 Okay, in this video, I'm going to talk about properties of real numbers. I'm just going to talk about two properties this time, the commutative property and the associative property. Now this is a little bit misleading. It looks like I only have two properties up here, but commutative and associative, they both kind of have subcategories, you can think of a mess. There's the commutative property of addition and the commutative property of multiplication. Also the commutative property of addition and the commutative property of multiplication. I'm going to show you the difference between the two. I know it only looks like two of them up there, but actually there's four different properties here. Okay, first thing I'm going to do is I'm going to talk about what their definitions are and then I'm going to give the algebra for them using variables and then I'm going to give number examples of them. Okay, first off the commutative property. Commutative property simply states that I can add or multiply two numbers in any order. It does not matter what order I add or multiply them in. So that gives us the difference between the commutative property of addition and the commutative property of multiplication. Again, I'll show you the difference between the two. So for the algebra portion, what I'm going to do is I'm going to show this, first the commutative property of addition, I'm going to show that first. Let's just use some random variables, it doesn't really matter what we use. Let's do something simple, A plus B. Okay, according to the commutative property if I have a number plus another number, in this case A plus B, it doesn't matter what order I add them in. So what I can do is I can actually switch these up. I can say B plus A is also going to give me the same thing. Okay, so that's the commutative property of addition. So that's with algebra, now let's do that with numbers. Let's use a couple of numbers for A and B, how about four and five? Four plus five is equal to five plus four. Okay, now these are pretty simple numbers to use. So if I take four plus five I get nine. On the other side of the equal side I get five plus four which is also nine. So we can see just with these real numbers that it doesn't matter what order I add the two numbers in, I'm still going to get the same answer. Okay, so that's the commutative property of addition. Now for the commutative property of multiplication. Okay, I'm going to use some of the same variables, but this time I'm going to multiply. So it's going to look very, very similar, but yet it's kind of a separate property. Okay, commutative property of multiplication A times B is equal to B times A. Now notice I didn't put a dot or a multiplication symbol between them. Remember that when we put two variables next to one another there's what we call implied multiplication. We just know that we are supposed to multiply these two numbers together. So A is a number, B is a separate number, and since they're right next to each other I've got to multiply those together. Alright, so A times B is equal to B times A. Commutative property tells us that it doesn't matter what order I multiply them in, I'm still going to get the same answer. Alright, so let's do this with numbers now. I used four and five for A and B, so let's just use the same numbers. Four times five. Now with numbers I can't just put them next to each other, that looks like a 45. So what I have to do is I have to use parentheses to denote that I am multiplying. I guess like what else I could use? I could use four dot five, make sure that dot is raised up a little bit so it doesn't look like 4.5, or I could use four times five. But these bottom two examples here we usually don't use those this late in algebra. The dot sometimes can look like a decimal, so we don't usually use that, and the x here, the multiplication symbol, well just like I said it looks like an x, so sometimes it looks like a variable. So these two we don't even use, so instead of using these two we use parentheses to tell us that two numbers are multiplying together. Anyway, so that's four and five, and again, commutative property of multiplication tells me it doesn't matter what order I multiply them in, so I'm going to switch these up, five times four, okay, four times five is 20, five times four is still 20, so you can see that it doesn't matter what order I have the numbers in, I'm still going to get 20. So those are, that's the first property, commutative, but there's two kind of subcategories, commutative property of addition and commutative property of multiplication. So there's two of them there, I know I didn't write it up there, but there are two of them there. Okay, so next is the associative property. Commutative property is when we have two numbers together, it doesn't matter what order we multiply two numbers, so commutative you can think of as, I can think of it this way, it's how we add or multiply two numbers, so that's commutative property as opposed to associative property we actually have groups of numbers, or a group of numbers, we'll call it group. Group of numbers, okay, so that's actually the big difference between the