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For the commutative property ...

According to wikipedia:

The word "commutative" is a combination of the French word commuter meaning "to substitute or switch" and the suffix -ative meaning "tending to" so the word literally means "tending to substitute or switch."

Therefore the choice of the word commutative to represent the concept of commutative property makes sense.

if you switch the order of the operands, you get the same result

a * b = b * a

What is the corresponding story for the associative property?

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In French, associer means making links and connections. Therefore, associative literally means tending to make links and connections. If $\star$ is an associative law, one has: $$(a\star b)\star c=a\star(b\star c).$$ With an associative law, you get the same result regardless of the pairwise associations.

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    $\begingroup$ Do you have any textual evidence for your etymological claims? $\endgroup$ – Rob Arthan Dec 21 '15 at 22:52
  • $\begingroup$ Not really, this is the explication my professors in Classe Préparatoire gave us. $\endgroup$ – C. Falcon Dec 21 '15 at 22:58
  • $\begingroup$ Well, the word associative can certainly be associated with latin ad-sociare (itself coming from ad=at/to and socius=companion), so loosely something like "connect to something to turn it into a companion" $\endgroup$ – justanotherhagman Dec 22 '15 at 13:28
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I think Hamilton came up with the term "associative," probably as a result of thinking about the (now-called) octonions that Cayley was writing to him about. If you'll allow me to speculate, I think it's likely it's because in parenthesized expressions such as $a(bc)$, two terms that are immediately concatenated together can be called "associated" in the standard English meaning of the term.

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    $\begingroup$ Indeed, it was Hamilton: "However, in virtue of the same definitions, it will be found that another important property of the old multiplication is preserved, or extended to the new, namely, that which may be called the associative character of the operation..." jeff560.tripod.com/a.html $\endgroup$ – Rahul Dec 22 '15 at 6:07
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I don't know if this is the origin, but I like to make an analogy to when two people are associates -- they're business partners (or have some similar relationship).

So when we write $(a + b) + c$, then $a$ and $b$ are hanging out together -- they're associates. But the associative property tells us that this is the same as $a + (b + c)$, when $b$ and $c$ are the ones hanging out together.

So colloquially, the associative property tells us that it doesn't matter how we group our terms (or factors) into pairs of associates -- who we associate with whom -- to perform the individual additions (or multiplications).

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    $\begingroup$ Indeed, some textbooks state that the grouping (or association) of the numbers does not affect the result. $\endgroup$ – skullpatrol Jun 11 '16 at 4:10

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