infinite number of friends. The winsome lass only needs to ask the occupant of Room #1 to move to #2, #2 to #4, #3 to #6, and so on: double your room number and move there. When the dust settles, all the odd-numbered rooms are free, and there’s an infinite number of those. Enough for you and your gaggle of friends.
So here’s the secret about infinity. Add one to infinity, you get infinity. Add googol to infinity, you get infinity. Add infinity to infinity, you guessed it, you get infinity again.
This is no boutique hotel with a limited number of rooms, the Infinity Lodge in Gangaipudupettai. This is a place for the hoi-polloi, because everyone and her brother-in-law is welcome. No wonder folks at the other Lodges look down their pretty noses at it.
But you need not. When you get there, I recommend Room #298712366123587324519138001847. Nice view from there. And have fun doubling that, when I show up with my infinity of friends.
Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. A Matter of Numbers will explore the joy of mathematics, with occasional forays into other sciences. Comments are welcome at firstname.lastname@example.org