Adding negative numbers example

find the sum of negative 15 plus negative 46 plus negative 29 now 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 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 let me draw a number line but 0 is going to be right over there and negative 46 is gonna 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 this distance right over here is going to be 46 and it's Direction is to the left that's why it is the number that is why we get to the number negative 46 the negative really tells you whether you're to the left or 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 same thing for negative 29 so once again let me draw my number line draw then oh that's I want to use the yellow again so again draw my number line let's say that this is 0 then negative 29 is gonna be roughly over here I'm estimating it and once again negative 29 is exactly 29 away from 0 so this length 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 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 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 to get us to negative 15 then we're gonna go 46 spaces to the left to get us to to get us to negative 15 plus negative 46 so let me draw that so then we're gonna go so then we're gonna go and we're gonna figure out what that number is in a second so then we're gonna go 46 to the left that's about that far this is really just this arrow I'm now placing it I'm starting off where the negative 15 started or whether if negative 15 left off and then I'm putting that arrow after that and I'm we don't know yet what number this gets us to we're gonna 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 all those 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 15 plus 46 plus 46 plus 29 plus 29 is equal to let's see 5 plus 6 is 11 11 Plus 9 is 20 carry the - 2 plus 1 is 3 3 plus 4 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 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 year is these you know this might look fancy but this is really just the length 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 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