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Video: Associative Operations

Tim Burnham

We explain the associative laws of addition and multiplication and go through a series of examples to show how they work. We also show how subtraction and division are not associative.

05:29

Video Transcript

In this video, we’re gonna talk about associativity and how the operations of addition and multiplication are associative but subtraction and division aren’t. The associative law of addition says that when you’ve got an expression that applies addition more than once, the result doesn’t depend upon how the pairs of terms are grouped. So for example, if you have one plus two plus three, it doesn’t matter whether you add the one and the two first and then add that to the three or whether you take one and then add the result of two plus three. Whichever way you do it, you’ll still get the answer six. So in general, 𝑎 plus 𝑏 plus 𝑐 gives the same result as 𝑎 plus 𝑏 plus 𝑐, and that gives the same result as 𝑎 plus 𝑏 plus 𝑐. And this works for all values of 𝑎, 𝑏, and 𝑐 whether they’re integers, fractions, decimals, positive, negative numbers, doesn’t matter. Now I’m sure you’ve all used this property before, even if you haven’t thought of it in quite this way or use these words to describe it. The fact that addition is associative can make adding up a bit easier.

For example, if you have to do this calculation: fifty-eight plus seventy-three plus twenty-seven. If we just did this in order, first of all we’d have to add fifty-eight to seventy-three, which is a hundred and thirty-one. And then we’d have to add the twenty-seven to get our answer of a hundred and fifty-eight. But this part here fifty-eight plus seventy-three wasn’t particularly easy to do in your head. But if we’d have used the fact that addition is associative, we could have done things in a slightly different order. Seventy-three plus twenty-seven is a hundred. So we’d have got fifty-eight plus a hundred, which gives the same answer: a hundred and fifty-eight. So spotting that we had digits that added up to ten or numbers that added up to a hundred just made our calculation a little bit easier. So using associativity of addition can just make your life a bit easier, give you a slightly simpler calculation to do.

Let’s move on to the associative law of multiplication. And that says when you have an expression that applies multiplication more than once, the result doesn’t depend on how the pairs of terms are grouped. Looks pretty familiar, huh? So for example, if you have to calculate four times five times six, you’ll get the same result if you do four times five and then multiply that by six or if you do four times the result of five times six. So that means that either twenty times six or four times thirty give you the same result, which gives you a result of one hundred and twenty in either case. And more generally, 𝑎 times 𝑏 times 𝑐 gives you the same result as 𝑎 times 𝑏 times 𝑐 and that gives you the same result as 𝑎 times 𝑏 times 𝑐. And again, this works for all values of 𝑎, 𝑏, and 𝑐 whether they’re integers, fractions, decimals, or positive or negative numbers.

Now let’s think about subtraction and division. Subtraction is not associative. So for example, if we think about ten take away five take away two, if we thought of this is as ten take away five take away two, that would give us five take away two and give an answer of three. But if we thought of it as ten take away five take away two, that would work out as ten take away three, which would give us an answer of seven. And clearly seven and three are not equal. And similarly division is not associative either. So if we took the example sixteen divided by four divided by two, if we grouped the first two terms together, we get sixteen divided by four, which is four. So that becomes four divided by two, which is two. But if we associate the second two terms together, four divided by two is two, so that gives us sixteen divided by two, which is equal to eight. And clearly, two and eight are different. So addition and multiplication are associative, but subtraction and division are not associative. Be careful though, addition and multiplication aren’t associative with each other. So if we try and calculate something like this, three plus five times seven, we’re mixing the operations. And you need to use the normal roles of the order of operations to evaluate that. So whether you use PEMDAS, BIDMAS, or BODMAS, whatever you use, that is required in this situation. So if I grouped the first two terms together, I’d have three plus five, which is eight. And that would be eight times seven, which is fifty-six. If I grouped the second two terms together, I would have three plus five times seven. And five times seven is thirty-five, so that gives me a total of thirty-eight. And as we all know, thirty-eight is the correct answer, fifty-six is the wrong answer. But the main point is, when we’re combining addition and multiplication, that associativity breaks down. It really does matter what order you do things in.

So to summarise then, addition is associative. Multiplication is also associative. But subtraction and division are not associative. And one last thing, associativity scales up to larger strings of terms. So for example, with addition, if there are pairs of terms that we think we can add together in our heads more easily, it’s okay to do those first because it doesn’t matter what order we add these things together in. And the same is true with multiplication.