PlanetMath: contraharmonic proportion

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (200)
Orphanage
Unclass'd
Unproven (511)
Corrections (9)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

contraharmonic proportion

(Definition)

Three positive numbers , , are in contraharmonic proportion, if the ratio of the difference of the second and the first number to the difference of the third and the second number is equal the ratio of the third and the first number, i.e. if

$\displaystyle \frac{m-x}{y-m} = \frac{y}{x}.$

(1)

The middle number

is then called the contraharmonic mean (sometimes antiharmonic mean) of the first and the last number.

The contraharmonic proportion has very probably been known in the proportion doctrine of the Pythagoreans, since they have in a manner similar to (1) described the classical Babylonian means:

$\displaystyle \frac{m-x}{y-m} = \frac{x}{x} \qquad ($ arithmetic mean $\displaystyle m)$

$\displaystyle \frac{m-x}{y-m} = \frac{x}{m} \qquad ($ geometric mean $\displaystyle m)$

$\displaystyle \frac{m-x}{y-m} = \frac{x}{y} \qquad ($ harmonic mean $\displaystyle m)$

The contraharmonic mean is between and . Indeed, if we solve it from (1), we get

$\displaystyle m = \frac{x^2+y^2}{x+y},$

(2)

and if we assume that $x \leqq y$ , we see that

$\displaystyle x = \frac{x^2+xy}{x+y} \leqq \frac{x^2+y^2}{x+y} \leqq \frac{xy+y^2}{x+y} = y.$

The contraharmonic mean

is the greatest of all the mentioned means,

$\displaystyle x \leqq h \leqq g \leqq a \leqq c \leqq y$

where

is the arithmetic mean,

the geometric mean and

the harmonic mean. It is easy to see that

$\displaystyle \frac{c+h}{2} = a$ and $\displaystyle \quad \sqrt{ah} = g.$

Example. The integer 5 is the contraharmonic mean of 2 and 6, as well as of 3 and 6, i.e. 2, 5, 6, are in contraharmonic proportion, similarly are 3, 5, 6:

$\displaystyle \frac{2^2+6^2}{2+6} = \frac{40}{8} = 5, \quad \frac{3^2+6^2}{3+6} = \frac{45}{9} = 5$

Note. Generalising (2) one defines the contraharmonic mean of several positive numbers:

$\displaystyle c(x_1,\,\ldots,\,x_n) := \frac{x_1^2+\ldots+x_n^2}{x_1+\ldots+x_n}$

There is also a more general type of mean:

$\displaystyle c^m(x_1,\,\ldots,\,x_n) := \frac{x_1^{m+1}+\ldots+x_n^{m+1}}{x_1^m+\ldots+x_n^m}$

Bibliography

1: DIDEROT & D'ALEMBERT: Encyclopédie. Paris (1751-1777). Electronic version in L'Encyclopédie de Diderot et d'Alembert.
2: HORST HISCHER: ``Viertausend Jahre Mittelwertbildung''. -- mathematica didactica 25 (2002). See also this.

"contraharmonic proportion" is owned by pahio.

(view preamble)

Also defines:

contraharmonic mean, antiharmonic mean

This object's parent.

Attachments:: comparison of Pythagorean means (Topic) by pahio

Log in to rate this entry.
(view current ratings)

Cross-references: mean, integer, easy to see, harmonic mean, geometric mean, arithmetic mean, Proportion, difference, ratio, numbers, positive
There are 2 references to this entry.

This is version 9 of contraharmonic proportion, born on 2008-02-14, modified 2008-02-17.
Object id is 10269, canonical name is ContraharmonicProportion.
Accessed 579 times total.

Classification:

AMS MSC:	01A17 (History and biography :: History of mathematics and mathematicians :: Babylonian)
	01A20 (History and biography :: History of mathematics and mathematicians :: Greek, Roman)
	11-00 (Number theory :: General reference works )
	62-07 (Statistics :: Data analysis)

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact