Adding & subtracting integers
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- Let's give ourselves a little bit more practice adding and subtracting positive and negative numbers. So, let's say that we have the statement, four is equal to negative eight minus blank. Like always, I encourage you to pause the video and see if you can figure out what blank would be to make the statement true. That negative eight minus blank is going to be equivalent to four. That four is equal to negative eight minus blank. So now, let's work it through together. To help us I will get a number line. So there's a number line right over there. And, one way to think about it is, I wanna start at negative eight, so this is negative five, negative six, negative seven, negative eight. I wanna start at negative eight and I wanna do something to negative eight to end up at four. To end up at four. So four is positive one, two, three, four. So that's four right over there. So, what do I need to do to get from negative eight to positive four? Well let's see. I have to move to the right one, two, three, four, five, six, seven, eight, nine, 10, 11, 12. I have to the right 12 spaces. Or another way to think about it, to get from negative eight to positive four I have to, I'm just gonna make sure I have some space up here, I have to add 12. Another way to think about it, to get from negative eight to zero, you have to add eight. Then to get from zero to four, you have to add another four. So you have to add eight plus four, or 12. But how do you do that? 'Cause they're saying negative eight minus something should be equal to four. Whatever you have over here, you want this to end up the equivalent to plus 12, so what can you do there? Well you could subtract a negative. If you subtracted negative 12, this would be equivalent to adding 12. So notice, let me rewrite it, four is equal to, let me just write it without my boxes there, is equal to negative eight minus negative 12. Well what happens when you subtract a negative? Well that ends up being adding the positive. So this is gonna be the same thing as negative eight plus positive 12, or four is equal to negative 8 plus 12, which we know is true. You add eight you get to zero, negative eight plus eight is zero, then you add another four, you get to positive four. So that one was pretty interesting. Let's do another one, we can't get enough practice, this is too much fun. All right. So let's say we had... let's say we had something blank minus negative two is equal to negative seven. How do we think about this? Well the first thing my brain wants to do is to simplify subtracting a negative. So if you subtract negative two that's the same thing as adding positive two. So this is an equivalent statement to say, something subtracting a negative two is the same thing as adding a positive two. Something plus two is equal to negative seven. And now we can get the number line out to think about it. So, get my number line out. So, my goal, my goal is I wanna get to negative seven. So there's negative five, negative six, negative seven right there. And I wanna add two to something to get to negative seven. So if I'm adding two that means I'm jumping two, I'm jumping two to the right, that's what adding two looks like. So what number do I need to start at that if I jump two to the right, if I add two I end up at negative seven? Well I would have to start two to the left of negative seven. If I start two to the left, one, two, if I started... Let me do this in a different color. If I start at this number right here and if I add two, plus one, plus two, I end up at negative seven. Well what number is this? This number that is two to the left of negative seven. Well it's negative nine. So this right over here is negative nine. Negative nine plus two, so if you start at negative nine and you move two to the right you end up at negative seven. Or, you could say, negative nine minus negative two, 'cause if you subtract negative two you're going to move to the right. If you subtract positive two, you move to the left. But then if you subtract negative two, you move two to the right, to negative seven. Another fascinating one. Let's do one more of these, I'm having maybe a little bit too much fun. All right. So let's say we have negative four is equal to blank minus nine. So let's get the number line out and always pause the video, see if you can try it on your own. But let's get my number line out and so let's think about it. Our goal, we wanna take some number subtract nine from it and get negative four. So we wanna get negative four and then there's some number that if we take nine away from it. So if you take nine away from something that's moving nine to the left. So, another way to think about subtracting nine, let me do this in a different color, subtracting nine. That's literally moving nine to the left. One, two, three, four, five, six, seven, eight, nine. So there's some number that we start with, we're gonna subtract nine and then we end up with negative four. We're gonna end up with negative four. Some number, that's the blank right over here, you subtract nine, you move nine to the left, that's gonna be equal to negative four. So another way to think about it is, this number is gonna be nine to the right of negative four. So let's think about what that number's gonna be. So this number is gonna be nine to the right of negative four. One, two, three, four, five, six, seven, eight, nine. It looks like positive five. So let's see if we're right about that. If we start as positive five and we subtract nine, so let's do that, one, two, three, four, five, six, seven, eight, nine. Yup, you subtract nine, you're gonna end up at negative four. So negative four is equal to five minus nine.