Tova Brown concludes her exploration of the Hilbert’s Hotel Paradox with a look at the cardinality of the real numbers.
You run a hotel with an infinite number of rooms. You pride yourself on accommodating everyone, even guests arriving in infinitely large groups — but some infinities are more infinite than others, as it turns out.
Tova Brown dives deeper into Hilbert’s Hotel Paradox, considering the difference between rational numbers and reals.
You run an infinitely large hotel, and are happy to realize that you can accommodate an infinite number of infinite groups of guests.
However, a delicate diplomatic situation arises when a portal to another universe opens, introducing a different kind of guest, in a different kind of group.
Can you make room for them all?
Tova Brown continues to examine Hilbert’s Hotel Paradox, pondering infinite sets of infinite sets.
As the proprietor of an infinitely large hotel, you pride yourself on welcoming everyone, even when the rooms are full. Your hotel becomes very popular among infinite sports teams, as a result.
Recruitment season presents a challenge, however, when many infinite teams arrive at once. How many infinite teams can stay in a single infinite hotel?
Tova Brown explores the second part of Hilbert’s Hotel Paradox. What’s infinity plus infinity?
Running an infinite hotel has its perks. Even when the rooms are full you can always find space for new guests, so you proudly welcome everyone who appears at your door.
When two guests arrive at once, you make room. When ten guests arrive, you accommodate them easily. When a crowd of hundreds appears, you welcome them all.
Is there no limit to your hospitality?
Tova Brown’s introduction to Hilbert’s Hotel Paradox, a riddle about the nature of infinity…
Once upon a time, there was a hotel with an infinite number of rooms. You might be thinking this is impossible, and if so you’re right. A hotel like this could never exist in the real world.
But fortunately we’re not talking about the real world, we’re talking about math. And when we do math we can make up whatever rules we want, just to see what happens.
Math Concepts: division as equal sharing, naming fractions, adding fractions, infinitesimals, iteration, limits
Prerequisite: able to identify fractions as part of a whole
This is how I tell the story:
Continue reading Infinite Cake: Don Cohen’s Infinite Series for Kids
[Feature photo above by Paolo Camera (CC BY 2.0) via Flickr.]
Half of our students were missing from this month’s homeschool teen math circle, but I challenged the three who did show up to wrap their brains around some large numbers. Human intuition serves us well for the numbers we normally deal with from day to day, but it has a hard time with numbers outside our experience. We did a simple yet fascinating activity.
First, draw a line across a page of your notebook. Label one end of the line $20 (the amount of money I had in my purse), and mark the other end as $1 trillion (rough estimate of the US government’s yearly overspending, the annual deficit):
Go ahead, try it! The activity has a much greater impact when you really do it, rather than just reading. Don’t try to over-think this, just mark wherever it feels right to you.
The kids were NOT eager to commit themselves, but I waited in silence until everyone made a mark.
This was a bit easier. Once they had committed to a place for a million, they went about that much farther down the line to mark a billion.
Denise Gaskins' Let's Play Math 