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Nit: while multiplication is commutative, the property you meant to use was symmetry. I'm sure it was a mistake and you know this, I'm just pointing it out to prevent confusion for others.

Edit: s/competitive/commutative/

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Huh. I heard (US high school algebra in a recent unspecified century) the fact that ab=ba called "the commutative property of multiplication", and the "symmetric property of equality" meant a=b <=> b=a, and didn't run across anything later on (comp sci from a department of the engineering college rather than the math department) inconsistent with that. Symmetry sounds like a reasonable term for both, but it sounds like there's a distinction I don't get?

https://en.wikipedia.org/wiki/Commutative_property

> In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.

Are you saying you disagree?

No, I had a brainfart and was thinking of associativity. I have no idea how it happened. Very embarrassing.

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