Today's Math Lesson is a Big One

Today's math lesson is brought to you by David Hilbert, a German mathematician who was one of the most influential mathematicians of the 19th and early 20th centuries. At the International Congress of Mathematicians in Paris in 1900, he set the tone for the new century by outlining the 23 most important unsolved problems in mathematics. One of the new ideas in mathematics that he embraced and supported was Georg Cantor's theory of sets and transfinite numbers, or more commonly known, the study of infinity.

One of the interesting ideas when thinking about infinite sets is their size. In general, sets are the same size if they can be put into a 1 - to - 1 correspondence. For instance, the set of the first 20 natural numbers and the set of students in the class are the same size if I can find a way to match them up one by one. So, Sally is #1, Jimmy is # 2, and so on, until the last student, Maria is #20. When you start to pair up infinite sets, it becomes a little less intuituve. For example the set of odd natural numbers (1, 3, 5, 7, ...) is exactly the same size as the set of all natural numbers:


1  3  5  7  9      2n - 1        199   . . .

1  2  3  4  5       n              100  . . .

Every number in the top row is paired with a unique number in the bottom row. Both rows go on forever. If someone challenges me to find the number in the top row that matches up to any natural number, I can do that. So the two sets must be the same size. That's when infinite sets become non-intuitive. We threw out half the numbers from the list (we threw out 2, 4, 6, 8, ...), but the new list is the same size as the old list. In fact, one of Cantor's most famous ideas is that the set of all positive fractions is also the same size as the set of natural numbers. Think about putting the fractions (reduced to lowest terms) in rows based on their numerator. Row 1 would be all the fractions with a 1 on top. Row 2 would be all the fractions with a 2 on top, and so forth. Note: 1 is 1/1, 2 is 2/1, etc. 2/2 doesn't show up because that's the same as 1 in the first row.

Row 1:     1   1/2   1/3   1/4   1/5    1/6  ...

Row 2:     2   2/3   2/5   2/7   2/9    2/11  ...

Row 3:     3   3/2   3/4   3/5   3/7    3/8   ....

Row 4:     4   4/3   4/5   4/7   4/9    4/11   ...

Row 5:     5   5/2   5/3    5/4   5/6    5/7  ....

Think of this as going on forever to the right and forever down as well. Every fraction would appear on this list somewhere. Now what we are going to do is assign the natural numbers to the fractions in an organized manner so that there is a 1- to - 1 correspondence.


Row 1:     1     1/2    1/3    1/4    1/5    1/6  ...  

               1     2      4      7      11      16



Row 2:     2      2/3    2/5    2/7   2/9    2/11  ...

               3      5       8     12      17



Row 3:     3      3/2      3/4    3/5   3/7    3/8   ....

               6      9       13      18



Row 4:     4      4/3     4/5     4/7    4/9    4/11   ...
              10     14        19



Row 5:     5      5/2      5/3    5/4   5/6    5/7  ....

                15       20

[Note: I don't know how well lined up they will be on your screen, but the first few elements in a fraction row should have a colored number below it.]

We are assigning the numbers by moving along diagonal lines starting in the upper left corner of the list. The diagonals all go from upper right to lower left. The first diagonal (in red) only contains the number 1 (which is labeled below it with a red 1). The second diagonal (in green) contains 1/2 (which is numbered 2) and 2 (which is numbered 3). Think of laying a ruler along that diagonal to hit the numbers in that group, then sliding the ruler to the right one number to get the next group. The third diagonal (in dark blue) contains 1/3, 2/3, and 3 (labeled 3, 4, and 5). The fourth diagonal is pink (7 - 10), the fifth is light blue (11 - 15). Every fraction will appear once on the list and will be paired (eventually) with a natural number. SO, the two sets are the same size.

To illustrate the idea of infinite sets, Hilbert devised the Grand Hotel, which has infinitely many rooms numbered, 1, 2, 3, 4, 5, and so forth forever.  One night the hotel is full (don't spend too long thinking about that - it will make your head hurt) when a car drives up with one person in it, who asks for a room. "No problem", says innkeeper Hilbert. "We will put you in room 1, put the person in room 1 in room 2, the person in room 2 in room 3, and so forth. Everybody will have a room to move to and we will still be full."

Just then a very big bus drives up with infinitely many guests on board. "Wonderful," says Hilbert. "Every guest already in the hotel will move into the room that is twice their current number. The person in room 1 moves to room 2, the person in room 2 moves to room 4, and so on. That will leave us infinitely many rooms empty for the new guests. The first one off the bus goes to room 1, the second one goes to room 3, the third to room 5, and so on. Still full!"

Unless you're the housekeeper, things are wonderful at Hilbert's Grand Hotel.