# Welcome to the Grand Hotel, Please Excuse our Appearance

Memo to All Guests

As you know, we recently embarked on an exciting renovation to expand the Grand Hotel. If you have stayed with us before, you may have noticed that your room numbers are now represented by real numbers, rather than just integers. Many guests have remarked how much they like the new numbering system.

However, we regret that additional renovations are necessary, and some rooms will be temporarily closed. During this time, we will need to move all guests to integer-numbered rooms.

To thank you for your patience, the infinity pool will be open extended hours.

D. Hilbert, Manager, Grand Hotel

To: Manager <dhilbert@grandhotel.com>
From: Assistant Manager <gcantor@grandhotel.com>
Subject: Are you sure?

David — We’re pretty full in the reals. I’ve been running the numbers and I’m not sure we’ll be able to fit all our guests in the integers. Can you let me know the plan.

Regards, Georg

To: Assistant Manager <gcantor@grandhotel.com>
From: Manager <dhilbert@grandhotel.com>
Subject: Re: Are you sure?

Georg, don’t worry — we never run out of rooms here. Remember when we had that termite infestation in the odd-numbered rooms? We just took the guest in room n and put them in 2n.

I mean, we even had to do this once when the reno was half done and we had guests in all the rational-numbered rooms. Not a problem then either.

David

To: Manager <dhilbert@grandhotel.com>
From: Assistant Manager <gcantor@grandhotel.com>
Subject: Re: Are you sure?

It would help me out a lot if you can write down what you are thinking.

To: Assistant Manager <gcantor@grandhotel.com>
From: Manager <dhilbert@grandhotel.com>
Subject: Re: Are you sure?

Georg, you know you’ll never get promoted if you don’t show initiative? But fine. Here’s what we’ll do.

First we’ll write down a list of all the real-valued room numbers, in any order you like.

Then we’ll enumerate the list, assigning an integer to each one: the first room in the list gets number 1, the second gets number 2, and so on down the list.

Now each real-valued room has an integer. Then we just give each guest the number of their room on the list. Simple as pie.

To: Manager <dhilbert@grandhotel.com>
From: Assistant Manager <gcantor@grandhotel.com>
Subject: URGENT — Call me

Hi David, it’s 3 AM and I’ve been calling and calling but there’s no answer. We have a big problem.

You remember the guest who checked in last night – Di Agonal? She had a strange request about her room.

I was making the list of rooms like you asked – Di looked over my shoulder and said, “I want a room with a unique number. Take the first digit of the first room on your list, and add 1; that will be the first digit in my room number. Then the 2nd digit of the 2nd room on the list, again adding 1; that will be the second digit. And so on.”

“What if the digit is 9?” I asked. “Then just turn it into a zero,” she said.

I didn’t think much about it — Di’s requested room was a real number just like all the others, and it happened to be available. I gave her the key and sent her on her way with the bellhop.

But when we went to move the guests into integer-numbered rooms, Di’s room number wasn’t on the list. The room number wasn’t the first element on the list, since the first digit didn’t match. And it wasn’t the second, since the second digit didn’t match. And so on. But my list had all the integers. Can it be that we have more rooms than integers?

I don’t mind telling you that Di is livid. And now more angry guests are coming to see me because they lost their rooms in the move. What do we do?

Memo to All Guests

Here at the Grand Hotel, we have always prided ourselves on having rooms available no matter who comes to our door.

Last night we let you down. Fortunately our friends at the Hotel Zermelo-Fraenkel were able to accommodate all of you.

Our former manager has taken full responsibility for last night’s events and has resigned. He asked me to convey his uncountable apologies to you.

As our way of making it up to you, complimentary prime rib will be served in the restaurant at 2, 3, 5, 7, and 11.

G. Cantor, Manager, Grand Hotel

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