Hi, welcome to Math Antics. In our last video, we learned what negative numbers are, and we learned that they're part of a special set of numbers called ‘integers’. Next, we need to learn how to do arithmetic with those integers. In this video, we’re gonna learn how to add and subtract integers. In the next video we’ll learn how to multiply and divide integers. Adding and subtracting whole numbers is pretty easy because there’s really only two possibilities: you’re either adding two positive numbers, or subtracting two positive numbers. But adding and subtracting integers is more complicated, because with negative numbers in the mix, there’s a lot more possibilities. Fortunately, you don’t need to memorize all these possibilities. .
Instead, we’re gonna learn some simple rules and strategies that will make the process a lot easier . Let’s start by learning two key rules that are really helpful to know when working with negative numbers. The first rule is this: Adding a negative is the same as subtracting a positive. To visualize what that means, suppose I give you positive two cupcakes, You’d be pretty happy, right? But what if I ‘give’ you negative two cupcakes instead? That’s the same as taking two cupcakes away from you. Would you still be happy? When it comes to math, this rule means that if you have a problem like this: .
5 + -1 (where you’re adding a negative number), you can just change it to subtraction. Instead of 5 + -1, you can just write 5 − 1. They’re different ways of writing the same thing. Okay, so rule #1 is: Adding a negative is the same as subtracting a positive. Now what about rule number #2? Well the second rule is this: Subtracting a negative is the same as adding a positive. Let’s say you have negative two cupcakes. Like if you have a debt… you OWE me two cupcakes. .
You wouldn’t be too happy having that negative, would you? But, what if I subtract that negative from you? Taking away the negative two made you more positive, right? Subtracting the negative gives you less debt! When it comes to math, this rule means that if you have a problem like 5 − (-1), where you’re subtracting a negative number, you can just change it to addition. Instead of 5 − (-1), you can just write 5 + 1, and that’s a problem you already know how to solve. Alright, now that you know our two rules, I want to show you that there’s really only four different types of problems (or 4 cases) when adding or subtracting integers. .
In case 1, you start with a positive number and you make it more positive. In case 2, you start with a positive number and you make it more negative. In case 3, you start with a negative number and you make it more negative. And in case 4, you start with a negative number and you make it more positive. The key to getting the right answer to the problem is to figure out which of these cases you’re dealing with, so let’s look at each case in more detail. In case 1, you start with a positive number and you make it more positive. The good news is that you already know how to do that! That’s just what regular addition is. For example, when you have the problem 7 + 3, .
You’re starting with a positive 7 and you’re making it more positive by adding positive 3 to it, to get the answer of positive 10. But, what if instead of 7 + 3, you're given the problem 7 − (-3)? That seems a lot harder, right? Well, if you remember rule #2, it’s not harder at all. In fact, it’s exactly the same problem because rule #2 says subtracting a negative is the same as adding a positive. We can change the “minus minus 3” into “plus 3” and we get the same answer: positive 10. So case 1 is just addition like you’re used to doing, but now that we have negative numbers, there’s a new way that it could be written that you need to be on the lookout for. Oh, and in this case (starting with a positive and making it more positive), you know your answer will always be positive because you stay on that side of the number line. .
In case 2, you start with a positive number, but you make it more negative. That is, you do something to it that makes its value smaller (or moves it to the left on the number line). But that’s just like regular subtraction like you already know how to do. For example, here we have the problem 8 − 5. We start with a positive 8 but we make it more negative by subtracting 5 from it. Subtracting 5 moves it to the left on the number line down to 3 as our answer. Our answer is still a positive number, but it’s less positive than it was before. And that raises an interesting question: What if we subtract even more, to move it further in the negative direction? .
Like, what if we subtract 8 instead of 5? That would take us all the way back down to zero on the number line. And before you knew about negative numbers, back when you thought the number line just stopped at zero, you probably thought that’s all we could take away. But watch this… Let’s start with positive 8 but then subtract 10 from it! Can we do that? Can we subtract a bigger number from a smaller one? Now we can! It just means that the answer we get will be a negative number. Starting with positive 8 and subtracting 10 takes us past zero, and down to negative 2 on the number line. So our answer is negative 2. As you can see, case 2 is a little more complicated, because the answer can be positive, negative or even zero, .
