In this video, we’re gonna talk about commutativity, or the commutative law of operations. Addition and multiplication are both operations that are commutative, but subtraction and division aren’t.
The commutative law of addition is when applying the operation of addition to two terms, you get the same result regardless of their order. For example, if I do three add five, I get eight. And if I do five plus three, I also get eight. And viewing that on a number line, that makes good sense. If I do three plus five, I start at zero. I’m gonna go for the three and then I’m gonna add five. So one, two, three, four, five, end up at eight. And if I do five plus three, I start off at five and I do one, two, three, I still end up at eight.
And more generally, if I add two numbers 𝑎 and 𝑏, I get 𝑎 plus 𝑏. But if I add them in the other order, 𝑏 plus 𝑎, I still get the result 𝑎 plus 𝑏. And that works for whatever values of 𝑎 and 𝑏 you use, whether that integers, fractions, decimals, positive, or negative numbers.
Now because the commutative law of addition means that you can add pairs of numbers in any order and get the same result, this scales up to longer strings of terms added together. And to show how that works, consider seventy-one plus three plus twenty-nine plus seven. Now if we just added those up in order, we’d have to add seventy-one to three first. So that will give us seventy-four plus twenty-nine plus seven. Now we got to add seventy-four and twenty-nine, which is a hundred and three. And then we got to add seven, and then that gives us an overall answer of one hundred and ten. But, by scanning that calculation beforehand, we could’ve seen that three and seven add together to make ten, and seventy-one and twenty-nine add together to make a hundred. So by swapping over the three and the twenty-nine, we get seventy-one plus twenty-nine plus three plus seven. Then, as we said, seventy-one plus twenty-nine is a hundred. A hundred plus three, that’s easy. It’s a hundred and three. And a hundred and three plus seven is a hundred and ten. So, by doing that swapping around, we’ve got to the same answer, but we had some easier steps to get there.
Now looking at the commutative law of multiplication, when applying the operation of multiplication to two terms, you get the same result regardless of their order. So for example, if we do four times six, we get twenty-four. Well, switching those around, if we do six times four, we also get twenty-four.
Or more generally, whether you do 𝑎 times 𝑏 or 𝑏 times 𝑎, you get the same result 𝑎𝑏. And again, that works for all values of 𝑎 and 𝑏, whether that integers, fractions, decimals, positive, or negative numbers. And when you think about multiplication visually, this still makes good sense. Four times six is the same as six times four. Whether we have four rows of six toy submarines or six rows of four toy submarines, we can group them into the same twenty-four toy submarines.
Now subtraction and division are not commutative because you can get different results depending on what order you arrange the terms. So for example, seven minus four is not the same as four minus seven and ten divided by five is not the same as five divided by ten.
So to summarize then, addition is commutative because 𝑎 plus 𝑏 gives you the same result as 𝑏 plus 𝑎 and that works for all values of 𝑎 and 𝑏. And multiplication is commutative because 𝑎 times 𝑏 equals 𝑏 times 𝑎 and again that works for all values of 𝑎 and 𝑏. And subtraction and division are not commutative, for example, because seven minus four is not the same as four minus seven. And also, for example, because ten divided by five is not the same as five divided by ten.