Associative Property

The associative property is a property of certain binary operations under which rearranging the parentheses in an expression made up of two or more instances of the same operator does not change the result, so long as the order of the operands themselves stays the same; for example (2 plus 3) plus 4 equals 2 plus (3 plus 4), and 2 times (3 times 4) equals (2 times 3) times 4, since addition and multiplication of real numbers are both associative. Associativity is not the same as commutativity, which concerns whether swapping the order of two operands changes the result: function composition and matrix multiplication are associative but generally not commutative, and operations such as subtraction, exponentiation and the vector cross product are not associative at all. Associative operations are common throughout mathematics, and many algebraic structures such as semigroups and categories explicitly require their operations to be associative; by contrast, the addition of floating point numbers in computer science is not associative, and the order in which such an expression is evaluated can noticeably affect rounding error.