Arithmonic
Mean
The New Mean
There is so much to investigate on this unexplored particular case of the Rational Mean.
The reader should not be surprised because of having not seen this operation neither in the mathematical literature or in any classroom.
From the evidence at hand, there are no precedents on the use of this operation from ancient times up to now.
Note: The image links lead to a webpage within this website.
The Arithmonic Mean
Given a set A of rational numbers.
If the numerators and denominators are equaled by groups or subsets, following an arbitrarily prefixed sequence s, as shown on the images to the left, then the set A becomes the set denoted: As.
The numerators and denominators to be matched have been highlighted in a light tone.
Thus, the rational mean of the fractions of the set As, is the Arithmonic Mean which appears denoted with the acronym AHm.
Depending on the type of set, there is a great variety of matching sequences that can be used, and each of them might produce a different result. Also, note that set A could have been transformed by equaling the first two numerators instead of the denominators.
It is not surprising you have never seen the Arithmonic Mean, neither in the mathematical literature nor in any classroom.
This operation may be seen as a fusion of two ancient concepts: arithmetic and harmonic means.
In some cases, depending on the set, the Arithmonic Mean might produce the same result as any of those means.
Also, notice that if we change the location of any fraction in set A then the Arithmonic Mean yields a different value. In this case not only matters the form of fraction, but also its location in the set.
The Arithmonic Mean is the key for the high-order rational processes for approximating roots that are shown on the page entitled ‘Number‘ on this website.