Adding and subtracting integers
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In this video, I want to familiarize ourselves with negative numbers, and also learn a bit of how do we add and subtract them. And when you first encounter them, they look like this deep and mysterious thing. When we first count things, we're counting positive numbers. What does a negative number even mean? But when we think about it, you probably have encountered negative numbers in your everyday life. And let me just give you a few examples. So before I actually give the example, the general idea is a negative number is any number less than the 0. And if that sounds strange and abstract to you, let's just think about it in a couple of different contexts. If we're measuring the temperature-- and it could be in Celsius or Fahrenheit. Let's just assume we're measuring it in Celsius. And so, let me draw a little scale that we can measure the temperature on. And so, let's say this is 0 degrees Celsius. That is 1 degree Celsius, 2 degrees Celsius, 3 degrees Celsius. Now let's say that it's a pretty chilly day, and it's currently 3 degrees Celsius. And someone who predicts the future tells you that it is going to get 4 degrees colder the next day. So how cold will it be? How can you represent that coldness? Well, if it only got 1 degree colder, we would be at 2 degrees. But we know we have to go 4 degrees colder. If we got 2 degrees colder, we would be at 1 degree. If we got 3 degrees colder, we would be at 0. But 3 isn't enough. We have to get 4 degrees colder. So we have to actually go one more below the 0. And that one below 0, we call that negative 1. And so, you can kind of see that the number line, as you go to the right of 0 on the number line, it increases in positive values. As you go to the left of the number line, you're going to have negative 1. Then you're going to have negative 2. And you're going to have negative 3. And you're going to have-- depending on how you think about it-- you're going to have larger negative numbers. But I want to make it very clear-- negative 3 is less than negative 1. There is less heat in the air then negative 1. It is colder. There's less temperature there. So let me just make it very clear. Negative 100 is much smaller than negative 1. You might look at 100, and you might look at 1, and your gut reaction is to say, wow, 100 is a much larger number than 1. But when you think about it, negative 100 means there's a lack of something. If it's negative 100 degrees, there's a lack of heat. So there's much less heat here then if we had negative 1. Let me give you another example. Let's say in my bank account today, I have $10. Let me do this in a new color. So let's say today I have $10. Now let's say I go out there-- because I feel good about my $10-- and let's say I go and spend $30. And for the sake of argument, let's say I have a very flexible bank, one that lets me spend more money than I have-- and these actually exist. So I spend $30. So what's my bank account going to look like? So let me draw a number line here. And you might already have an intuitive response. I will owe the bank some money. So let me write this over here. So tomorrow, what is my bank account? So you might immediately say, look, if I have $10, and I spend $30, there's $20 that had to come from someplace, and that $20 is coming from the bank. So I'm going to owe the bank $20. And so, in my bank account, to sure how much I have, I could say $10 minus $30 is actually negative $20. So in my bank account tomorrow, I'm going to have negative $20. So if I say I have negative $20, that means that I owe the bank. I don't even have it. Not only do I have nothing, I owe something. It's going in reverse. Here, I have something to spend. If my $10 is in my bank, that means the bank owes me $10. I have $10 that I can use to go spend. Now all of a sudden, I owe the bank. I've gone in the other direction. And if we use a number line here, it should hopefully make a little bit of sense. So that is 0. I'm starting off with $10. And spending $30 means I'm moving 30 spaces to the left. So if I move 10 spaces to the left-- if I only spend $10, I'll be back at 0. If I spend another $10, I'll be at negative $10. If I spend another $10 after that, I will be at negative $20. So each of these distances-- I spent $10, I'd be at 0. Another $10, I'd be at negative $10. Another $10, I would be at negative $20. And this whole distance right here is how much I spent. I spent $30. So the general idea, when you spend or if you subtract or if we're getting colder, you would move to the left of the number line. The numbers would get smaller. And we now know they can get smaller than 0. They can go to negative 1, negative 2, they can even go to negative 1.5, negative 1.6. You're getting more and more negative the more you lose. If you're adding-- if I go and get my paycheck, I will move to the right of the number line. Now with that out of the way, let's just do a couple of more pure math problems. And just think about what it means if we were to say 3 minus 4. So once again, this is exactly the situation that we did here with the temperature. We're starting at 3, and we're subtracting 4. So we're going to move 4 to the left. We go 1, 2, 3, 4. So that gets us to negative 1. And when you're starting to do this, you really understand what a negative number means. I really encourage you to visualize the number line, and really move along it depending on whether you're adding and subtracting. Now let's do a couple more. Let's say I have 2 minus 8. And we'll think about more ways to do this in future videos. But once again, you just want to do the number line. You have a 0 here. Let me draw the spacing a little bit. So we have 0 here. We're at 1, 2. If we're subtracting 8, that means we're going to move 8 to the left. So we're going to go 1 to the left, 2 the left. So we've gone 2 to the left to get to 0. We have to move how many more to the left? Well, we've already moved 2 the left. To get to 8, we have to move 6 more to the left. So we're going to have to move 1, 2, 3, 4, 5, 6 more to the left. Well, where is that going to put us? Well, we were at 0. This is a negative 1, negative 2, negative 3, negative 4, negative 5, negative 6. So 2 minus 8 is negative 6. 2 minus 2 would be 0. When you're subtracting 8, you're subtracting another 6. So we'd go up to negative 6. We go 6 below 0. Let me do one more example. And this one will be a little less conventional for you, but hopefully it will make sense. And I'll do this in a new color. Let me take negative 4 minus 2. So we're starting at a negative number, and then we're subtracting from that. And if this seems confusing, just remember the number line. So this is 0 right here. This is negative 1, negative 2, negative 3, negative 4. So that's where we're starting. Now we're going to subtract 2 from negative 4. So we're going to move 2 to the left. So if we subtract 1, we will be at negative 5. If we subtract another 1, we are going to be at negative 6. We are going to be at negative 6. So this is negative 6. Let's do another interesting thing. Let's start at negative 3. Let's say we have negative 3. Instead of subtracting something from that, let's add 2 to it. So where would this put us on the number line? So we're starting at negative 3, and we're adding 2, so we're going to move to the right. So you add 1, you become negative 2. But if you add another 1, which we have to do, you become negative 1. You move 2 to the right. So negative 3 plus negative 2 is negative 1. And you can see for yourself but this all fits our traditional notion of adding and subtracting. If we start at negative 1, and we subtract 2, we should get negative 3-- kind of reverse of this thing up here. Negative 3 plus 2 gets us there. Then if we start then, we subtract 2, we should get back to negative 3, and we see that happens. If you start at negative 1 right over here, and you subtract 2-- you move 2 to the left-- you get back to negative 3. So hopefully, this starts to give you a sense of what it means to deal with, or add and subtract, negative numbers. But we're going to give a lot more examples in the next video. And we're actually going to see what it means to subtract a negative number.