Many mathematical functions are methods for operating on two numbers to get a new number. When working with more than two numbers, you operate on one pair of numbers at a time until the list is exhausted. If you always get the same answer no matter how you pair off the numbers, the function is associative. If you can get different answers depending on how you pair off the numbers, the function is non-associative.

## Addition

Given A + B + C, does it matter which addition you perform first? For example, if A = 8, B = 4, and C = 2, is (8 + 4) + 2 equivalent to 8 + (4 + 2)? You can answer this question by representing A, B and C with groups of blocks (or beads). You can either add eight blocks to four blocks, and then add two blocks to that total -- or you can add four blocks to two blocks, and then add eight blocks to that total. The resulting grand pile will be the same in both cases: 14 blocks. It does not matter which numbers you add first because addition is associative.

## Multiplication

Multiplication also is associative. For example, consider the formula for the volume of a box: Volume = length x width x height. You can either multiply the length times the width, and then multiply that result by the height -- or you can multiply the width times the height, and then multiply that result by the length. If a box is 8 inches long, 4 inches wide, and 2 inches high, multiplying (8 x 4) x 2 = 8 x (4 x 2) = 64. It does not matter how you associate or group the numbers that contribute to the final answer.

## Subtraction

Subtraction, on the other hand, is not associative. You can see this if you take the same numbers used above, but substitute a minus sign for the preceding signs. The resulting sequences are (8 - 4) - 2 followed by 8 - (4 - 2). If you carry out the subtraction, you will see that (8 - 4) - 2 = 2, but 8 - (4 - 2) = 6. The two sequences are not equivalent. It matters how you group or associate numbers that you are going to subtract from each other, so subtraction is not associative.

## Division

Division also is not associative. Again take the numbers used above, but substitute a division sign for the preceding signs. The resulting sequences are (8 / 4) / 2 followed by 8 / (4 / 2). If you carry out the division, you will see that (8 / 4) / 2 = 1, but 8 / (4 / 2) = 4. The two sequences are not equivalent. It matters how you group or associate numbers that you are going to divide, so division is not associative.

#### References

- The Math Dude: What is the Associative Property?
- Crewton Ramone's House of Math: The Associative Property of Addition
- National Security Agency.gov: Concept Development Unit: Do You Commute? Do You Associate? The Commutative and Associative Properties and How They Apply to Addition and Multiplication
- Math Warehouse: Definition of Associative Property

#### Resources

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