# Thread: Set of 13 real numbers

1. ## Set of 13 real numbers

Hi, I'm having difficulty getting started with this problem:

Show that given any set of 13 distinct real numbers, there will be two elements in the set (x and y) such that
$0 < \frac{x-y}{1+xy} \leq 2 - \sqrt{3}$

I really don't know where to start, but this is a course in combinatorics, and the pigeonhole principle is supposed to be used somewhere (probably in the part about sets)

1) How does the set having 13 elements come into play? I know that if we're talking about integers, then it would probably be related to some modulo thing (forcing 2 elements to have the same residue) but we're talking about real numbers

2) What is so special about the expression $\frac{x-y}{1+xy}$? I mean, is this an expression of a property? Like distance formula or something? The closest thing I can get to is the difference of inverses (i.e. $\frac{1}{y} - \frac{1}{x} = \frac{x-y}{xy}$, but I don't know where the "1+" part is coming from.

3) What's the significance of $2 - \sqrt{3}$? Could it be replaced by anything else?

Basically, I get what's important, but I don't get what's connecting them ... any help with those connections would be great

2. ## Re: Set of 13 real numbers

Originally Posted by Bingk
Hi, I'm having difficulty getting started with this problem:

Show that given any set of 13 distinct real numbers, there will be two elements in the set (x and y) such that
$0 < \frac{x-y}{1+xy} \leq 2 - \sqrt{3}$

I really don't know where to start, but this is a course in combinatorics, and the pigeonhole principle is supposed to be used somewhere (probably in the part about sets)

1) How does the set having 13 elements come into play? I know that if we're talking about integers, then it would probably be related to some modulo thing (forcing 2 elements to have the same residue) but we're talking about real numbers

2) What is so special about the expression $\frac{x-y}{1+xy}$? I mean, is this an expression of a property? Like distance formula or something? The closest thing I can get to is the difference of inverses (i.e. $\frac{1}{y} - \frac{1}{x} = \frac{x-y}{xy}$, but I don't know where the "1+" part is coming from.

3) What's the significance of $2 - \sqrt{3}$? Could it be replaced by anything else?

Basically, I get what's important, but I don't get what's connecting them ... any help with those connections would be great

Define $x_i:=tan(\alpha_i)$ , $\alpha_i \in (-\frac{\pi}{2},\frac{\pi}{2})$ for all $1\leq i \leq 13$ and use Dirichlet's principle.