Signs of sums on a number line | Integers: Addition and subtraction | 7th grade | Khan Academy

Let's give ourselves some intuition and then some practice adding negative numbers. So, let's start with negative 11 plus negative 3.

So, first we can visualize what negative 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. If we're talking about negative 11, you could think about that as 11 to the left of 0, or if you start at 0 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 negative 3. So, if I start at negative 11 and I add negative 3, I'm going to go 3 more to the left. I'm going to 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 negative 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 negative 11 plus 3. What would that look like? Well, we have the same negative 11 going on right over here, which we already drew. But now, let me do plus 3 in a different color, and let me do it in green so we don't get confused.

So, negative 11 plus 3. If we start at negative 11 right over here, but now we're adding 3. So, now I'm going to go to the right. 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 less to the left than negative 11 was. But that's interesting; I'm still going to get a negative value.

Now let's imagine if I were to have positive 11 plus negative 3. Think about what would happen there. So, positive 11, let me do this in this red color. It might look something like this. Positive 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 negative 3 to that. I'm going to add negative 3. So, where I left off here, I'm now going to go 3. Am I going to go to the right or to the left? What's negative 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 positive 11 plus 3, well, I have that positive 11 in red already over there, and then if I were to add 3, it would get me even more positive. 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 sign, so for 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 negative 11 plus 3, negative 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 0 than 3 is moving in the right direction. And then when we did 11 plus negative 3, it was the other way around.

Hopefully, we've built up some intuition, and now what I'm going to do is tackle these exact same problems, but we're going to do it with a number line that actually has the numbers 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 negative 11 plus negative 3, what would that get us? And I'll try to remember the colors I just used.

Negative 11, I can start at 0 here, and I'm going to go to the left 11. And I get to—whoops, I'm a little bit too far. I'm going to go all the way negative 5, negative 10, right over here. Negative 11 gets me right over there.

And then to that, I am going to add negative 3. So, to that, I'm going to do a different color. Let me do this carefully. To that, I am going to add negative 3. So, I start there, and I'm going to go three more to the left. So, one, two, three. So, I am going to end up right over there.

And where do I end up? That is at negative 14. So, this is equal to negative—I'm just in a neutral color—it's equal to negative 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 positive 11. So, let me do that in red. So, positive 11 would look like that. That's that right over there.

And then I am going to add three. So, I'm going to add 1, 2, 3, and I get to positive 14. So, notice here I ended up 14 to the right of 0. Here I ended up 14 to the left of 0.

But now let's look at the scenarios where we had mixed signs. So, let's say we have negative 11 plus 3. What do we think that is going to be equal to? So, we already have drawn negative 11. That is negative 11 right over there.

And now we want to think about plus 3. So, we're going to start right over here, and I'll do it a little bit higher so we can see ourselves. And then plus 3, we're going to go 3 to the right from that point. So, one, two, three. Where do we end up?

Well, this right over here is negative 8. So, we are now at negative 8. Now, let's do it the other way around. If we had positive 11 plus negative 3, what is that going to be equal to?

Well, we've already drawn positive 11 here, but now we're adding negative 3. Negative 3 is in this like salmon color looking thing. They're very close but different. So, now we're going to add negative 3 to positive 11.

So, we start here, but we're going to go 3 to the left because it's negative. So, one, two, three. And where do we end up? We end up at positive 8.