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Signs of sums on a number line

Let's build intuition for the sign of sum involving negative and positive numbers. We'll decide whether each sum ends to the left or right of 0. Then we'll evaluate exactly to see whether the intuition worked out. Created by Sal Khan.

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Video transcript

- [Instructor] Let's give ourselves some intuition and then some practice adding negative numbers. So let's start with -11 plus -3. So first we can visualize what -11 looks like on a number line like this. I intentionally have not marked off any of the numbers. We can just build an intuition for what's going to happen here. So if we're talking about -11, you could think about that as 11 to the left of zero, or if you start at zero right over here, we are going to go 11 to the left, which I'm just estimating. I have nothing marked here, but let's just say that gets us right about there. Now to that, I am going to add -3. So if I start at -11 and I add -3, I'm gonna go three more to the left. I'm gonna go three more even more in the negative direction. So it's going to go something like that, and I'm going to end up even more negative, even further from zero, further to the left from zero than I was at -11. You might already be guessing what that is, but we'll think about that more in a little bit. Let's do another example. Let's say, -11 plus 3, what would that look like? Well, we have the same -11, going on right over here, which we already drew, but now let me do plus 3 in a different color, let me do it in green so we don't get confused. So negative 11 plus 3. So if we start at -11 right over here, but now we're adding 3. So now I'm gonna go to the right. So notice, here, I'm still ending up left of zero on the number line, so I'm still going to end up in a negative place, but it is less negative. It's left, it's less to the left that negative 11 was. But that's interesting, I'm still gonna get a negative value. Now, let's imagine if I were to have +11 plus -3, think about what would happen there. So +11, lemme do this in this red color, it might look something like this, +11 would look something like that. I'm starting at zero and I go 11 to the right and now I'm going to add -3 to that, I'm going to add negative three. So where I left off here, I'm now going to go three. Am I going to the right or to the left? What's a -3? I'm going to go to the left over here. So it might get us someplace right around there. So I went 11 to the right and then I go 3 to the left. So I'm still going to end up to the right. I'm still going to have a positive value, but it's going to be less positive than that 11 over there. Now, let's give one more scenario and this is one that you're probably familiar with for many years. If I had +11 plus 3. Well, I have that +11 in red already over there. And then if I were to add 3. it would get me even more positive and I think you know what that probably is if you go even more positive. So there's a couple of interesting patterns. If we are adding numbers of the same signs, if we're adding two negatives, we still end up with a negative. If we add two positives, we still end up with a positive. But when we're adding numbers of different signs, you actually end up taking on the sign of whichever one is further from zero. So for example, when I took -11 plus 3, I still end up with an answer that is to the left of zero, something that is negative because 11 is further to the left of zero than 3 is moving in the right direction. And then when we did 11 plus -3, it was the other way around. Hopefully we've built up some intuition and now what I'm gonna do is tackle these exact same problems, but we're gonna do it with a number line that actually has the number marked off and we can actually compute what these are going to be. So let me delete all of this and then let me give ourselves, let's mark these things off, and then let's do the same ones over again. So if I were to say -11 plus -3, what would that get us? I'll try to remember the colors I just used. - 11, I can start at zero here and I'm going to go to the left, 11, and I get to, whoops, I went a little bit too far. I'm gonna go all the way -5, -10, right over here, - 11, gets me right over there. And then to that I am going to add -3. So to that I'm going to, different color, let me do this carefully. To that, I am going to add -3. So I start there and I'm gonna go three more to the left. So 1, 2, 3. So I am going to end up right over there. And where do I end up? That is at -14. So this is equal to negative, lemme do this in a neutral color, it's equal to -14. Now, let's, actually, let's do this in different orders just to see what's going on here. Let's do the, essentially, something very similar, but let's go to the right of the number line. Let's do 11 plus 3. So 11 plus 3, we could start with +11. So let me do that in red, so +11 would look like that, that's that right over there. And then I am going to add 3. So I'm going to add 1, 2, 3, and I get to +14. So notice, here I ended up 14 to the right of zero, here I ended up 14 to the left of zero. But now let's look at the scenarios where we had mixed signs. So let's say we have -11 plus 3. What do we think that is going to be equal to? So we already have drawn -11. That is negative 11, right over there. And now we want to think about plus 3, so we're gonna start right over here, and I'll do it a little bit higher so we can see ourselves. And then plus three, we're gonna go three to the right from that point. So 1, 2, 3. Where do we end up? Well, this right over here is -8, so we are now at -8. Now let's do it, let's do the other way around. If we had +11 plus -3 what is that going to be equal to? Well, we've already drawn +11 here, but now we're adding -3. - 3 is in this, like, salmon color looking thing. They're very close, but different. So now we're going to add -3 to +11, so we start here, but we're gonna go three to the left 'cause it's negative. So 1, 2, 3. And where do we end up? We end up at +8.