In mathematics, the three classical Pythagorean means are the arithmetic mean (A), the geometric mean (G), and the harmonic mean (H). They are defined by:

Each mean has the following properties:

• Value preservation:
• First order homogeneity:
• Invariance under exchange: for any and .
• Averaging:

These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians because of their importance in geometry and music. The harmonic and arithmetic means are reciprocal duals of each other for positive arguments () while the geometric mean is its own reciprocal dual.