Archive for September, 2010

Theorem 36: the Cantor set is an uncountable set with zero measure

September 30, 2010

This week’s post is by Laura Irvine, who is just about to start her second year reading Mathematics at Murray Edwards College, Cambridge.  Thanks very much, Laura!

First of all, what is the Cantor set?

To form the Cantor set, start with the closed interval [0,1] (this means 0 and 1 are included in the interval) and remove the middle open third of the interval (i.e. remove (\frac{1}{3}, \frac{2}{3}) where the curved brackets mean the interval is open, so \frac{1}{3} and \frac{2}{3} are not themselves in the interval).  You should be left with two disjoint closed intervals: [0, \frac{1}{3}] and [\frac{2}{3}, 1].  I’m going to call this step the first iteration.

Then do the same thing to each of these intervals: remove the middle third of each to get the new intervals as [0, \frac{1}{9}], [\frac{2}{9}, \frac{1}{3}], [\frac{2}{3}, \frac{7}{9}], [\frac{8}{9}, 1].  Then remove the middle third of each of these intervals.  Keep repeating this process and the Cantor set is the set of all the points in the interval [0,1] that are never removed.  So the first few steps of the process look like this:

The Cantor set

Do you understand how this set is formed?  Can you think of some points that are in the Cantor set?

Well, 0 will never be removed: the first closed interval after the n^{\textrm{th}} iteration would be [0, \frac{1}{3^n}] so 0 will be in the infinite intersection of the first interval of each step.  The other interval endpoints are points in the set too.  It turns out that there are also points that are in the set that aren’t interval endpoints.

An interesting and, in my opinion, rather surprising property of the Cantor set is that it has measure 0, despite being an uncountable set!  The fact it is uncountable means there is no way of writing all the numbers in the Cantor set in a list.

So, intuitively, this is saying that if all the points in the Cantor set were lined up next to each other, the line would have length 0 and yet there are infinitely many points in the set.  How weird is that?!



More maths on Radio 4

September 22, 2010

Quick advance warning of a couple of forthcoming maths programmes on BBC Radio 4.

One is In Our Time, discussing imaginary numbers.  I’ve written about In Our Time before; I’ll listen out for this programme.  It should be available to download as a podcast after it is broadcast tomorrow.

The other is a repeat of Marcus du Sautoy’s A Brief History of Mathematics, a series of short programmes about various key characters in mathematics.


Theorem 35: the best rational approximations come from continued fractions

September 8, 2010

Like Theorem 31, this post is based on a session that I did with some school students.  And as I did there, I encourage you to try the activities for yourself (if you haven’t done them before).  I think that the students enjoyed the session; I hope that you enjoy the post!

Here’s a strange looking thing (if you haven’t seen anything like it before):

1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + ...}}}}}

What can you say about it?  Can you compute it?  Does it make sense?

That was an infinite thing (that’s what the dots were supposed to indicate).  Here are the first few finite chunks of it:

1, 1 + \frac{1}{1}, 1 + \frac{1}{1 + \frac{1}{1}}, 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1}}}, …

What can you say about them?  Any patterns?  Can you explain any patterns?  Can you predict what the nth one will look like?  I recommend also looking at the differences between consecutive terms in the sequence.  Can you say anything interesting about them?