Archive for the ‘Guest post’ Category

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?!


Theorem 9: Bachet’s duplication formula

October 6, 2009

This week’s guest author is James Cooper.  Thanks, James!

Bachet’s Equation and Geometry

Today’s blog entry concentrates on Diophantine Equations — problems posed in terms of whole numbers, and connected problems over the rationals.  Typically these problems are very easy to understand but difficult to solve.  Their solution often involves leaving the safe world of the integers and using tools and techniques from other areas of mathematics before “projecting” the answer back into whole numbers.  The example I’m going to describe today will use ideas from algebra and geometry.

Fix some integer c.  What are the rational solutions of the equation

y^2 - x^3 = c?             (\star)

By rational solution, I allow x and y to be fractions, but not arbitrary reals.  So we are asking for the difference between a square and a cube to be a certain fixed integer.  This is known as Bachet‘s equation, and we will see that its Geometric interpretation is the key to generating solutions.