Associative property of multiplication

Basic multiplication is a pairwise operation. To multiply more than two numbers, you must combine them in pairs successively until all the numbers have been used. For example, to multiply <math>X = 3 \times 4 \times 5</math> you first pick two consecutive numbers, say 3 and 4, and multiply them: <math>X = 12 \times 5</math>. Now there are just two numbers remaining, which you can multiply to get the final answer: <math>X = 60</math>.

The associative property of multiplication is the fact that the answer does not depend on how the pairings are done. For example, we could have started with <math>4 \times 5 = 20</math>, and then done <math>X = 3 \times 20 = 60</math>. We use parentheses to indicate the order of multiplication used. For example, writing <math>(3 \times 4)\times 5</math> means that you first multiply <math>3 \times 4</math> then multiply the result by 5. The associative property is expressed by the formula <math>(3\times 4) \times 5 = 3 \times(4 \times 5)</math>.

With more than three numbers, there can be many ways to do the multiplication. For example, with 4 numbers one of the ways is <math>2 \times 3 \times 4 \times 5 = ((2 \times 3) \times 4 )\times 5 = (6 \times 4 )\times 5 = 24\times 5 = 120</math>.

The associative property of multiplication is different from the commutative property. For example, multiplication of matrices has the associative property but is not commutative. If <math>A</math>, <math>B</math>, and <math>C</math> are three matrices, then <math>A\times(B \times C)=(A\times B) \times C </math> but <math>A\times B \neq B \times A </math>.