Algebra 1 – Module 5, Topic 2 – Properties of Real Numbers 
Hey guys, welcome to Algebra 1. We're reaching that point in the course where it's about time to start solving multi-step linear equations, but before we get there I want to make sure you have a good understanding of the properties of real numbers. Now, you've seen some of this before in Pre-algebra, but a little of it may be new to you. You ready? Let's go.
Now, before we start talking about the properties of real numbers, I want to make sure you know exactly what a real number is. The definition of a real number, it really can be a small number, say 1. It can be a large number, 2,000. It can be negative, -5. It can be positive, 4. It can be a fraction. It can be a decimal. It can be rational, which means it has a decimal that will terminate, say 2.5. it could be irrational, pi for example. Goes on and on forever, never repeats, never ends. Basically a real number is everything that's not an imaginary number. I know that sounds a little funny, but there are such things as imaginary numbers and you'll see more of that when you get to Algebra 2. For right now just know that a real number is just about all of the numbers that we're going to deal with in this course. 
Real numbers have properties that you need to know about. One of them, for example, we've used so far, the Distributive Property which tells me that if I have 3 real numbers, we're going to say that they're r, s, and t just to call them variables so we know we can replace them with any value that we wanted to. If I were given a setup like this where I have r multiplied by the quantity (s+t), I know that I need to multiply r times the first term, so rs, and add that to the product of r times the second term, rt. We've handled some of that before. That's one of the ones, like I said, that isn't new to you.
Another one's the Associative Property. The Associative Property tells me that if I'm adding numbers, in this case the Associative Property of Addition, I can group together the terms in any way I want when I'm adding. For example, if I were adding these 3 terms, r, s, and t, and I wanted to add r and s together first and then add t, I get exactly the same answer if I added s and t together first, and then added r. You believe me? Let me show you an example.
Let's say r is 1, s is 2, t is 3. Let's just use 1, 2, 3. Now let's say on the left side I'm going to start out by adding r and s together first, but on the right I want to add s and t together first. Let's see if we get the same answer. Let's give us a little more room to work up here. So okay, on the left, 1+2, I know that's 3. Then 3+3, I know that's 6. Now on the right it's telling me first add the 2+3. So 2+3, I know that's 5. 1+5 is also 6. That's the Associative Property. You can group together the terms however you want when you're adding and it doesn't change the value on either side. 
Now multiplication. Basically the same idea except this time I'm multiplying. If I wanted to multiply r and s together and then multiply that product times t, I get the exact same answer if I multiply s and t together first, and then multiply that times r. Let me show you this one too. Let's get some of this work out of the way here so we have some space. Let's say again r, s, and t are 1, 2, and 3. Here we have 1x2x3. 1x2x3. Let's say on the left I want to multiply 1 and 2 together first, and on the right I want to multiply 2 and 3 together first. 1x2, that's 2. Then 2x3, that's 6. Now on the right side. Get that just to keep it separate. Scoot that out of the way. We have some space here. On the right side, 2x3, that's 6. Then 1x6 is also 6.
That's the Associative Property of Multiplication which, like I said just tells you you can group together terms when you're multiplying however you want and in whatever order you want, and it doesn't change anything. All right. Let's keep moving. 
Commutative Property. What that tells you is for addition you can add terms together in whatever order you want. It doesn't change anything. Then for multiplying, you can multiply in whatever order you want. It doesn't change anything. Let me show you one. Let's say this time r is 1, s is 2.
That's telling us 1+2 is the same as 2+1, and that's true right? 1+2 is 3. 2+1 is 3. Changing the order of addition doesn't change the answer. Then, like I said, for multiplication same idea. Let's say now it's 1x2 and 2x1. 1x2, that's 2. 2x1, that's also 2. Changing the order of multiplication doesn't change your answer. Let's keep going. 
Identity. This one may be new, but you may have seen it in Pre-algebra. The Identity Property of Addition tells me, if I see right here r+0=r, adding 0 to a term doesn't change its value. We know that intuitively, but this is actually putting it on paper mathematically. For example, let's say this time r is 4. 4+0 is 4. Adding 0 to anything doesn't change its value. 
Now for multiplication. Let's think about this for a second. What could you multiply a number by and not change its value at all? Got it? 1. Multiplying a number by 1 doesn't change the value of that number. That's the Identity Property of Multiplication. Tells me that any real number times 1 is just that real number. That's just the answer I get back. For example, 4x1 is still 4. That's the Identity Property of Multiplication. Let's keep moving.
Inverse. The Inverse Property of Addition, what that tells me is if I have a real number and I add its opposite to it the answer is 0. For example, let's say r is still 4. 4+(-4), so 4 plus its inverse, is 0. That's the Inverse Property of Addition. Adding the inverse, adding a numbers inverse to it, is 0. All right.
