In mathematics, the three classical Pythagorean means are the
arithmetic mean
In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The colle ...
(AM), the
geometric mean
In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...
(GM), and the
harmonic mean
In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. It is sometimes appropriate for situations when the average rate is desired.
The harmonic mean can be expressed as the recipro ...
(HM). These
means were studied with proportions by
Pythagoreans and later generations of Greek mathematicians
because of their importance in geometry and music.
Definition
They are defined by:
:
Properties
Each mean,
, has the following properties:
; First order
homogeneity:
; Invariance under exchange:
: for any
and
.
; Monotonicity:
;
Idempotence:
Monotonicity and idempotence together imply that a mean of a set always lies between the extremes of the set.
:
The harmonic and arithmetic means are reciprocal duals of each other for positive arguments:
:
while the geometric mean is its own reciprocal dual:
:
Inequalities among means
There is an ordering to these means (if all of the
are positive)
:
with equality holding if and only if the
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
In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and ...
,
, and reciprocal duality (
and
are also reciprocal dual to each other).
The study of the Pythagorean means is closely related to the study of
majorization In mathematics, majorization is a preorder on vectors of real numbers. Let ^_,\ i=1,\,\ldots,\,n denote the i-th largest element of the vector \mathbf\in\mathbb^n. Given \mathbf,\ \mathbf \in \mathbb^n, we say that \mathbf weakly majorizes (or ...
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, so 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
Nicomachus of Gerasa ( grc-gre, Νικόμαχος; c. 60 – c. 120 AD) was an important ancient mathematician and music theorist, best known for his works ''Introduction to Arithmetic'' and ''Manual of Harmonics'' in Greek. He was born in ...
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
Archytas (; el, Ἀρχύτας; 435/410–360/350 BC) was an Ancient Greek philosopher, mathematician, music theorist, astronomer, statesman, and strategist. He was a scientist of the Pythagorean school and famous for being the reputed found ...
:
The name "harmonic mean", according to
Iamblichus
Iamblichus (; grc-gre, Ἰάμβλιχος ; Aramaic: 𐡉𐡌𐡋𐡊𐡅 ''Yamlīḵū''; ) was a Syrian neoplatonic philosopher of Arabic origin. He determined a direction later taken by neoplatonism. Iamblichus was also the biographer of ...
, 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).
Curiosity
The smallest pairs of different natural numbers for which the arithmetic, geometric and harmonic means are all also natural numbers are (5,45) and (10,40).
See also
*
Arithmetic–geometric mean
*
Average
*
Golden ratio
*
Kepler triangle
References
External links
*{{MathWorld, urlname=PythagoreanMeans, title=Pythagorean Means, author=Cantrell, David W.
Means