commutative law, in mathematics, law holding that for a given binary operation (combining two quantities) the order of the quantities is arbitrary; e.g., in addition, the numbers 2 and 5 can be combined as 2+5=7 or as 5+2=7. More generally, in addition, for any two numbers a and b the commutative law is expressed as a+b=b+a. Multiplication of numbers is also commutative, i.e., a×b=b×a. In general, any binary operation, symbolized by ∘, joining mathematical entities A and B obeys the commutative law if A∘B=B∘A for all possible choices of A and B. Not all operations are commutative; e.g., subtraction is not since 2−5≠5−2, and division is not since 2⁄5≠5⁄2.
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