# commutative law

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commutative law
a law asserting that the order in which certain logical operations are performed is indifferent.
[1835-45]

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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.

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in mathematics, either of two laws relating to number operations of addition and multiplication, stated symbolically: a + b = b + a and ab = ba. From these laws it follows that any finite sum or product is unaltered by reordering its terms or factors. While commutativity holds for many systems, such as the real or complex numbers, there are other systems, such as the system of n × n matrices or the system of quaternions, in which commutativity of multiplication is invalid. Scalar multiplication of two vectors (to give the so-called dot product) is commutative (i.e., a·b = b·a), but vector multiplication (to give the cross product) is not (i.e., a × b = −b × a). The commutative law does not necessarily hold for multiplication of conditionally convergent series. See also associative law; distributive law.

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Universalium. 2010.