# 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 .

*The Columbia Electronic Encyclopedia,* 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.

**See more Encyclopedia articles on: **Mathematics