## Sunday, February 6, 2011

### LaTeX in Blogger

I've already used it in the previos post, but not mentioned it: HERE you find a great way to post LaTeX in your blog. It's very simple, just enter the formulas and copy the html-code into your blog.
The editor itself is great. It is fast and got many features like shortcuts to important commands and auto-closing brackets. Also, the result itself (the image) as well as the html-code is presented just in time!

Another important thing: it's open-source, so you can use it without any limitations (at least I haven't read something different yet).

## Saturday, February 5, 2011

### You think there are many rational numbers? You're wrong.

I want to share something very interesting with you that told me my calculus teacher the other day.

Take the rational numbers in the closed interval from 0 to 1. It's easy to see, that this set is given by

$\mathbb{Q} \cap \left [ 0,1 \right ] = \left \{ 0, 1/1, 1/2, 2/2, 1/3, 2/3, 3/3, 1/4, ... \right \}$

So the numbers in this set are countable. Let's call them

$x_0, x_1, x_2, ...$

Now let $\varepsilon > 0$. Cover every $x_n \in \mathbb{Q} \cup \left [ 0,1 \right ]$ with an interval of length $\frac{\varepsilon}{2^n}$, that is

$I_n = \left [ x_n - \frac{\varepsilon}{2^{n+1}}, x_n + \frac{\varepsilon}{2^{n+1}} \right ]$

It follows that the measure of the union of the Intervals fulfills:

$\mu\left ( \bigcup_{n=0}^{\infty} I_n \right ) \leq \sum_{n=0}^{\infty}\frac{\varepsilon}{2^n} = \frac{\varepsilon}{1-\frac{1}{2}} = 2\varepsilon$

So what we see is: if we chose the epsilon small enough, the rational numbers, including the intervals around them, cover nearly no space in the interval from 0 to 1! That means, the space is coverd almost entirely by the irrational numbers.
I think this fact shows us very well what the difference between "countable" and "uncountable" sets is, and how much more irrational numbers than rational numbers there are even though they are both infinite many.