The power means are defined for a set of real numbers, *a*_{1}, *a*_{2}, …, *a _{n}*:

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

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 addition (*p* = 1), harmonic 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*represent differences such as |

_{i}*x*–

_{i}*y*|, then the operations are

_{i}*p*-norms:

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