Now, multiplication is a little different. The Inverse Property of Multiplication tells you if you multiply a number by its reciprocal, the answer is 1. This is what I mean. Let's say rx1/r=1. Now I'm going to use some numbers here so I can really prove this to you, so you really know what I mean. Let's get this out of our way. Get rid of that. Let's slide that a little bit. Get some more space. Let's say, again, r is 4. Multiplying 4 by its reciprocal, so by 1/4, that answer is going to be 1. Let me show you.
Remember back earlier in the course we said that we could always make a whole number a fraction by giving it a denominator of 1? Doesn't change its value and it just kind of helps us work with fractions. Now I'll look at this like a multiplication problem of 4/1x1/4. I'm just going to multiply straight across to get this answer. 4x1, that's 4. Denominators, 1x4, that's 4. If my final answer 4 divided by 4, I know that answer is 1. Any number times its reciprocal its answer is 1. That's the Inverse Property of Multiplication. All right, doing good. Let's keep going. 
Closure. This one may be new. What Closure tells us, the Closure Property of Addition, is if I add 2 real numbers together, my answer is always a real number. I'll never be able to add 2 real numbers together and get an imaginary number. For example, let's say r is 4, s is 5. I know 4 is a real number, and I know 5 is a real number. When I add those together I get 9. 9 is also a real number. That's all that means. That's all that means.
Closure Property of Multiplication. That tells me if I multiply 2 real numbers together, the answer is also real. Here we go. Let's still say r is 4 and s is 5. 4x5. I know 4 is a real number, I know 5 is a real number. Product of 4 and 5 is 20. 20 is also a real number. That's all that means. Any time you're going to multiply 2 real numbers together the answer is also a real number. All right. Now let's practice a little bit.
See we have a set of equations here, 2 equations. They're asking us which property justifies the work? I know it looks like a lot. There's a lot of terms, and variables. There's a lot going on. Just take a deep breath. Step back, look at it, and we're just going to break it down to figure out what's going on here. If I look at my first equation I see I have 2x+3 on the left, and on the right I have (4x+1) and I'm adding x to that. Second equation I still have 2x+3 on the left, so the left sides of these equations are the same. But the right side, something's changed.
You see here that they switched the order? Now (4x+1), it's still in parenthesis, but it's the second group listed. It's listed towards the back. That x term, that's listed up at the front. I'm still adding things together, but they changed the order of addition. Think back. Which property of real numbers said that you can change the order when you're adding and it didn't change anything? Remember? It's one of those ones you've seen in Pre-algebra. It's the Commutative Property of Addition. That's exactly what's going on here. You're changing the order of what you're doing over here. Changing the order of your addition, but it's not changing the answer for these. All right, good job. Let's keep moving.
Example 2. This one's a little different. Which property justifies the work between step 2 and step 3? We've got to kind of look at this in parts, look at it individually. We want to look at what's going on from step to step so we can figure out which property that we're going to need to answer this question. Let's start breaking this down. Step 1 starts out by giving me the equation. X+4=10. Step 2, things have changed a little bit. 
I have x+4 on the left, but they added a -4. I still have that 10 on the right, but they've added a -4. File that in your memory. Keep a hold of that. Then I see step 3. I have x+0. It seems like what must have happened here, 4+(-4). That's where the 0 came from. Then I have 10+(-4), which is 6. I'm going to stop there because this is what they want me to focus on. Here I have 4 and I added a -4. It's really the left side is the action you want to watch on this one. If you had a 4 and you added a -4 and your answer was 0, do you remember what property said that if you had a number, and you add its opposite the answer was 0?
That one was Inverse Property of Addition. That is the property that justifies the work from step 2 to step 3. I have 4. I added its inverse and then the answer was 0. That's what allowed me to move from step 2 to step 3. All right. Doing good. All right. It's time for you to try one. 
Now they're asking you to figure out which property justifies the work between step 3 and step 4. I'm going to set you up a little bit and I'm going to let you go. I want you to pause me, see how you do, and then press play again when you want to know the answer. 
I see for step 3 we had x+0=6. Step 4 I have x=6. It looks like what happened here, the action took place on the left side. I want you to see if you can figure out which property tells you that adding 0 to a real number doesn't change the value of that real number. Press pause, think about that for a minute, and then press play when you want to compare answers. 
Let's see how you did. Adding 0 to a real number doesn't change the value of the real number. That is the Identity Property of Addition. That is the property that allowed us to move from step 3 to step 4. Adding 0 to a real number doesn't change the value of that real number. All right. Good job. Let's keep going. 
We have a few more properties of real numbers and then I'm going to let you practice again for a little bit. One of the properties is the Reflexive Property, which basically means that any real number is equal to itself. Right? We already know that really. For example, 4=4. That's true. That's all the Reflexive Property means, is that any real number is equal to itself, and that's it. Keep moving. 
Multiplicative Property of 0. What this property tells me is that the product of any real number and 0 is always 0. That's what that means. Let's say rx0 is 0. That's the Multiplicative Property of 0. Multiplying a real number times 0 is 0. Keep moving. Got a couple more. 
