Adding negative numbers example

Find the sum of negative 15 plus negative 46 plus negative 29. To do this, let's just first visualize what each of these numbers look like. So I'm going to draw a number line for each of them. Negative 15 might look something like this. So if this is 0 and let's say that this is negative 15, I could represent negative 15 as-- if I'd like to-- an arrow that points from 0 to negative 15. The length of the arrow is the absolute value of this. It's the distance from 0. So the length here is 15. And the negative says that we are pointing to the left. So the absolute value is 15. That's the length of that arrow. Let's do the same thing for negative 46. So once again, let me draw my number line. 0 is going to be right over there. And negative 46 is going to be someplace over here. Negative 46. Notice, same exact idea. The distance between negative 46 and 0-- or another way to think about that is the absolute value of negative 46-- this distance right over here is going to be 46. And its direction is to the left. That is why we get to the number negative 46. The negative really tells you whether you're to the left or to the right of 0, but the absolute value says how far are you to the left or the right of 0. And then, finally, let's do the same thing for negative 29. So once again, let me draw my number line. I want to use the yellow again. So draw my number line. Let's say this is 0. Then negative 29 is going to be roughly over here. I'm estimating it. And once again, negative 29 is exactly 29 away from 0. So this length right over here is 29, and it is to the left. That is why it is negative 29. If it was positive 29, it would be 29 to the right of 0. So we've represented all of these numbers. And you can see what their absolute values are like. And now let's think about what happens if you add them. So one way to think about adding these numbers is if you added these arrows, if you put this arrow on top of this arrow or to the left of this arrow, if you started where this arrow leaves off-- and then you put this green arrow, and then you put the orange arrow. So let's do that. Let's draw that. It's going to be a longer arrow now. So we're going to start at 0. First, we have the negative 15. So we're going to move 15 spaces to the left to get us to negative 15. Then we're going to go 46 spaces to the left to get us to negative 15 plus negative 46. So let me draw that. So then we're going to go-- and we're going to figure out what that number is in a second. So then we're going to go 46 to the left. That's about that far. Really just this arrow, I'm now placing it. I'm starting off where the negative 15 left off, and then I'm putting that arrow after that. And we don't know yet what number this gets us to. We're going to have to do a little math here. And that's actually the point of this problem. But we know what the length of this arrow is. The length of this arrow is 46. It is 46 to the left, and we know the length of this magenta arrow is 15. And then, finally, we have this orange arrow. The orange arrow, which we know has a length of 29. Although, it's 29 to the left. It looks like that. It has a length of 29. So when all is said and done, where are we? Where do we end up on our number line? Well, the total length that we are to the left is going to be 15 plus 46 plus 29. But it's going to be to the left, so it's going to be negative. So what we can really view this as, since all of these have the same sign, is we could say, look, this is the same thing as the absolute value of negative 15, which is 15, plus the absolute value of negative 46, which is 46, plus the absolute value of negative 29, which is 29, but then take the negative value of it. So let's just add that. 15 plus 46 plus 29 is equal to, let's see-- 5 plus 6 is 11. 11 plus 9 is 20. Carry the 2. 2 plus 1 is 3. 3 plus 4 is 7. 7 plus 2 is 9. So you get 90. So this entire length here is 90. The entire length, if you add up the arrows, you get a 90. But it is not 90 to the right. So if it was 90 to the right, these would all be positives. And then we would just get a positive 90. But this is 90 to the left. So when you add these guys up, you don't get to positive 90. You get to negative 90. So one way to think about it is these are all the same sign. So this is going to be equal to the negative of the absolute value of negative 15 plus the absolute value of negative 46 plus the absolute value of negative 29. And the reason why we're writing this here-- this might look fancy, but this is really just the length of this purple arrow. This absolute value of negative 46, that's really just the length of the green arrow. 46. And this is just the length of the orange arrow. So this is 15 plus 46 plus 29, gave us 90, but it is to the left. So that is why it is negative 90.