 Alright, now let's take a look at the second of our arithmetic operations, which is subtraction. So everything you need to know about the addition and subtraction of integers can be found in the following. First off, the definition of the additive inverse of A. Again, we can read this as negative A, but it's much better to read it as the additive inverse of A, because this reminds us of something very important, namely that A plus its additive inverse is going to always give us zero. The other thing that we need to know is that the addition of integers is associative and commutative. I can rearrange the addition of integers. And then finally, because I'm talking about subtraction, I actually do need a definition of subtraction, and so I'm going to take our definition of subtraction for the whole numbers and replace it with integers. So if A, B, and C are integers, no longer whole numbers, if they are integers where C is equal to A plus B, then A is equal to C minus B, and conversely. If I have this, I also have this sum. And all of these things are everything that there is to know about integer addition and subtraction. If you understand these three ideas, if you understand this definition and this definition, and make use of the associativity and commutativity of integer addition, then you know how to add and subtract integers. All those other rules that you may have learned for integer addition and subtraction all come back to these things, and you don't need those rules if you know the definitions and this one property. Just to emphasize that, let's take a look at a problem we'll find using only the definition of additive inverse, the associativity and commutativity of integer addition, definitions, and whole number arithmetic. Let's find 8 minus the additive inverse of 5, defend your steps. And again, the reason that the defend your steps is here is that when you defend your steps, you're going to have to say why you're allowed to do what you're doing. And if you're using anything besides what you're allowed to do, that step is going to identify that you're doing something you're not allowed to do. So here's a quick wrong answer to this question. 8 plus negative 5 is 8 plus 5, and that's 13. And well, this value is correct, but the problem is, well, we haven't even defended our steps. But if we were to do that, what allows us to write 8 minus negative 5 is 8 plus 5 is not a definition, is not associativity and commutativity, is not a definition of the arithmetic operations, and is not whole number arithmetic. It's another property of integer arithmetic that we're not allowed to use. So here is a wrong answer to the question. So let's collect what information we do now. So on the one hand, because we're dealing with this additive inverse of 5, well, we can use our definition of additive inverse. What we know, 5 plus its additive inverse gives you 0. Because we're dealing with a subtraction, then I'll pull up the definition of subtraction. If I have integers where c is equal to a plus b, then I can rewrite that as a subtraction, a is equal to c minus b. And these are things that I know and I can make use of. So let's take a look at what we can do. So here's a good common strategy in math. If you don't know what something is, give it a name. I'm going to figure out what 8 minus negative 5 is. So I'll give it a name, I'll call this x. So let x equal 8 minus the additive inverse of 5. And well, here I have a statement that says, if I have a subtraction, I could rewrite it as an addition. So if I have 8 equals 8 minus negative 5, I can rewrite that by the definition of subtraction as 8 is equal to x plus additive inverse of 5. Now, the only addition that we have involving additive inverse of 5 is this one. I know that 5 plus its additive inverse gives me 0. So if I had a 5 here, I could do something with it. So let's add 5 to both sides. We're allowed to do that as part of our whole number arithmetic. And so let's see. Well, I do know by the definition of additive inverse that additive inverse of 5 plus 5 is equal to 0. Well, strictly speaking, I wrote this down. Remember that integer arithmetic is associative and commutative. I can reverse the sum. Additive inverse of 5 plus 5 gives me 0. So my definition of additive inverse gives me x plus 0 over on the right-hand side. And I can do a whole number arithmetic to simplify the expressions. 8 plus 5 is 13. x plus 0 is x. And finally, remember that x was the same thing as 8 minus negative 5. So that's equal. So if I see the one, I could replace it with the other. Well, here's an x. So I can replace it with 8 minus negative 5. And there's my solution. 13 equals 8 minus negative 5. And I've defended my steps. I've invoked my definition of subtraction. I've invoked my definition of additive inverse. This is whole number arithmetic. The solution to this problem is everything here. You cannot answer this question with anything less than these steps.