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'contraharmonic proportion'
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| Title of object: |
contraharmonic proportion |
| Canonical Name: |
ContraharmonicProportion |
| Type: |
Definition |
| Created on: |
2008-02-14 13:28:51 |
| Modified on: |
2008-02-14 13:28:51 |
| Classification: |
msc:11-00, msc:62-07 |
| Defines: |
contraharmonic mean |
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Content:
Three positive numbers $x$, $m$, $y$ are in {\em 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
\begin{align}
\frac{m-x}{y-m} = \frac{y}{x}.
\end{align}
The middle number $m$ is then called the {\em contraharmonic 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:
$$\frac{m-x}{y-m} = \frac{x}{x} \qquad (\mbox{arithmetic mean }m)$$
$$\frac{m-x}{y-m} = \frac{x}{m} \qquad (\mbox{geometric mean }m)$$
$$\frac{m-x}{y-m} = \frac{x}{y} \qquad (\mbox{harmonic mean }m)$$
The contraharmonic mean $m$ is indeed between $x$ and $y$. Indeed, if we solve it from (1), we get
$$m = \frac{x^2+y^2}{x+y},$$
and if we assume that\, $x \leqq y$, we see that
$$x = \frac{x^2+xy}{x+y} \leqq \frac{x^2+y^2}{x+y} \leqq \frac{xy+y^2}{x+y} = y.$$ |
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