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 coll ...
(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 (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
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, size, ...
:
; 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,
, 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 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 i ...
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 f ...
:
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
Hippasus of Metapontum (; grc-gre, Ἵππασος ὁ Μεταποντῖνος, ''Híppasos''; c. 530 – c. 450 BC) was a Greek philosopher and early follower of Pythagoras. Little is known about his life or his beliefs, but he is sometimes ...
. The Pythagorean means also appear in
Plato's
Timaeus Timaeus (or Timaios) is a Greek name. It may refer to:
* ''Timaeus'' (dialogue), a Socratic dialogue by Plato
* Timaeus of Locri, 5th-century BC Pythagorean philosopher, appearing in Plato's dialogue
*Timaeus (historian) (c. 345 BC-c. 250 BC), Gree ...
. Another evidence of their early use is a commentary by
Pappus.
The term "mean" (μεσότης, mesótēs in Ancient Greek) appears in the
Neopythagorean
Neopythagoreanism (or neo-Pythagoreanism) was a school of Hellenistic philosophy which revived Pythagorean doctrines. Neopythagoreanism was influenced by middle Platonism and in turn influenced Neoplatonism. It originated in the 1st century BC a ...
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
In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7 ...
*
Golden ratio
*
Kepler triangle
A Kepler triangle is a special right triangle with edge lengths in geometric progression. The ratio of the progression is \sqrt\varphi where \varphi=(1+\sqrt)/2 is the golden ratio, and the progression can be written: or approximately . Squa ...
References
External links
*{{MathWorld, urlname=PythagoreanMeans, title=Pythagorean Means, author=Cantrell, David W.
Means