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 <math>X = 3 + 4 + 5</math> you first pick two consecutive numbers, say 3 and 4, and add them: <math>X = 7 + 5</math>. Now there are just two numbers remaining, which you can add to get the final answer: <math>X = 12</math>.

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 <math>4 + 5 = 9</math>, and then done <math>X = 3 + 9 = 12</math>. We use parentheses to indicate the order of addition used. For example, writing <math>(3 + 4)+ 5</math> means that you first add <math>3 + 4</math> then add the result by 5. The associative property is expressed by the formula <math>(3+ 4) + 5 = 3 +(4 + 5)</math>.

With more than three numbers, there can be many ways to do the addition. For example, with 4 numbers one of the ways is <math>2 + 3 + 4 + 5 = ((2 + 3) + 4 )+ 5 = (5 + 4 )+ 5 = 9+ 5 = 14</math>.

The associative property of addition is different from the commutative property.