The Rational Mean

In my last post I mentioned the mediant of a pair of fractions and its application in what is called the Farey Sequence. I mentioned that the property was considered a "curiosity" in mathematical number theory. Then I was called to task for this usage by Domingo Gómez Morín who has spent a considerable amount of time trying to convince people that the mediant, or rational mean as he prefers to call it, is a valuable mathematical tool. He has some nice work on a web page where he shows that the rational mean is a) a general case of both the harmonic and arithmetic means as well as the geometric (not shown on his page) and golden mean, and b) an excellent tool for finding roots (both real and complex). He has thought through all this for so long that it would be a travesty for me to try to explain his work, so check it out for yourself. And thanks to Mr Morin for the reminder...he had sent me a link before and I have a note to look more deeply into it... now I shall.