The restriction to positive numbers is necessary to avoid division by zero. For instance, if we tried to take the harmonic mean of the set we would be trying to calculate , which is obviously problematic.
The harmonic mean is a part of a frequently used inequality, the Arithmetic mean-Geometric mean-Harmonic mean inequality. The Inequality states that for a set of positive numbers :
In the line of power of means, the harmonic mean is the mean of the -1 power.
Harmonic mean is commonly used to find the average of rates.