Depending on the numbers you’re subtracting. And it’s even more complicated than that because there’s also another way this type of problem can be written, thanks to rule #1. Rule #1 says that adding a negative is really the same as subtracting a positive. And that means our original subtraction problem (8 − 5) could have been written like this instead (8 + -5). But either way it’s written, you’re starting with a positive and making it more negative. Now on to case 3… In case 3 you start with a negative number and you make it more negative. That’s really just the opposite of case 1 where we started on the positive side of the number line and made it more positive. For example, let’s say that you’re given the problem: -7 + -3. That means that you start with negative 7 and then you make it even more negative by adding negative 3 which gives you negative 10 as the answer. .
And as you might have guessed by now, thanks to our rules, there’s another way we could write this type of problem. Because of rule #1, we know that adding a negative 3 is the same as subtracting 3. So this problem could also be written like this: -7 − 3. But either way it’s written, in this case you’re starting with a negative number and making it even more negative. And because of that, you know that your answer will always be negative for case 3 since you stay on the negative side of the number line. And finally, in case 4, you also start with a negative number, but this time you’re gonna make it more positive. Here’s an example of a problem like that: -8 + 5. This problem starts with negative 8 but then we add positive 5 which makes it more positive, or moves us to the right on the number line, up to -3 which is our answer. .
Our answer is still negative, but it’s less negative than what we started with. Ah, but this looks similar to case 2, right? Imagine what would happen if you made it even more positive. You could add 8 (instead of 5) and that would move you all the way up to zero. And if you added even more, like -8 + 10, that would take you to the positive side of the number line and give you the answer of positive 2. So just like in case 2, the answer here can be positive, negative or even zero depending on the numbers you‘re adding. And, just like before, there’s another way this type of problem can be written. Remember rule #2? …that subtracting a negative is the same as adding a positive. Because of that rule, -8 + 5 could also be written as -8 − (-5). .
That’s a LOT of negatives, but thanks to rule #2, we can write our “minus minus” as a “plus”. But either way it’s written, in this case, we start negative and move in the positive direction. Okay, now that we’ve learned our two rules and we’ve seen that there’s really just four types of problems (or 4 different cases) that you’ll need to solve, I want to give you a basic strategy that will help when you’re adding or subtracting integers. This strategy involves asking 3 simple questions: The first question is, “Am I starting with a positive or a negative number?” This question is easy to answer because all you have to do is look at the sign in front of the first number of the problem. If there’s no sign, then you’re starting with a positive number. But if there’s a minus sign, then you’re starting with a negative number. .
Once you know what you’re starting with, the next question to ask is, “Am I making it bigger or smaller?” In other words, is what you’re adding or subtracting going to make the value move in the positive direction of the number line or the negative direction? If you’re subtracting a positive or adding a negative (Rule #1) then you're making the value smaller, which is moving to the left on the number line. If you’re adding a positive or subtracting a negative (Rule #2) then you're making the value bigger, which is moving to the right on the number line. And the last question that you need to ask is, “Will my answer be positive, negative, or zero?” The answer to this will depend on which case you have and what values you’re working with. If it’s a case 1 problem, you know the answer will always be positive, and if it’s a case 3 problem, then you know the answer will always be negative. But you’ll remember that in cases 2 and 4, .
The answer could change between positive, negative or zero depending on the difference between the two numbers that you’re working with. So asking these three questions will help you visualize what’s happening in the problem that you’re trying to solve, and it’ll help you get the right answer. But the most important thing that you can do to get good at integer arithmetic is to practice. It’s important that you try a lot of problems that you have an answer key for so that you know if you’re doing it right. And don’t get discouraged if it’s confusing at first. You’ll get it if you practice! And you might want to re-watch this video a few times since there’s so much that it covers. As always, thanks for watching Math Antics, and I’ll see ya next time! Learn more at www.mathantics.com .