two, is commutative property is how we look at two numbers, how we add or multiply two numbers, associative property is how we add or multiply a group of numbers, okay, let's do it with the algebra and with numbers to get an idea, or a better idea of what these two properties are. Okay, so I'm going to use some variables again, I'm going to use a and b, so let's use something else, I'll use x, y and z, those are other common, famous variables that are used, okay, so we have x plus y plus z, so here's my group of numbers, so again two numbers a, b and group of numbers x, y and z, so you can see the definite difference between commutative and associative, all right, so if you remember your order of operations PEMDocs, parentheses, exponents, multiplication division, addition, subtraction, whenever you have addition here I'm supposed to add these from left to right, okay, so what I'm going to do is I'm actually going to take x and y and I'm going to add those together first, so I'm using parentheses here to denote do that first, all right, take x plus y and then add z, associative property of addition tells me it doesn't matter what order I add these numbers in, okay, so I can take x plus y plus z but instead of adding the first two together, what I'm going to do is I'm going to add the last two together, so I'm going to use parentheses here to denote to add the last two together first, all right, now what does that do, how does that affect these numbers, well again associative property tells us it doesn't affect anything, we're still going to get the same answer no matter what numbers we use, so let's try this with numbers to see what it looks like, so let's do some simple numbers, I used four and five so let's go even simpler than that, we'll do x is one, y is two and z is three, so in this case I have a little close together here, one plus two plus three, okay, add those together, one plus two is three, one plus two is three, three plus three is going to be six, okay, so you can see here that this left side with numbers added in that's going to be six, so let's see what the other side looks like, so I'm basically moving my parentheses around a little bit, okay, so I got one plus two plus three right there at the edge of the screen, okay, two plus three, parentheses do that first, two plus three is five, five plus one is six, left side was six, the right side is six, so we can see with numbers this associative property holds true, okay, the left side was six, the right side was six, it doesn't matter what order I add these groups of numbers with, alright, so that is the associative property of addition, now let's go with the associative property of multiplications, so I'm going to be doing my next one, I use the same variables, so I'm going to take x times y times z, now same thing with PEMDA, same thing with addition, I do, if I'm just multiplying, I do everything from left to right, so in this case I have to take x times y, I have to do that first and then take that time z, okay, that's the proper way to do multiplication if all you're doing is multiplication, alright, associative property of multiplication tells me that honestly it doesn't matter what order I multiply all these numbers in, I can order, multiply the first two together or x times y times z, I can multiply the last two together first, so it does not matter what order I multiply my groups of numbers in, as long as the only thing I'm doing is multiplying, alright, so that's with variables, let's now do this with numbers, so it makes a little bit more sense, alright, so I used 1s, 2s and 3s, I'll use those again, 1 times 2, okay I am using the dot here, I don't want to do too many parentheses, so I'm just doing the dot here, 1 times 2 and then z is 3 here on the outside equals 1 on the outside, 2 times 3 on the inside, again I know I said earlier not to use that dot, but in this case I don't want to do parentheses over and over again, so I am actually using the dot there, make sure that dot is raised up so you don't mistake that for a decimal, anyway back to the numbers, 1 times 2 is 2, 2 times 3 is 6, on this other side 2 times 3 is 6, 6 times 1 is 6, the left side is 6 and the right side is 6, so we can see associative property holds true, doesn't matter what numbers I use, doesn't matter how I multiply them, doesn't matter what order, okay, I'm always going to get 6 with the numbers 1, 2 and 3, alright so a little bit of summary, those are 2 properties of real numbers, you can almost call it 4 properties, we have the commutative property of addition, commutative property of multiplication and again that property states that it doesn't matter what order you add or multiply 2 numbers, which is different from the associative property, okay, associative property of addition, associative property of multiplication, that tells us it doesn't matter what order you add or multiply group of numbers, it could be 3 or 4 or 5 numbers it doesn't really matter, in this case I'm just using 3, okay so those are 2 properties of real numbers, again the commutative property of addition and of multiplication and the associative property of addition and of multiplication.