• 1.2: Associative Law of Addition and Multiplication

    Photo of a pile of money in different currencies.

    Just after closing, you accidentally knock over the cash register at your aunt's coffee shop, and all of the day's money spills out onto the floor. You carefully put the five-dollar bills into one pile, all of the one-dollar bills into another pile, and all of the quarters, dimes, nickels, and pennies into their own piles. Then you add up the total. Thankfully, you check your math, and it matches the day's receipt: $1,500.00 exactly. Would you have tallied up this same amount if you had placed both five- and one-dollar bills into a single pile and grouped all of the coins into another separate pile? Would having a different grouping alter their count?

    Here is another puzzling story: Average Joe loves math. He is known far and wide for his love of averaging. Whenever you hand him two numbers (say 2 and 10), Average Joe will add them and then divide by 2 (to give 6). If you were to hand him the numbers 5 and 15, he would happily average them to produce the number 10. But what happens if one day Joe's friend Above-Average Sharon hands him three numbers, say 2, 10, and 14? Will Average Joe first compute the average of 2 and 10, yielding 6, and then compute the average of 6 and 8? Or will he first average 10 and 14, yielding 12, and then compute the average of 12 and 2? Does it matter which grouping Average Joe uses to compute his averages?

    In our first story, the change in grouping will not affect the final result, but it does in our second Average Joe story! Certain mathematical combinations or operations depend on how the parts or numbers are grouped, and sometimes they do not.

    The associative property is the adjective we use to describe the operations that are unaffected by grouping. More specifically, the associative law of addition and multiplication tells us that no matter how we group or "associate" the numbers we add or multiply, the outcome remains the same.

    This may sound awfully similar to the commutative property, but it expresses a different (although similarly useful) property. In arithmetic calculations, we often place parentheses around a set of numbers to indicate we are associating or grouping them together. We always carry out the calculation in parentheses first. For example, 5+(2+1)=5+3=8. When we calculate this, we first calculate 2+1=3. Then, we add 5 to get 8. We can also write (5+2)+1=8. Here, we first calculate 5+2=7 and then add 1 to get a sum of 8. We get the same result regardless of how we group the numbers.

    This same law works for multiplication. When we can compute (2\times 2)\times 3=12. We can change the grouping and write 2\times(2\times 3)=12. The placement of the grouping does not change the answer.