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Equivalent expressions with negative numbers

Understand that we can commute and associate terms as long as we keep the signs with the terms. We can also apply the additive identity property and the additive inverse property. Created by Sal Khan.

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Video transcript

- [Teacher] We're asked which of the following expressions are equivalent to two minus 9.4 plus zero plus 3.71? And we need to pick two answers. So pause this video and see if you can have a go at it before we do this together. All right, now let's look through the choices. So this first choice, we have 9.4. Here we subtracted 9.4. Now when you subtract a number, you can also view it as adding its opposite. So we could rewrite this top thing as two plus negative 9.4. Notice when we subtract 9.4, that's the same thing as adding negative 9.4 so we could write it this way. And when you are adding numbers, and that's actually this property or this principle that we just used, it's really useful because when you're adding numbers like this, it doesn't matter what order you add them in. When you have some subtraction involved, now all of a sudden the order does matter. But when you're adding like this, it doesn't matter the order that you are doing it with. So we can swap these numbers any way we want. We could write this, for example, as negative 9.4 plus two plus 3.71. We could ignore the zero or put that anywhere obviously. Plus zero, something like that. But what we see over here, they don't have a negative 9.4. They have a positive 9.4. So I really don't know how I can reconcile this first thing in this right over here, so I'm gonna rule that one out. Now this one over here, you have a two. We have a two there. We have a 3.71. We have a 3.71 there and both of those things are just being added. And now we have a, and now from that we are subtracting 9.4. And so here they are subtracting 9.4 and it is indeed the case here, you have to be very careful with subtraction if, we wouldn't wanna make it 9.4 minus something else. But as long as we're only subtracting the 9.4, then the swapping of this order does work and you could even verify that with the numbers over here. You could do two minus 9.4 plus 3.71 or you could do two plus 3.71 minus 9.4. So I like this choice. Now let's look at this one. This has 3.71, 3.71. It has the two, it has the two, and now it's adding 9.4. Now we already said we could rewrite subtracting 9.4 as adding negative 9.4, but we can't rewrite that as just adding 9.4. If this was like this, if it had a negative there, then this would work, but it's not that. And so I'm going to rule that one out and I'm guessing this one's going to work, but let's see. So this one put parentheses around it, so it's really just telling us what we want to do first. So it says first do the two minus 9.4 which if you just went left to right, you would've done anyway. And then plus zero plus 3.71 and zero plus 3.71, obviously that's just going to be 3.71. So of course, this does seem very reasonable to do. That you could go two minus 9.4 plus 3.71. Two minus 9.4, the zero doesn't matter here, and then plus 3.71. So I like that choice. Let's do another example. So once again, we wanna come up with an equivalent with equivalent expressions, so let's look at this one. Actually before I even look at the choices, let's just recognize that if you subtract a number, it's the same thing as adding its opposite. So for example, this could be rewritten as negative 6/5 plus 1/2 plus the opposite of negative 8/5 which is positive 8/5. That's another way we could rewrite it. That might be helpful. Let's see, over here we have 1/2 which we see right over there. We have plus 8/5 and interestingly that looks like the one that we did in the second version right over here, and then we have minus 6/5. So is minus 6/5 the same thing as this over here? Well, let's think about it. What we're really doing when we write this negative 6/5 out front, that's the same thing as, we could do that as plus negative 6/5 and that's going to be the same thing as subtracting the opposite of it. Or another way of thinking about it is subtracting positive 6/5. Let me write it this way, subtracting positive 6/5. It's actually hard to tell because I put that I have, actually let me write it this way. If we just took this version and if we were to use a commutative property to swap the orders, we could write this as 1/5 plus 8/5 and then plus negative 6/5. All I did is swap the order and now we know that this last part could be rewritten. So it's 1/5 plus 8/5. Instead of plus negative 6/5, that can be rewritten as negative or minus I should say, minus positive 6/5. Why can I do that? Because subtracting a number is the same thing as adding its opposite. And so all of these are equivalent and this is exactly what they have written here, this last scenario. So I like that one. Now let's see, this one has a negative 6/5 and it has a plus 8/5 and a plus 1/2. So all they did is they took this version which we knew is equivalent and they swapped the order that we are doing. They actually swapped the order of the 1/2 and the 8/5. And when you're dealing with addition like this, you can move things around, the commutative property. You can do your addition in different orders and that's all they're doing. Actually this is not just commutative, this is also associative. They're literally putting that parentheses there and say hey, let's just do, and let's just do the those first two first. Let's swap these two and then of course, even if you're going left to right, you would do these first two even if the parentheses weren't there. So all of this is definitely equivalent to negative 6/5, swapping the 1/2 and the 8/5. So 8/5 plus 1/2. So I like that choice too. And so we had two choices, so these are probably not going to work and let's see why they don't real fast. They both have the negative 6/5. This has the plus 1/2, but it has minus 8/5 instead of minus negative 8/5. It could have plus 8/5 here, but not minus 8/5. So we rule that one out. Negative 6/5, and let's see. Then it's adding negative 8/5 instead of subtracting negative 8/5. So I don't like that, and then it subtracts 1/2 instead of adding 1/2, so it's no good on multiple dimensions. Let's do one last example. Which of the expressions are equivalent? As always pause and see if you can figure that out. So let's see, so actually before I look at the choices, I'm gonna see if there's different ways to rewrite this. We could rewrite this as 1.7 minus 8.33. Why can I do that again? Because subtracting a number is the same thing as adding its opposite, and then I could say plus. I could write this as 8.33 and if I wanted to I could even write this, instead of minus 1.95, I could even write that as plus negative 1.95 if I choose 'cause once again, subtracting a number is the same thing as adding its opposite. Now I'm tempted to try to compute this because we could just, when we're doing addition like this and I could write it like this if I want everything to be addition. 1.7 plus negative 8.33 plus 8.33 plus negative 1.95. When everything is expressed in addition, we know that the order doesn't matter. The commutative property tells us that. And even the associative property tells us how we put the parentheses doesn't matter, and so we could actually do that first and though these two would cancel out. Actually maybe we will have to do that. That choice looks like that, but let's just go step by step. So this has 1.7 and 1.7. Let me do this in a different color. One point, that's not a different color. 1.7, 1.7, they're subtracting 8.33. That's subtracting 8.33 here, so that's equivalent to this version right here. Then they're subtracting it again. No, this does not look good because even when we try to compute it, we know that these two terms will cancel out each other, so I don't like that one. Let's see, this one. If we take this one here and we were to reorder it because this is all addition of different integers here, but if we were to reorder it, we could reorder it as negative 8.33, negative 8.33, plus 8.33, plus 8.33, plus 1.7, plus 1.7, and then we have plus negative 1.95 which we know we can write as minus 1.95, minus 1.95, because subtracting a number is the same thing as adding its opposite, and that's what they have here. They put these parentheses. But if you just went left to right, I should say, with this one, you're going to get the same thing. So I like that choice there. Now this choice, if you just canceled these two terms out, you're left with a 1.7 plus negative 1.95 which we already know is the same thing as minus 1.95, so I like this choice as well. Now why won't this one work? Let's see, 1.95. They're trying to change the order, but you can add a negative 1.95 and then put it out front. But this is a positive 1.95, so I'm already not liking this. And then they're subtracting 1.7. There's no reason why you should subtract 1.7, so this is definitely not looking good.