An Etymological Dictionary of Astronomy and Astrophysics

A number whose reciprocal is the → arithmetic mean of the reciprocals of a set of numbers. Denoted by H, it may be written in the discrete case for n quantities x₁, …, x_n, as:

1/H = (1/n) Σ(1/x_i), summing from i = 1 to n.

For example, the harmonic mean between 3 and 4 is 24/7 (reciprocal of 3: 1/3, reciprocal of 4: 1/4, arithmetic mean between them 7/24). The harmonic mean applies more accurately to certain situations involving rates.

For example, if a car travels a certain distance at a speed speed 60 km/h and then the same distance again at a speed 40 km/h, then its average speed is the harmonic mean of 48 km/h, and its total travel time is the same as if it had traveled the whole distance at that average speed. However, if the car travels for a certain amount of time at a speed v and then the same amount of time at a speed u, then its average speed is the arithmetic mean of v and u, which in the above example is 50 km/h.

See also: → harmonic; → mean.