Pythagorean means explained

In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians^[1] because of their importance in geometry and music.

Definition

They are defined by:

Properties

Each mean, $\backslash operatorname$, has the following properties:

First-order homogeneity:

\operatorname{M}(bx _{1,} \ldots, bx _{n)} = b \operatorname{M}(x _{1,} \ldots, x _{n)}

Invariance under exchange:

\operatorname{M}(\ldots, x _{i,} \ldots, x _{j,} \ldots) = \operatorname{M}(\ldots, x _{j,} \ldots, x _{i,} \ldots)

for any

i and j .

Monotonicity:

a \leq b → \operatorname{M}(a,x _{1,x} _{2,\ldots} x _{n)} \leq \operatorname{M}(b,x _{1,x} _{2,\ldots} x _{n)}

Idempotence:

\forall x, M(x,x,\ldots x) = x

Monotonicity and idempotence together imply that a mean of a set always lies between the extremes of the set:$$\backslash min(x\_1,\backslash ,\; \backslash ldots,\backslash ,\; x\_n)\; \backslash leq\; \backslash operatorname(x\_1,\backslash ,\; \backslash ldots,\backslash ,\; x\_n)\; \backslash leq\; \backslash max(x\_1,\backslash ,\; \backslash ldots,\backslash ,\; x\_n).$$

The harmonic and arithmetic means are reciprocal duals of each other for positive arguments,$$\backslash operatorname\backslash left(\backslash frac,\backslash ,\; \backslash ldots,\backslash ,\; \backslash frac\backslash right)\; =\; \backslash frac,$$

while the geometric mean is its own reciprocal dual:$$\backslash operatorname\backslash left(\backslash frac,\backslash ,\; \backslash ldots,\backslash ,\; \backslash frac\backslash right)\; =\; \backslash frac.$$

Inequalities among means

There is an ordering to these means (if all of the

x _{i} are positive)$$\backslash min\; \backslash leq\; \backslash operatorname\; \backslash leq\; \backslash operatorname\; \backslash leq\; \backslash operatorname\; \backslash leq\; \backslash max$$with equality holding if and only if the x _{i} are all equal.

This is a generalization of the inequality of arithmetic and geometric means and a special case of an inequality for generalized means. The proof follows from the arithmetic–geometric mean inequality,

\operatorname{AM} \leq max , and reciprocal duality (min and max are also reciprocal dual to each other).

The study of the Pythagorean means is closely related to the study of majorization and Schur-convex functions. The harmonic and geometric means are concave symmetric functions of their arguments, and hence Schur-concave, while the arithmetic mean is a linear function of its arguments and hence is both concave and convex.

History

Almost everything that we know about the Pythagorean means came from arithmetic handbooks written in the first and second century. Nicomachus of Gerasa says that they were "acknowledged by all the ancients, Pythagoras, Plato and Aristotle."^[2] Their earliest known use is a fragment of the Pythagorean philosopher Archytas of Tarentum:

The name "harmonic mean", according to Iamblichus, was coined by Archytas and Hippasus. The Pythagorean means also appear in Plato's Timaeus. Another evidence of their early use is a commentary by Pappus.

The term "mean" (μεσότης, mesótēs in Ancient Greek) appears in the Neopythagorean arithmetic handbooks in connection with the term "proportion" (ἀναλογία, analogía in Ancient Greek).

Smallest distinct positive integer means

a	b	HM	GM	AM
5	45	9	15	25
10	40	16	20	25	a	5	10
b	45	40
HM	9	16
GM	15	20
AM	25	25-->

Of all pairs of different natural numbers of the form (a, b) such that a < b, the smallest (as defined by least value of a + b) for which the arithmetic, geometric and harmonic means are all also natural numbers are (5, 45) and (10, 40).^[3]

See also

• Arithmetic–geometric mean
• Average
• Golden ratio
• Kepler triangle

Notes and References

1. Book: Heath, Thomas. History of Ancient Greek Mathematics.
2. Book: Gerasa.), Nicomachus (of . Introduction to Arithmetic . 1926 . Macmillan . en.
3. Virginia Tech Mathematics Department, 39th VTRMC, 2017, Solutions, part 5