Means and operations

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

power mean

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

arithmetic mean

harmonic mean

geometric mean

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

root mean square

These are all examples of quasi-arithmetic means:

quasi arithmetic mean

The power means suggest power operations:

power operation

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


harmonic addition

root sum squared

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

p norm

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

Euclidean norm

Reciprocal addition is part of reciprocal arithmetic, see here.