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Associative and commutative properties of addition with negatives

The commutative and associative properties apply to addition, but not to subtraction. Rewriting subtraction as addition of the opposite is a superpower that suddenly lets us use the commutative and associative properties (without changing the value of the expression) in expressions that previously had subtraction involved. Created by Sal Khan.

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Video transcript

- [Instructor] What we're going to do in this video is evaluate this pretty hairy expression and we could just try to do it. We could go from left to right, but it feels like there might be a simpler way to do it. I'm adding 13 here and then I'm subtracting 13. I'm, I have a negative five here and then I have a positive five. You might be tempted to say, "Well, maybe I could change the order of, that I'm adding and subtracting things." Well, if we were just adding a bunch of numbers you could change the order. For example, we know that nine plus seven is equal to 16, and if I just change the order the commutative property tells us, well, seven plus nine, it applies to addition. If I swap the order here it's still going to be equal to 16, but that's not true if I do nine minus seven or seven minus nine. If I change the order, I'm not getting the same result. The commutative property does not apply to subtraction. This top expression is positive two. This bottom expression is negative two. So I can't just use the commutative property here to change the order with which I'm adding and subtracting because I have subtraction here. But what if I could rewrite this expression, so it only involves addition? "How can you do that, Sal?" You are probably thinking. And the key realization is when you subtract a number it's the same thing as adding the opposite. For example, if I have six minus three, that is the same thing as adding the opposite of positive three, which is negative three. Or if I had six minus negative three, subtracting a number is the same thing as adding its opposite. So what I could do is all of these places where I'm subtracting a number, instead I could just rewrite it so I'm adding the opposite. So let me do that. So I can rewrite this expression as negative 5 plus 13. So far, I'm only adding. Here, I'm subtracting all of a sudden. Minus 21, subtracting a number. Well, I can rewrite that as adding its opposite. So subtracting, lemme just another color. Subtracting a number is the same thing as adding its opposite. All right, let me keep going. Then I'm adding a five. Remember, I'm just trying to make this, so I'm just adding a bunch of things instead of adding and subtracting. I have the plus 21. Now again, I am subtracting a number, so I can rewrite that as adding, I'm subtracting a positive 13. I can rewrite that as adding a negative 13. And then last but not least, over here I am subtracting again. I'm subtracting a number, so I can rewrite that as adding the opposite of this number. So I'm now adding positive 11. Subtracting negative 11 is the same thing as adding positive 11. Now, why did I do this? Well, now I can use a commutative property. All I'm doing is I'm adding a bunch of numbers now so I can swap the order in with, with which I add. So I could now rewrite it. Let's see, I have a negative five, and now let me add this positive five next. So add the positive five, and then I have, I'm adding a positive 13, and to that I can add the negative 13. Remember, the only reason why I can now swap the order is 'cause I'm only adding a bunch of integers. Next, I have this negative 21. So let me circle that. I'm adding a negative 21, adding negative 21, and then I could add the positive 21, which is right over there. And then last but not least, I add 11. Now, why was all of this super useful? Well, now look what happens. Things start to simplify a lot. Not only when I'm doing addition can I use a commutative property, can I change the order, but I can also use the associative property very easily. So I could start to say, all right, let me add these two first and I could also add these two. I can pair these up, and that's useful because these cancel out with each other. They're opposites. If I take a negative five plus a positive five, that's a zero. A 13 plus a negative 13, that's a zero. A negative 21 plus 21, that's a zero. And so what am I left with? I am just left with a positive 11. So hopefully you see that if I can rewrite subtraction as adding the opposite, I can now use the commutative and the associative properties to really simplify things which is really useful the rest of your mathematical careers. I encourage you in your own time, go left to right with this original expression and you'll see that you get this exact same result. It's just going to actually take you a lot more time.