The Zero Product Property. Similar to the Multiplicative Property of 0, but a little bit different. Zero Product Property, say that 3 times fast, the Zero Product Property tells us that if I have a product, so if I multiply 2 numbers together, 2 real numbers, in this case r and s, and the answer is 0, that means that either r was 0, or s was 0. That's all the Zero Product Property means. If you multiply 2 real numbers together and your answer was 0, then 1 of those real numbers had to be 0.
For example, let's say r is 4 again. It's the favorite number today. In order for me to multiply something by r and the answer to be 0, it had to be 0 that I used. That's all that means. You have a product of 2 numbers. Your answer is 0. 1 of those numbers has to be 0. That's all that means. All right. Moving right along.
Symmetric Property. What that means here, if r=s, then s=r. That's all that means. Symmetric Property. Basically you can flip around your equation and it doesn't change the value at all. For example here I'm going to use a variable that's going to represent a real number. I'm going to say if x=4 then 4=x. That's all that means. Those 2 real numbers have the same value, and I can write that equation in either order that I want and it doesn't change its meaning at all. For example, x=4 is the exact same thing as 4=x. All right. Doing good. 
Transitive Property. This is actually one of my favorites. I like puzzles and it just kind of reminds me of trying to figure out a puzzle here. What the Transitive Property means is if r=s, and s=t, then r=t. Let's kind of pause, think about that for a second. If r=s, and s=t, then r=t. Let me show you what I mean. Let's say if r=s, okay we'll take that. We'll believe that. Now they're saying s=t. So that means, basically, this. R=s, okay but now they're saying s=t, so I'm going to take the s away and replace it with t. Let's make that look like a t here. All right. Replace that with t. That's telling me then that r must equal t. 
That's the Transitive Property. Sometimes you may hear it called the chain rule. It means basically the same thing. I think about it sometimes like cutting out the middle man. If r=s, and s=t, let's just cut out s. Cut out the middle. R=t. All right. Doing good. Got an example for you. 
Which property justifies the work? Remember kind of the steps we took when we were solving these a minute ago. Step back from it for a minute. Take a deep breath and then get ready to attack it. It looks like on the first equation I have 5x+3=2x+9. Second equation I have 2x+9=5x+3. Okay, so what actually happened here? What's the difference between these 2 equations? Can you see they just swapped the order that they wrote it? First the 5x+3 came first then the 2x+9 was second. Then they just reversed the order. Which property told me when working with an equation that I can swap what's on the sides of my equation?
Got it. That was the Symmetric Property. That's what they were trying to show you right here, that you could swap the left and right side of your equation without changing the meaning of your equation. All right. Let's keep going. 
Use the Zero Product Property to explain why the solution to 5x=0 is 0. Now this is one I want you to try. I want you to look back and see exactly what that Zero Product Property said, and then I want you to look at this equation, this 5x=0, and tell me why the answer has to be 0. I want you to use the Zero Product Property to explain it to me. Take a minute, and know when you're explaining things in math it's a mixture of words and numbers. You're kind of going to need to show me a few things, and then maybe write out a couple of things. Take your time. Take a stab at it. Then press play when you're ready to compare. 
Let's see how you did. I'm going to show you how I kind of thought through it, and as long as your thinking process was similar to mine, we're not going to think about it exactly the same way, but as long as we have the same general idea then you know you got it right. The Zero Product Property says that if I multiply 2 numbers together, 2 real numbers together, and the answer was 0, then 1 of the numbers had to be 0. Right? I'm going to write that down here.
Zero Product Property. Remember that that said if rs=0, so if the product of 2 real numbers was 0, then r=0 or s=0. 1 of these numbers had to be 0 if the product of those numbers is 0. Then, looking at the equation that they gave us, this 5x=0, if the product of these 2 real numbers is 0, then I know 1 of these numbers has to be 0. Let's split them up so we can consider these 2 numbers separately. This is really the product of 5 and x right? 
5, that's kind of like our r. Matter of fact, let's get a few things out of the way to really see this clearly. Let's move that up a bit. Move that over. We do want the multiplication too. All right. Here we go. Zero Product Property. Rs=0. I'm going to write that right above here as soon as I get the pen back. All right. Rs=0. All right.
Zero Product Property says that if rs is 0, then either r is 0 or s is 0. Let's look at this. 5 is like our r here. Right? If r is 5 then I know it can't be 0. Right? 5 does not equal 0. S is our unknown here, and s, in this case, is x so s is going to have to have a value of 0 because the only way that I can multiply 5 times a number and that number be 0, and that product be 0, is if that number was 0. The number that I multiplied 5 by. X must be 0 because r=5 in this case. First real number is 5 because they gave that to us. That second real number has to be 0 if that product is 0. All right.
Okay, guys. You've reached the end of reviewing your properties of real numbers. I hope you have a good understanding of what we reviewed here. It's going to really come in handy when we start solving those multi-step linear equations. Hope to see you soon. Bye.