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Clearly, the word conjugate or conjugation is used for a myriad of different concepts across mathematics and even in science (see the Wikipedia page).

Its meaning can range from the fraction used to rationalize a denominator in pre-algebra, to the $gNg^{-1}$ action in group theory. Complex numbers have conjugates, and harmonic functions can have “harmonic conjugates”.

Surprisingly, I can’t find an explanation of the origin of the word “conjugate” to describe all these cases, nor an explicit meaning or mathematical definition of the word itself.

What is the history of this word, and what (if any) unifying concepts tie the examples together?

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The word, simply put, means "coupled". In Spanish, you have the word "cónyuge" for the husband or wife of a person. Here we usually have a certain concept (the conjugates of a root, the harmonic conjugate), or operation (elements in the same orbit of an action) and we like to identify elements related under this concept.

I wouldn't think there is a unifying (mathematical concept) that tie these examples together. At least, I wouldn't bet on it.

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    $\begingroup$ There is also "conjugal relations" in English. $\endgroup$ – Suzu Hirose Nov 30 '14 at 4:29
  • $\begingroup$ @SuzuHirose Indeed. $\endgroup$ – Pedro Tamaroff Nov 30 '14 at 4:31
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    $\begingroup$ It comes from the Latin conjux, giving rise to the stem conjug- which means spouse. $\endgroup$ – guest196883 Nov 30 '14 at 4:35
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    $\begingroup$ If there's a unifying concept, it's "operations with order 2". $\endgroup$ – Chas Brown Nov 30 '14 at 4:47
  • $\begingroup$ Now try to compare it with the word "dual". $\endgroup$ – Mehrdad Nov 30 '14 at 7:06

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