Why is it called commutative property?

I can see why distributive is called distributive (distribute whatever you are multiplying to everything within the brackets).

Associative because when the same associative operator appear in a row, you can change around the numbers.

But why is the commutative property called commutative?

• I've always wondered this, but apparently never enough to ask! +1 – Matt Samuel Dec 19 '18 at 2:48
• I think of it like a commuter train. They move around or switch places. Well, not when I was in elementary school when I was expected to know those terms, as I didn't even know the word commuter, but later in high school I did, when no one cared about those properties anymore, lol. – Octopus Dec 19 '18 at 5:24
• As often, the best source for this kind of questions is Earliest Known Uses of Some of the Words of Mathematics : see COMMUTATIVE and DISTRIBUTIVE. – Mauro ALLEGRANZA Dec 19 '18 at 16:01
• @Octopus lol I never cared until it kept reoccuring when studying math in university (algebra, fields, subspaces etc.) – Jay Patel Dec 19 '18 at 18:28

3 Answers

From the Wikipedia article "Commutative Property", under History and Etymology:

The first recorded use of the term commutative was in a memoir by François Servois in 1814, which used the word commutatives when describing functions that have what is now called the commutative property. The word 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." The term then appeared in English in 1838 in Duncan Farquharson Gregory's article entitled "On the real nature of symbolical algebra" published in 1840 in the Transactions of the Royal Society of Edinburgh.

• Nice etymology! I'll just add "commutare" from Latin, meaning, "to exchange," from which the French "commuter" is derived. – Christopher Marley Dec 19 '18 at 2:50

https://www.merriam-webster.com/dictionary/commutative

https://www.merriam-webster.com/dictionary/commutation

Commutative, from commutation, which means exchange, trade, or replacement according to the first 2 definitions. The commutative property says that the order in which the operation is carried out does not matter. You can exchange/trace factors or addends and still arrive at the same product or sum.

https://www.thefreedictionary.com/commutative

Definition one of the adjective form gives "involving substitution, interchange"

So you just switch or commute the two addends or factors and get the same sum or product!

• I like this a lot since I speak english. (Commutative, communication meaning exchange). The other explanation uses french / latin words. – Jay Patel Dec 19 '18 at 3:05
• I see the word in legal context, " a death sentence has been commuted to life imprisonment". Looks like here also it means replacement. – P Vanchinathan Dec 19 '18 at 3:07

Seems pretty reasonable since commute means to move around or change places, approximately. One of the most basic ways of moving elements around is to switch their order.

I guess another word could have been used. Of course "abelian" is used for groups, after Niels Henrik Abel.