BYU-Idaho Online Learning
Video Transcript
[This transcript is currently a work in progress.]
Male: We're now to the point in math where we're dealing with letters as numbers, where we're going to combine things together, and we're going to put things together. We got to know what the rules of the game are, the laws and the properties that have existed with numbers all the time, but to make sure we don't mess up when we've got letters involved, we need to identify what properties we can actually do. What can we get away with when we're doing stuff? Let's look at some numbers first of all. If we look at 3 plus 5, we know that's exact same thing as 5 plus 3. That's eight and that's eight. We can switch them around. Look at this, if there's 3 times 5, we know that's the same as 5 times 3. We can switch those around. What if we did 3 minus 5, 5 minus 3? Are those the exact same thing? Can we switch two numbers around a subtract sign? No way, but wait a second, we have done something. If we view this as, instead of 3 minus 5, 3 plus a negative 5, and we switch it around a plus sign, it becomes negative 5 plus 3. Are those two things the same? They are indeed, but that one is not the same. That's good to know. What about 3 divided by 6 and 6 divided by 3? At least one of them comes out nice. That's a two, and that's not a two. Oh, that's not good. So that's not the same. Of course, we could change this into 3 times 1/6 and move it around a times sign, 1/6 times 3, that works. Moving them around a times sign works. Moving them around a plus sign works, switching orders. What about exponents? What if we took like 2 to the 5 and 5 to the 2? Can we switch them around exponentially? Two times 2 times 2 is 8, times 2 is 16 times 2 is 32 over here. This guy is 25, that's 5 times 5. Nope, doesn't happen. We can't do it there either. We can switch whenever there is a plus or a times. We can switch order. These rules hold for all numbers. You could try them with anything. It gives it a special name to differentiate it from those things that we can't. Anytime you have addition or multiplication, it's called commutative or commuting. It is the commutative property, or the commutative law of addition and multiplication. Now it comes from the word "commute", to drive back and forth. You're switching in order, driving back and forth. That's the commutative property of multiplication or addition. Great, that's one of them. Let's look at the next one. We take something like 3 plus 4 plus 5, with parentheses around those two, and go 3 plus 4 plus 5 with the parentheses around the other two. Should that make any difference? Of course it shouldn't. Remember back when we had piles of M and M's or Skittles, and you have a pile with 3 and 4 and 5. If you combine these two piles together, first 9 and then add the 3 to it, it shouldn't make any difference if you move the parentheses around one way or the other. What about multiplication, does that work? Three times 4 times 5. Moving parentheses like that. Well, 3 times 4 times 5 better be the same as 3 times 4 times 5. Well, if we try subtraction, you'll notice, oh, 3 minus 4 minus 5, these two aren't the same. This is 3 minus a negative 1, that's 4. This is negative 1 minus 5, that's negative 6. Those are not the same. What about division? Let's try 30 divided by 6 divided by 2, so we get something coming out here and do that. Compare it to 30 divided by 6 divided by 2. This, we get 30 divided by 3 is 10, here we have 5 divided by 2 and it doesn't work at all. Again, for multiplication and addition, it works, but for subtraction, division, exponents, it's not going to work. This is a special law that you can just, if you keep the exact same order, you can actually move parentheses, so again, with multiplication and addition, multiplication and addition, you can actually move parentheses. Holy cow, you can move parentheses. And this is called the associative law because it associates. This 4 first associates with the 3 or it can first associate with the 5, either one of them. It's the associative property of addition and multiplication. Okay, good. Those are the major laws and properties. There are a couple more things we need to identify. There are numbers that are invisible. For example, sticking with addition and multiplication, if we take any number such as 6 and we add zero to it, we get 6. Seven plus zero equals 7. Eight thousand three hundred and ninety-four plus zero equals 8,394. Notice that this stayed exactly the same, which means that that guy is basically invisible. That invisible thing has a very specific name. It is called the identity. So, invisible number is, that means identity, that's its name. In this case, adding zero is the identity. Well, what does that in multiplication, 6 times what equals 6? Well, it's 1. Seven times 1 equals 7, 8,394 times 1 equals 8,394. In this case, 1 is the identity with multiplication. Zero is the identity with addition. With addition, it's zero. With multiplication, it's the number 1, and that's the identity. And the final one, we've all heard the word "canceling out", like 5, and you add it to a negative 5 or 5 minus 5, it's like, oh, they cancel out. What they really did was add to be an invisible number. Oh, look at that. Good thing we put that up there. That means that these guys are opposites, or called inverses. When they add up to be the invisible number, they're additive inverses. Same thing if you took like 5 times 1/5. You multiply those guys and you get the invisible number. They're inverses. However, these guys are multiplicative, meaning with multiplication inverses, the familiar terms, additive inverses, are opposites. Multiplicative inverses are, see if you remember this word, reciprocals, where they actually undo each other. So these are all the laws. This is what canceling out really means, is that they really add to zero, or if you're multiplying the canceling out, they are multiplying to one, or dividing to one, if you prefer to say it that way. Let's look and make sure we can identify some of these in some expressions. All right, here are a few examples using variables and equations where we're going to identify which of these laws were being used. Here, 3 plus X equals X plus 3. They switched order on us around a plus sign. That means they commuted, commutative property of addition. That's what just happened in there. Let's look at this one. Two, parenthesis, 5 X minus 7 Y equals 2 parenthesis negative 7 Y plus 5 X. What happened there? Did they move the parentheses? Nope, they have parentheses, but we're not associative property because they didn't move the parentheses. What they did, they switched these two order, imagining that as a plus a negative 7 Y. They moved them around. Again, commutative property of addition. You can switch order if you view it as a plus sign. Let's try this one. Two parenthesis 5 X plus 7 Y parentheses equals parentheses 5 X plus 7 Y parentheses 2, oh my goodness, they did some switching. They took this whole unit and switched places with that whole unit. That's commutative because they switched order, but they switched it around this times sign, being next to each other. That's commutative of multiplication. Let's look at this one. Did they do the same thing here? Three plus 8 X plus 9 Y, 3 plus 8 X plus 9 Y. Notice I didn't switch any order. We're not commutative, but the parentheses moved, which meant this was the associative property of, these are all plus signs, associative property of addition. Let's write up just a couple more so you can see this. If I have a number 5 X and I add it to zero, and I get 5 X, I've used the identity of addition, or additive identity. If I take 6 X plus a negative 6 X , 6 X minus 6 X and that goes away, these guys are opposites. I've used additive inverses. 3 P times 1/3p, this is using multiplicative inverses, or reciprocals, even with letters in them. If I take 7 P plus zero equals, if I take 7 P times 1 and I get 7 P, I've used the multiplicative identity. And there you go, those are the properties.
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