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Commutativity is important for e.g. parallel programming. See e.g. <https://youtu.be/dPK6t7echuA>.

In the context of linear algebra, it would also be good for folks to know that matrix multiplication (/linear transformation composition) is not commutative.

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Isn’t it more often associativity that is required for parallel programming? Being able to parallelize “a+b+c+d” only requires associativity to order the operations as “(a+b) + (c+d)”. Sure, there’s additional benefits for memory locality if you can rearrange the term to “(a+c) + (b+d)”, assuming that the four terms are stored in contiguous memory and each sum is computed with vector operations, but that’s not strictly required for parallelization.

In parallel programming, subcomputations may complete out of order. The ability to combine those subcomputations in any order can be pretty useful!

That, though, is associativity, not commutativity.

In FP arithmetic, operations are commutative, not associative.

If you receive b before a and combine them as b*a, that's commutative. Both properties are useful in parallel programming, but you may only need one in a specific case, depending on your application.

Both are important. I replied to a comment about commutativity, so I focused on that.

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