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
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
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