Adding numbers with different signs

Find the sum 15 plus negative 46 plus 29. Let's just think about this first part over here where we have 15 plus negative 46. And we'll worry about the plus 29 later on. So let's just do 15-- let me do that in a different color-- plus negative 46. I'll do it in that orange. Let me draw a number line here, just so we can properly visualize what is going on. That's my number line. And we're starting at 15. So let's draw a 0 over here. And so we're starting at 15. 15 could be right over here. So this is 15. Let me draw a big fat arrow to signify this is 15. 15 has an absolute value of 15, so the length of this arrow would be 15. Now we're adding negative 46 to that 15. This is equivalent to 15 minus 46, which means we are going to move 46 spots to the left of 15. A negative sign or a minus means we're moving to the left on the number line. So we're going to move 46 to the left. We're starting at 15, and we are going to move 46 to the left. So the length of this arrow right here is going to be 46, and we're moving to the left. This is the negative 46 that we're adding to the 15. We're going to end up at some point over here. That point is clearly 0. Because we were 15 to the right. Now we're going to move 46 to the left. So we're definitely going to be to the left of 0. It's definitely going to be a negative number. And we can even think about the absolute value of that negative number. We can just visualize it. This yellow arrow has a length of 15. This orange arrow has a length of 46. The blue arrow that I'm about to draw, which is the sum of these two, is going to have this length right over here. And just visually, how can we figure out the length of this blue part if we know the length of this orange part and we know the length of this yellow part? Well, it's just going to be the difference. It's just going to be the difference of these two. So the absolute value of the sum is going to be the difference between this length, 46, and 15. Let me just figure that out, 46 minus 15. 6 minus 5 is 1. 4 minus 1 is 3. So the length of this is going to be 31, and it's going to be 31 to the left of 0. So this is going to be negative 31 right over here. We know that this first part over here is negative 31, and then to that we are going to add 29. So we're going to add 29. Let me do that in another color. What does that mean? That means that we're going to start at negative 31, and we're going to move 29 to the right. We're adding 29, so we're going to move 29 to the right. So maybe that gets us right about there. I'm trying to draw an arrow of length 29. I can draw a cleaner looking arrow than that. I'll do it right over here, actually. So then we're going to move 29 over to the right. That's the 29 part. Now this is a positive 29, and so how do we figure out what this is? This is 29 right here that we're adding. This is going to land us right over here on the number line. So how do we figure out what number that is? Well, once again, we can just visualize it. And eventually you won't have to draw number lines and stuff, but I think it'll be useful here. We're starting at negative 31. We're adding 29 to it, so it's going to make it less negative. But we're still adding less than 31, so we're not going to get all the way back to 0. We're still going to have a negative number. But how could we figure out the absolute value of that negative number, its distance? Well, once again, that little white part right there, that white part plus this 29, is going to equal 31, if you just think of absolute value, if we don't think about the signs, if we just think about the length. Or another way to think about it, 31 minus 29 will give us the length of that white part. And of course, it's going to be negative, because the negative number here is larger than the positive number. And we're adding the 2. If we do 31 minus 29, you could borrow and all of that, but that's clearly just going to be equal to 2. You could say that's 11. This is a 2. You subtract, this is equal to 2. But since it's negative 31 plus 29, it's going to be a negative 2. We're still going to stay negative. We haven't moved far to the right enough to pass 0. So this right here is going to be negative 2. Or another way to think about it, the length of this white bar, the absolute value, is going to be 2. 2 plus 29 is 31. But we're operating to the left of 0, so it's negative 31. This is negative 2. Anyway, hopefully you found that useful. And let me make it clear. Our final answer is negative 2.