Pythagorean Means

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 because of their importance in geometry and music.

Definition

The three Pythagorean means are defined by the equations
\begin
\operatorname \left( x_1,\; \ldots,\; x_n \right) &= \frac, \\ pt \operatorname \left( x_1,\; \ldots,\; x_n \right) &= \sqrt \text \\ pt \operatorname \left( x_1,\; \ldots,\; x_n \right) &= \frac .
\end

Properties

Each mean, $\operatorname$, has the following properties for positive real inputs:

; First-order homogeneity: $\operatorname(bx_1, \ldots, bx_n) = b \operatorname(x_1, \ldots, x_n)$
; Invariance under exchange: $\operatorname(\ldots, x_i, \ldots, x_j, \ldots) = \operatorname(\ldots, x_j, \ldots, x_i, \ldots)$ for any $i$ and $j$.
; Monotonicity: if $a \leq b$ then $\operatorname(a,x_1,x_2,\ldots x_n) \leq \operatorname(b,x_1,x_2,\ldots x_n)$
; Idempotence: $M(x,x,\ldots x) = x$ for all $x$

Monotonicity and idempotence together imply that a mean of a set always lies between the extremes of the set:
$\min(x_1, \ldots, x_n) \leq \operatorname(x_1, \ldots, x_n) \leq \max(x_1, \ldots, x_n).$

The harmonic and arithmetic means are reciprocal duals of each other for positive arguments,
$\operatorname\left(\frac, \ldots, \frac\right) = \frac,$
while the geometric mean is its own reciprocal dual:
$\operatorname\left(\frac, \ldots, \frac\right) = \frac.$

Inequalities among means

There is an ordering to these means (if all of the $x_i$ are positive)
$\min \leq \operatorname \leq \operatorname \leq \operatorname \leq \max$
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, $\operatorname \leq \max$, 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." 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" (Ancient Greek μεσότης, ''mesótēs'') appears in the Neopythagorean arithmetic handbooks in connection with the term "proportion" (Ancient Greek ἀναλογία, ''analogía'').

Smallest distinct positive integer means

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).Virginia Tech Mathematics Department
''39th VTRMC, 2017, Solutions''
part 5

See also

* Arithmetic–geometric mean
* Average
* Golden ratio
* Kepler triangle
* QM-AM-GM-HM inequalities

Notes

References

External links

*{{MathWorld, urlname=PythagoreanMeans, title=Pythagorean Means, author=Cantrell, David W.

Means
Greek mathematics