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Welcome to Algebra I: Properties of Real Numbers

26 minutes

Hey guys, welcome to Algebra 1. We're reaching that point where it's time to solve multi-step linear equations, but first, I want to make sure you understand the properties of real numbers. You've seen some of this in Pre-Algebra, but some may be new to you. You ready? Let's go. Okay. Before talking about the properties of real numbers, you should know 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. Or 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 funny, but there are imaginary numbers. You'll see them when you get to Algebra 2. But a real number is all of the numbers that we'll deal with in this course. Real numbers have properties you must know. One that we've used is the distributive property, which says, if I have three real numbers-- we'll call them r, s, and t; variables we know we can replace with any value that we want to. If I were given this setup, 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 that before. That's one of the ones that isn't new to you. Another one's the associative property. The associative property tells me that if I'm adding numbers-- here, the associative property of addition-- I can group them together in any way I want when adding. For example, if I were adding these three terms, r, s, and t, and I wanted to first add r and s, and then add t, I'd get exactly the same answer if I added s and t together first, and then added r. 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. On the left side, I'm going to start out by adding r and s together. On the right, I'll add s and t first. Is it the same answer? Let's get 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. 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 them together however you want when adding. 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'd get the exact same answer if I multiplied s and t together first, and then multiplied that times r. Let me show you this one. Let's get 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 multiply 1 and 2 together first, and on the right I multiply 2 and 3 together first. 1x2, that's 2. Then 2x3, that's 6. Now on the right side-- just to keep it separate, let's scoot that away. 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 tells you you can group together terms when multiplying however you want, 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. And then for multiplying, you can multiply in whatever order and 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-- I see right here r+0 = r. Adding 0 to a term doesn't change its value. We know that intuitively, but this is 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? You got it? One. Multiplying a number by 1 doesn't change the value of that number. That's the identity property of multiplication; it says any real number times 1 is just that real number. That's 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 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 4--still 4. 4+(-4), so 4 plus its inverse, is 0. That's the inverse property of addition. Adding the inverse, adding a number's 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 rx 1/r = 1. I'm going to use some numbers here to prove this to you, so you'll see. Let's get this out of our way, get rid of that. Let's slide that a little bit, get some 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 earlier we said we could make a whole number a fraction by giving it a denominator of 1? Doesn't change its value and it helps us work with fractions. I'll look at this like a multiplication problem of 4/1 x 1/4. I'll multiply straight across to get this answer. 4x1, that's 4. Denominators, 1x4, that's 4. For 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 two real numbers together, my answer is a real number. I'll never add two 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 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 says if I multiply two real numbers together, the answer is also real. Here we go. Let's say r is 4 and s is 5. 4x5. 4 is a real number and 5 is a real number. The product of 4 and 5 is 20. 20 is also a real number; that's all that means. When you multiply two real numbers together, the answer is also a real number. All right, now let's practice a little bit. We have a set of equations here, two equations. They're asking which property justifies the work? I know it looks like a lot-- there's a lot of terms and variables. Take a deep breath, look at it, and we'll break it down to figure out what's going on. 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 parentheses, but now 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've changed the order of addition. Think back. Which property of real numbers said you can change the order when adding without changing anything? Remember? It's one 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-- 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 look at this in parts, individually. Look at what's going on from step to step to figure out which property we'll 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 have that 10 on the right, but they've added a -4. File that in your memory, keep ahold of that. Then I see step 3. I have x+0. It seems like what 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 I'll focus on. Here I have 4 and I added a -4. It's really the left side, is the action to watch here. If you had 4, added -4, and your answer was 0, do you remember what property said that if you add a number and its opposite, the answer was 0? That one was the 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. Figure out which property justifies the work between step 3 and step 4. I'll set you up and let you go. I want you to pause, see how you do, then press play to hear 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 was on the left side. 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, 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 that real number. That is the identity property of addition. That property allowed us to move from step 3 to step 4. Adding 0 to a real number doesn't change its value. All right, good job! Let's keep going. We have more properties of real numbers. Then I'll let you practice again for a bit. One property is the reflexive property, which states 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-- 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 different. The zero product property-- say that three times fast-- the zero product property states that if I have a product-- if I multiply two real numbers together, here, 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 two real numbers together and your answer was 0, one of those real numbers had to be 0. For example, let's say r is 4 again. It's the favorite number today. For me to multiply something by r and get 0, I must have used 0. That's all that means. You have a product of two numbers. Your answer is 0. One number 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. You can flip around your equation without changing the value at all. Here, I'm going to use a variable to represent a real number. I'm going to say if x = 4 then 4 = x. That's all that means. Those two real numbers have the same value. The equation can be in either order that I want. 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 one of my favorites. I like puzzles, and it reminds me of figuring out a puzzle here. What the transitive property means is if r = s, and s = t, then r = t. 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 that r must equal t. That's the transitive property. You may hear it called "the chain rule." It means the same thing. It's 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. Which property justifies the work? Remember the steps we took when we were solving these a minute ago. Step back from it. Take a deep breath, 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 two equations? 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 the sides of my equation? You got it--that was the symmetric property. That's what they're showing you right here, that you could swap the left and right sides 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." This is one I want you to try. Look back and see what that zero product property said, and then look at this equation, this 5x = 0, and tell me why the answer has to be 0. Use the zero product property to explain it to me. Take a minute. When explaining things in math, you mix words and numbers. You need to show me a couple things, then write out a couple things. Take a stab at it. Then press play when you're ready to compare. Let's see how you did. I'll show you how I thought through it. If your thinking process was similar to mine-- we won't think about it the same way-- but if we have similar ideas, you got it right. The zero product property says that if I multiply two numbers together-- two real numbers together-- and the answer was 0, then one of the numbers had to be 0, right? I'll write that down here. Zero product property. Remember that that said if rs = 0, so if the product of two real numbers was 0... then r = 0 or s = 0. One of these numbers had to be 0 if the product of those numbers is 0. Then, looking at the equation they gave us, this 5x = 0, if the product of these two real numbers is 0, then one of these numbers must be 0. Let's split them up to consider them separately. This is really the product of 5 and x, right? 5, that's kind of like our r. 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'll write that right above here when I get the pen. All right, rs = 0. All right, zero product property says 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 x 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. The first real number is 5; they gave us that. That second real number must 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 understand of what we reviewed here. It will come in handy with multi-step linear equations. See you soon. Accessibility provided by the U.S. Department of Education.

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In this program, students learn about the properties of real numbers. The basic properties of real numbers are used to determine the order in which to simplify math expressions. Part of the "Welcome to Algebra I" series.

Media Details

Runtime: 26 minutes

Welcome to Algebra I
Episode 1
31 minutes
Grade Level: 7 - 12
Welcome to Algebra I
Episode 2
25 minutes
Grade Level: 7 - 12
Welcome to Algebra I
Episode 3
18 minutes
Grade Level: 7 - 12
Welcome to Algebra I
Episode 4
17 minutes
Grade Level: 7 - 12
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Episode 5
22 minutes
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Episode 6
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Episode 7
24 minutes
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Episode 8
15 minutes
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Episode 9
25 minutes
Grade Level: 7 - 12
Welcome to Algebra I
Episode 10
16 minutes
Grade Level: 7 - 12