Associative property of addition

Basic addition is a pairwise operation. To add more than two numbers, you must combine them in pairs successively until all the numbers have been used. For example, to add you first pick two consecutive numbers, say 3 and 4, and add them: . Now there are just two numbers remaining, which you can add to get the final answer: .

The associative property of addition is the fact that the answer does not depend on how the pairings are done. For example, we could have started with , and then done . We use parentheses to indicate the order of addition used. For example, writing means that you first add then add the result by 5. The associative property is expressed by the formula .

With more than three numbers, there can be many ways to do the addition. For example, with 4 numbers one of the ways is .

The associative property of addition is different from the commutative property. For example, addition of matrices has the associative property but is not commutative. If , , and are three matrices, then but .