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In mathematics, the three classical Pythagorean means are the
arithmetic mean In mathematics and statistics, the arithmetic mean ( ), arithmetic average, or just the ''mean'' or ''average'' is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results fr ...
(AM), the
geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...
(GM), and the
harmonic mean In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means. It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments. The harmonic mean ...
(HM). These
mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
s were studied with proportions by
Pythagoreans Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the Ancient Greece, ancient Greek co ...
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 Homogeneity and heterogeneity are concepts relating to the Uniformity (chemistry), uniformity of a Chemical substance, substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, ...
: \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 Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
: 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 Inequality may refer to: * Inequality (mathematics), a relation between two quantities when they are different. * Economic inequality, difference in economic well-being between population groups ** Income inequality, an unequal distribution of in ...
and a special case of an inequality for
generalized mean In mathematics, generalised means (or power mean or Hölder mean from Otto Hölder) are a family of functions for aggregating sets of numbers. These include as special cases the Pythagorean means (arithmetic mean, arithmetic, geometric mean, ge ...
s. 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 Nicomachus of Gerasa (; ) was an Ancient Greek Neopythagorean philosopher from Gerasa, in the Roman province of Syria (now Jerash, Jordan). Like many Pythagoreans, Nicomachus wrote about the mystical properties of numbers, best known for his ...
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 Iamblichus ( ; ; ; ) was a Neoplatonist philosopher who determined a direction later taken by Neoplatonism. Iamblichus was also the biographer of the Greek mystic, philosopher, and mathematician Pythagoras. In addition to his philosophical co ...
, was coined by Archytas and
Hippasus Hippasus of Metapontum (; , ''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 credited with the discovery of the existence of irra ...
. The Pythagorean means also appear in
Plato Plato ( ; Greek language, Greek: , ; born  BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...
'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 In mathematics, the arithmetic–geometric mean (AGM or agM) of two positive real numbers and is the mutual limit of a sequence of arithmetic means and a sequence of geometric means. The arithmetic–geometric mean is used in fast algorithms f ...
*
Average In colloquial, ordinary language, an average is a single number or value that best represents a set of data. The type of average taken as most typically representative of a list of numbers is the arithmetic mean the sum of the numbers divided by ...
*
Golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...
* Kepler triangle * QM-AM-GM-HM inequalities


Notes


References


External links

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