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.
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.
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.
Have you made a New Year’s resolution to spend more time with your family this year, and to get more exercise? Problem-solvers of all ages can pump up their (mental) muscles with the Annual Mathematics Year Game Extravaganza. Please join us!
For many years mathematicians, scientists, engineers and others interested in math have played “year games” via e-mail. We don’t always know whether it’s possible to write all the numbers from 1 to 100 using only the digits in the current year, but it’s fun to see how many you can find.
Use the digits in the year 2016 to write mathematical expressions for the counting numbers 1 through 100. The goal is adjustable: Young children can start with looking for 1-10, middle grades with 1-25.
You must use all four digits. You may not use any other numbers.
Solutions that keep the year digits in 2-0-1-6 order are preferred, but not required.
You may use a decimal point to create numbers such as .2, .02, etc., but you cannot write 0.02 because we only have one zero in this year’s number.
You may create multi-digit numbers such as 10 or 201 or .01, but we prefer solutions that avoid them.
My Special Variations on the Rules
You MAY use the overhead-bar (vinculum), dots, or brackets to mark a repeating decimal. But students and teachers beware: you can’t submit answers with repeating decimals to Math Forum.
You MAY use a double factorial, n!! = the product of all integers from 1 to n that have the same parity (odd or even) as n. I’m including these because Math Forum allows them, but I personally try to avoid the beasts. I feel much more creative when I can wrangle a solution without invoking them.
Once again, Nrich.Maths.org and Plus.Maths.org are offering a selection of activities to encourage your students’ mathematical creativity, one for each day in the run-up to Christmas. Click one of the images below to visit the appropriate Advent Math Calendar page.
“Wild Maths” puzzles and articles for teens and up. When you get to the +Plus Magazine website, you can tell which links are live by the drop shadow under the picture. One link becomes live each day — so come back tomorrow and discover something new!
You know you’re a math teacher when you see a penny in the parking lot, and your first thought is, “Cool! A free math manipulative.”
My homeschool co-op math students love doing math with pennies. They’re rather heavy to carry to class, but worth it for the student buy-in.
This month, I’m finishing up the nearly 150 new illustrations for the upcoming paperback edition of my Let’s Play Math book. I’m no artist, and it’s been a long slog. But a couple of the graphics involved pennies—so when I saw that penny on the ground, it made me think of my book.
And thinking of my book made me think it would be fun to share a sneak peek at coming attractions…
The Penny Square: An Example of Real Mathematics
Real mathematics is intriguing and full of wonder, an exploration of patterns and mysterious connections. It rewards us with the joy of the “Aha!” feeling. Workbook math, on the other hand, is several pages of long division by hand followed by a rousing chorus of the fraction song: “Ours is not to reason why, just invert and multiply.”
Real math is the surprising fact that the odd numbers add up to perfect squares (1, 1 + 3, 1 + 3 + 5, etc.) and the satisfaction of seeing why it must be so.
Did your algebra teacher ever explain to you that a square number is literally a number that can be arranged to make a square? Try it for yourself:
Gather a bunch of pennies—or any small items that will not roll away when you set them out in rows—and place one of them in front of you on the table. Imagine drawing a frame around it: one penny makes a (very small) square. One row, with one item in each row.
Now, put out three more pennies. How will you add them to the first one in order to form a new, bigger square? Arrange them in a small L-shape around the original penny to make two rows with two pennies in each row.
Set out five additional pennies. Without moving the current four pennies, how can you place these five to form the next square? Three rows of three.
Then how many will you have to add to make four rows of four?
Twenty-five is a square number, because we can arrange twenty-five items to make a square: five rows with five items in each row.
Each new set of pennies must add an extra row and column to the current square, plus a corner penny where the new row and column meet. The row and column match exactly, making an even number, and then the extra penny at the corner makes it odd.
Can you see that the “next odd number” pattern will continue as long as there are pennies to add, and that it could keep going forever in your imagination?
The point of the penny square is not to memorize the square numbers or to get any particular “right answer,” but to see numbers in a new way—to understand that numbers are related to each other and that we can show such relationships with diagrams or physical models. The more relationships like this our children explore, the more they see numbers as familiar friends.
The Penny Birthday Challenge: Exponential Growth
A large jar of assorted coins makes a wonderful math toy. Children love to play with, count, and sort coins.
Add a dollar bill to the jar, so you can play the Dollar Game: Take turns throwing a pair of dice, gathering that many pennies and trading up to bigger coins. Five pennies trade for a nickel, two nickels for a dime, etc. Whoever is the first to claim the dollar wins the game.
Or take the Penny Birthday Challenge to learn about exponential growth: Print out a calendar for your child’s birthday month. Put one penny on the first day of the month, two pennies on the second day, four pennies on the third day, etc. If you continued doubling the pennies each day until you reach your child’s birthday, how much money would you need?
Warning: Beware the Penny Birthday Challenge! Those pennies will add up to dollars much faster than most people expect. Do not promise to give the money to your child unless the birthday comes near the beginning of the month.
A Penny Holiday Challenge
The first time I did pennies on a calendar with my homeschool co-op class was during December, so we called it the Penny Christmas Challenge:
How many pennies would you need to cover all the days up to the 25th?
I told the kids that if their grandparents asked what gift they wanted for Christmas, they could say, “Not much. Just a few pennies…”
The Penny Square, Dollar Game, and Penny Birthday Challenge are just three of the myriad math tips and activity ideas in the paperback edition of Let’s Play Math: How Families Can Learn Math Together and Enjoy It. Coming in early 2016 to your favorite online bookstore…
Tova Brown concludes her exploration of the Hilbert’s Hotel Paradox with a look at the cardinality of the real numbers.
