Associative property of multiplication

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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 X = 3 \times 4 \times 5 you first pick two consecutive numbers, say 3 and 4, and multiply them: X = 12 \times 5. Now there are just two numbers remaining, which you can multiply to get the final answer: X = 60.

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 4 \times 5 = 20, and then done X = 3 \times 20 = 60. We use parentheses to indicate the order of multiplication used. For example, writing (3 \times 4)\times 5 means that you first multiply 3 \times 4 then multiply the result by 5. The associative property is expressed by the formula (3\times 4) \times 5 = 3 \times(4 \times 5).

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

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 A, B, and C are three matrices, then A\times(B \times C)=(A\times B) \times C but A\times B \neq B \times A .

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