commutative law

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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commutative law

Two closely related laws of number operations. In symbols, they are stated: a + b = b + a and ab = ba. Stated in words: Quantities to be added or multiplied can be combined in any order. More generally, if two procedures give the same result when carried out in arbitrary order, they are commutative. Exceptions occur (e.g., in vector multiplication). See also associative law, distributive law.