 Okay, today we're going to talk about the properties of real numbers. These are just four properties of the real numbers. There are many more, but today we're going to look at additive identity, multiplicative identity, additive inverse, and multiplicative inverse. Those are the four we're going to talk about today. And what we're going to do is I'm going to talk about the algebra first, and then I'm going to talk about the numbers. The algebra is just talking about these principles, talking about these properties, excuse me, with variables, and then I'm going to talk about them with numbers, just plug in numbers for the variables. Okay, so the additive identity is first, this is a very, very basic property. It basically says if I have a number plus zero, that's always going to simply just equal that number. A lot of these properties that we define, they're very simple properties. But again, in mathematics we always want to define everything no matter how simple it is. Okay, so the additive identity basically states if you have a number plus zero, you're always going to get simply just that number. And again, I'm using n to represent that number. You're going to be using a little bit of algebra. Okay, so now for the numbers, so I'm going to use a number for n. It doesn't really matter what I use. In this case I'll use three, plus zero is equal to three. So if I take a number plus zero, nothing's really going to happen. I'm simply just going to get three. And that is the additive identity. And I know this property is very basic, very simple, something you already know. But again, what we're doing is we're taking the very basic properties and we're writing them down. So again, simple stuff that you know, we're just labeling them as different properties. All right, next we have the multiplicative identity. This one is kind of similar to the additive identity. But again, we're going to use multiplication instead of addition. Multiplicative identity, I use a different variable. So I'll use, in this case, I'll use m. So m just represents a number. It doesn't really matter what number it is. If I take a number and I multiply times one, I'm going to get that same number, m. Okay, so the additive identity, take a number plus zero and you get that same number. For the multiplicative identity, take a number times one and you're going to get that same number, okay? So again, a little bit of similarity there, but the difference is addition and multiplication, okay? So there's the algebra with it, now I'll do numbers with it. Let's try something negative. How about negative two times one is simply equal to negative two. So even with negative numbers, if you multiply times one, you're still going to get the same thing. Negative two times one, you simply get negative two, same thing. Okay, next is the additive inverse in mathematics. When we talk about inverses, it usually means opposite. But in mathematics, it's a little bit different when we say opposite. Sometimes opposite numbers, opposite operations, there's a little bit difference there. Anyway, additive inverse, this one is if you take a number. So for example, if I take, I'll use the same variables as before. If you take n plus the opposite additive inverse. So we're taking a number plus it's opposite. Notice I'm using parentheses here. The opposite of n, whatever that number is, okay? If I take a number plus it's opposite, I'm always going to get zero. So that's your additive inverse. Take a number plus it's opposite, I always get zero. All right, so let's throw some numbers in there now. What number did I use before? I used three, so let's use that same number. Three plus a negative three is going to equal zero. Now again, if you look at it with the numbers, that's very simple. Three plus negative three is zero, everybody knows that. Now when you look at it with the algebra, that's a little bit different since we're introducing variables into this. But again, simple property that we know, we're just defining it as the additive inverse, okay? Last but not least, we have the multiplicative inverse. As you can well imagine, we're going to be multiplying here. I'm going to use the same variables before with multiplicative. I'm going to use n. But this time when I multiply, I'm going to use the inverse. Now the inverse of multiplication is, the opposite of multiplication is actually division. So I'm going to take a number times, now you'll see the division here in just a second, a number times it's reciprocal. Now remember reciprocal that vocabulary where you take a number and you flip it. You flip it over that fraction bar, okay? So m times one over m times the reciprocal, okay? Now when I talked about division, this fraction bar is also known as division, so that's where the inverse comes in, that's where division comes in, okay? Equals, so m times one over m is going to equal one. So if I ever have a number times it's reciprocal, I'm always going to get one. All right, so let's put numbers to this what I used last time. I used negative two last time for m, so I'll do that same thing here. Negative two times one over negative two. And two on top, two on bottom, those will cancel. Negative, two negatives makes it positive, so everything's going to cancel to one, okay? So those are our four properties of real numbers. One of the, four of the properties of real numbers. Additive identity, multiplicative identity, additive inverse, and multiplicative inverse.