Here's a very interesting link showing an operation similar to averaging but with fractions or rational numbers and it has relationships to a wide variety of mathematical structures in a very simple form:

The Rational Mean
http://mipagina.cantv.net/arithmetic/rmdef.htm

You can see it closely related to a weighted average of rational numbers:

Notice that if we computed the average of a set of integers, we could rewrite this as a rational mean like this:

2+3+4=9
mean is 9/3=3

As a rational mean we have:

(2/1)+(3/1)+(4/1)=(2+3+4)/(1+1+1)=9/3=3

We can rewrite these as weighted averages by using integer multiples. For example, weighting the 2 and 3 with double significance we have:

(2*2/1*2)+(3*2/1*2)+(4/1)=(2*2+3*2+4)/(1*2+1*2+1)=14/5 (=2 4/5)

Anyway he shows many examples of simple sequences that generate quite a diverse array of irrational numbers.

Also, here's an interesting link that I found the above from:

Gap Theory
http://linas.org/art-gallery/farey/continue/continue.html