 Okay, today we're going to talk about two quick properties. These are the final two properties of real numbers that I'm going to talk about. The closure property and the density property. And again, just like my previous videos, I'm going to start with the algebra using variables with them and then do examples with numbers. These are two pretty simple properties of real numbers. And again, as I've stated in my previous videos, these are going to be, seem very simple, very, very basic properties. Stuff that you already know, but again, we're taking the stuff that we already know and defining them as mathematical properties just to make it easier for us. Okay, so the closure property. Closure property simply states that if you add two numbers together or if you multiply two numbers together, you're going to get a real number. Actually, I should be more specific. If you have two real numbers and you add them together, you get a real number. If you have two real numbers and you multiply them together, you're also going to get a real number. Okay, so what does that look like with the algebra? Come up with some variables. We'll just use A and B. A plus B is equal to C. And this doesn't really make a whole lot of sense at this point with the variables, but it just means A and B, if they're real numbers, then when you add them together, you're simply just going to get another real number. A, B, and C are all real numbers in this case. So that's the closure property for addition. So then the closure property for multiplication, A times B is equal to C. Actually, I can't use C here because when you add two numbers together, you get a certain number. But then when you multiply those two numbers, you don't get that same C. I'm going to choose a different variable. Let's go with D. Okay, now again, this doesn't make a whole lot of sense with the variables. Let's put numbers in there. A, B, and C. I'm just going to use the numbers 7, 8, and 9. Why not? Okay, so this would be 7 plus 8 is equal to, okay, well, I can't use 9 in there. So 7 plus 8 is going to be equal to 15. Okay, so if I take two real numbers, add them together, I'm going to get a real number. And then A times B is equal to D. So this is the closure property for multiplication. This would be 7 times 8. Notice I'm using the parentheses to denote multiplication. 7 times 8 is 56. So D in this case would be 56. So again, take two real numbers, multiply them together, you get another real number. Again, very, very basic. We all understand this. We all know this. We're just taking these answers and defining them as simple property. Closure property of addition. Closure property of multiplication. All right, now onto the density property. Density property simply states that between any two real numbers is another real number. And it's kind of hard to do a variable, so instead I'm going to do this with just a number line. So forget the algebra, forget the numbers. I'm just going to do a number line for this one. Okay, so I'll start my number line here and then here. Again, number lines go on forever, so I'm going to include arrows on both sides. Okay, so I got to choose two real numbers. And again, for this demonstration, I'm going to choose whatever I want to. It doesn't really matter what I choose. For example, I'll choose zero and one. Okay, those are two real numbers, zero and one, two very basic real numbers. Okay, now density property states that between these two numbers, there's always going to be another real number. Okay, so one number that is between this, let's say just right halfway in between is 0.5. There we go. And now that's pretty easy to understand. There's always going to be a number between two numbers, but you can keep going with this density property. So for example, if I want to go, if I choose two other real numbers, so example, zero and 0.5, if those are my two new real numbers, then there's always going to be a real number between them. So between zero and 0.5, oh heck, we don't have to go right in between. Zero and 0.3 is between zero and 0.5, so that follows along with the density property. And I can continue to go with this. So I can choose zero and 0.3, and between there is 0.1, and there's another real number in between them, and I can keep going, keep going, keep going. I can choose any two real numbers. It doesn't matter how close they are. I can choose 0.1 and 0.3 to get 0.2, I'll run out of room here, but you get the idea that it doesn't matter what two numbers I choose, I can always get in between those two numbers. So if I choose something that's a little bit closer, 0.1, 0.2, between that, I'm going to go up with my numbers here, 0.15, okay, that's between 0.1 and 0.2, and I can just keep going, keep going, keep going. And that's what's called the density property. If I have any two real numbers, there's always another real number in between them. And those are the two properties, the two last properties for real numbers. We have the closure property, closure property of addition, closure property of multiplication, and we have the density property, and those are the two last ones for now.