 Alright, so let's take a look at the addition and subtraction of the integers. Now, in order to proceed, we want to go back to our idea that the negative symbol is an indicator that we're looking at the additive inverse of a quantity. And all of the problems we'll be taking a look at involve this idea that we're going to prove a particular property. So for example, in this case, we want to prove that the negative of quantity x plus 3 is the same thing as negative x plus negative 3. And again, this might seem to be an algebra problem, but with an understanding of what the basic idea of the additive inverse is, this requires no special treatment. As soon as you understand what this means, this is just an extension. So let's take a look at that. So by definition, whatever this thing is is going to be the additive inverse of x plus 3. And what that means is that I can write down an equation. The additive inverse of x plus 3 plus x plus 3 is going to be 0. That's always guaranteed to be true. And we're going to employ a duck proof here. And what I want to do is I want to trade out this expression negative x plus 3 with something that still gives me a true equation. So now I have this x plus 3 equals 0. I know that if I add the additive inverse to x and the additive inverse of 3, I still get a sum of 0. Here's additive inverse of x plus x, additive inverse of 3 plus 3. They're going to add together to get 0. And if I compare my two equations, first equation, second equation, all of this is the same in both equations, which means that what's left over also has to be the same thing. So that tells me that this expression negative x plus 3 must be the same as negative x plus negative 3, and there's my proof. Well, this is actually how we derive more general properties of the integers in general if I have two integers, the additive inverse of the quantity A plus B is the same as the additive inverse of A plus the additive inverse of B. Now, for future reference, it's important to keep in mind that this is a theorem, it is a property of the integers. It's something that we have proven at some point, and it's something that we use. If we're being asked to prove something about the integers, it's important to determine whether or not it is allowable to use these integer properties. For example, consider this problem. We want to evaluate and prove a particular statement. Now, you might know from previous courses how to add integers, but for the moment imagine that you've never actually learned these rules, that you're being exposed to these concepts for the first time. If you have never learned the rules, if this is the first time seeing it, as long as you understand what the concept of negative 8 means, then evaluating this is not a problem. Now, we might want to try a duck proof to try and figure out what that is. So the idea is that we would like to say something that involves this expression 3 plus negative 8. And while 3 doesn't give us any problems, that negative 8 does give us problems. And the one thing I know is that negative 8 is the additive inverse of 8. So that tells me the additive inverse of 8 plus 8 gives me 0. So here's a nice little equation that I can start with. 3 plus negative 8, that's what I'm looking at. If I add 8 to that, these two terms here add to 0. And so this entire expression over on the left-hand side is going to simplify down to 3. Now my goal is to write down another equation that is very similar to this, where the only difference is that instead of 3 plus negative 8, I have something else. And so what I want to do is I want to write a new equation that includes this plus 8 that includes the equals 3, but it has something else in this place here. So now I stare at this equation and say, well, what I have is something plus 8 equals 3. And I might think about that for a moment, and I ask the question, well, what can I put in that blank space? And after a little bit of thought, it should be apparent that what you can put in there is negative 5. And so again, going back to an earlier idea, this negative 5 here will cancel out with 5 of the 8, leaving 3, which is what I have there. So I now have two equations, and these two equations are almost the same. The only difference is one of them has 3 plus negative 8, the other one has negative 5. So that tells us that those two things have to be the same thing. Again, this is our basic duck proof. This expression has the property that if I add 8, I get 3. Well, this has the property that if I add 8, it gets 3. And so these two expressions act in exactly the same way. And so I'm going to say that they are in fact equal that 3 plus negative 8 is in fact equal to negative 5. Well, let's take a look at another subtraction that we'll see is going to be related to that one. Take a look at 3 minus 8. And again, if I were to read this as I ordinarily read subtractions, what it seems like I'm trying to do is I'm trying to remove 8 objects from a set of 3 objects, but I can't do that. So I can't deal with the subtraction as it stands. However, we can try a duck proof, and we're going to start with 3 minus 8 and find a property of it. And again, the thing that we might notice here is the thing that's giving us trouble is this minus 8. Well, if I add 8 back, well, if I subtract 8 and then return 8, I should get what I started with. Never mind that someplace in the middle we've done something that's a little bit peculiar. So that says that I should be able to say 3 minus 8 plus 8 is equal to 3. And again, I want to write down a new equation where everything except for this 3 minus 8 is the same. So I want the plus 8 equals 3, but instead of 3 minus 8, I have something plus 8 equals 3. So I'll keep the plus 8, keep the 3, and what can I put in the parentheses? And again, we think about this a little bit, and it turns out that we can put in a negative 5 inside the parentheses. And if I compare my two equations, they are the same plus 8 equals 3 plus 8 equals 3. The only difference between these two equations is one of these has 3 minus 8, and the other one has negative 5. And so we apply our duck proof, and this 3 minus 8 has the same property as minus 5, so 3 minus 8 and negative 5 have to be the same thing. Now, notice that the last two examples gave us two statements, 3 plus negative 8 equals negative 5, and also 3 minus 8 is equal to negative 5. And again, duck proof, both of these are equal to negative 5. Well, if they're both equal to negative 5, they must be equal to each other, and so that tells me 3 plus negative 8 must be the same thing as 3 minus 8. And this is an example for more general property of the integers. Anytime I add the additive inverse, it's the same as subtracting the number. Well, let's see what else we can do with this. So let's take a look at the problem 3 minus 8, negative 6. And again, as before, if I want to evaluate and prove this expression, then I might want to build an expression that includes 3 minus negative 6. So I might start with 3 minus negative 6, and think about that for a moment. If I subtract 3, that'll get rid of this. And if I add negative 6, that'll get rid of the minus negative 6. And the nice thing about that is the value of that expression that is going to be 0. So that gives me the equation 3 minus negative 6 minus 3 plus negative 6 is 0. So check it out, we have 3 minus 3 plus negative 6 minus negative 6. And so after all the dust settles, everything cancels out. I get a sum of 0, and there's my first equation. Now I'd like to write another equation where everything is the same except for this thing that I'm trying to evaluate. So I'm going to keep the minus 3 and the plus and negative 6, except I have something else here, still equal to 0. So it's going to look something like that. And let's think about that. So if I want to get 0 here, well, I'm subtracting 3, so I'm going to want a 3 in here. I'm adding a negative 6. Well, I know that negative 6 is the additive inverse of 6. So if I add a 6 to it, if I have a 6 here plus negative 6, that's going to be 0. And if I have a 3 in here, minus 3 is also going to be 0. So what goes inside the parentheses must be a 3 plus 6. And so now this gives me two equations where almost everything is the same except for this little bit here. And so if I compare those two bits, 3 minus negative 6 and 3 plus 6, they must be the same thing. They have to be equal to each other. And so that tells me that 3 minus negative 6 is the same as 3 plus 6, otherwise known as 9. And two things have happened here. One, we have actually evaluated 3 minus negative 6, plus we've also proven that the evaluation is correct. Now, just a quick note, the proof is this entire frame here. Everything here must be included as part of the proof. Otherwise, you don't have a proof. You have a bold assertion of a fact that may or may not be true. Now, this last is another example of a more general property that we can ultimately rely on, which is that if I have two integers a and b, a minus a negative b is the same as a plus b.