Means and operations

The power means are defined for a set of real numbers, a₁, a₂, …, a_n:

The best-known of these are the arithmetic, geometric, and harmonic means, with p = 1, p = –1, and  p → 0:

The other well-known power mean is the root mean square with p = 2:

These are all examples of quasi-arithmetic means:

The power means suggest power operations:

These include arithmetic addition (p = 1), reciprocal addition (p = –1), and the root sum squared (p = 2):

In order to avoid roots of negative numbers, the absolute value is often taken: | a_i |. If the a_i represent differences such as | x_i – y_i |, then the operations are p-norms:

The best-known norm is the Euclidean norm or distance (p = 2):

Reciprocal addition is part of reciprocal arithmetic, see here.