Maths Moments

My intention was to carry on the little narrative I began with the librarian faced with the two piles of books and to tell you what happens in the next room. But perhaps in preparation for our meeting at The Hope I'll record some of the things I was thinking of including.

I think the maths moments I hope we might include will focus on two topics
(1) number and infinity
(2) shapes and the nature of space

This post will deal with the first one.

(1) Number and infinity
(a) I quite like the fact that you can calculate exactly how many books there are in the library given that the number of pages is limited. It's a large number but one can also calculate from this the number of hexagons.
(b) But it's still possible to make infinite books by combining the books in the library. So for example given ten books numbered 0 to 9, any number could represent a new book where I read the ten books in the order given by the number. For example 134115 means the book got by reading Book 1 followed by Book 3 followed by Book 4 followed by Book 1 again then Book 1 again then Book 5. It’s almost like using the first library to build a second library with infinitely many books.
(c) But then perhaps you come across a third library where the books are all labeled with fractions. Does this library have more books in than the second library where all the books are labeled with whole numbers? This was Cantor’s great discovery that actually these two infinities have the same size because there is a way to pair up all the books in each library. It’s called Cantor’s diagonal slash. There is a very graphic way to show how to pair these numbers up which I’ve done in one of our workshops and I’ve also done with the workshops I’ve created with Complicite. I wonder if one could explore also using the audience as books. The Cantor argument depends on arranging the fractions in a big infinite two dimensional grid. Here is the script I’ve written yesterday for the TV programme I’m making:

Cantor needed to count all the fractions in a systematic way. To do this he started by arranging all the fractions in an infinite grid. The first row contained all the fractions with 1 on the bottom. [See these fractions being laid out] In the second row came all the fractions with 2 on the bottom. [Next see these fractions] Carrying on like this, the 6th row for example would contain all the fractions with 6 on the bottom.

Every fraction appears somewhere in this grid. Where’s 5/7…go to the 5th column of the 7th row. [See this being lit up] But how can we count these fractions. How can we pair up the whole numbers with this infinite grid of fractions. [Perhaps see the whole numbers appear along the top of the screen]

Cantor’s snake is the key. Imagine a line snaking back and forward diagonally through the fractions [see the line running through the fractions] then by pulling this line straight we can match up every fraction with one of the whole numbers. [Now see the fractions being pulled up by the line and aligning with the whole numbers. We can even have the whole numbers running off the screen to the left so that we gradually see more and more fractions pairing up.]

So the fractions are the same sort of infinity as the whole numbers.
(d) But then there comes a library with books labeled with infinite decimal numbers, like pi=3.14159… or e=2.71828… Each library could even have its own librarian. Almost like the librarians showing off to each other about whose library has the most books. It turns out that this librarians library has an infinity which is much bigger than the fractional library or the first infinite library. There is a nice way to show this physically as well, an exercise which I’ve done before with us and Complicite.
(e) One of the big mysteries of the twentieth century was: Is there an infinite library with strictly fewer books than the infinite decimal library but strictly larger than the infinite fractional library. The problem to sort this out was called the Continuum Hypothesis. In the middle of the twentieth century a rather surprising answer was reached: Both answers could be true. There was an entirely consistent mathematical world where there was such a library. And an equally consistent mathematical world where no such library existed. It was a discovery that was almost as shocking as the discovery of non-Euclidean geometries.