Dive into the world of writing mathematical expressions from verbal statements. Understand the importance of parentheses in expressions, and how they can change the meaning of calculations.
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- [Voiceover] What I hope to do in this video, is give ourselves some practice interpreting statements and writing them as mathematical expressions, possibly using parentheses. So let's get started. And for any of these statements, if you get so inspired, and I encourage you to get so inspired, pause the video and see if you can write them as mathematical expressions. So this first one says 700 minus 19, divided in half. So we could say, another way to think about divided in half is divided by two, so we could write this as 709 minus 19, and we're going to do that first, so that's why I put the parentheses around it, divided by two, or divided in half. That's one way that we could write this. Now the next one, and once again, pause it if you get inspired, and I encourage you to. Three times the sum of 56 and seven. So it's gonna be three times the sum of 56 and seven. So the sum of 56 and seven, we want to take that first, so it's going to be 56 plus the seven, that's the sum of 56 and seven, and then we want to do three times that. We want to do three times this sum. So we could write it like that. Another way we could write it, when you're dealing with parentheses, and you're going to see this more and more as you get into more and more fancy algebra, I guess you could say, but what I'm about to show you isn't so fancy, is, you don't have to write the multiplication sign here. You could just write three, and then open parentheses, 56 plus seven, and this, too, is three times the sum of 56 and seven. And you want to be very careful, because you might be tempted to maybe do it without the parentheses, so you might be tempted to do something like this, three times 56 plus seven, but this one isn't, obviously, three times the sum of 56 and seven. In fact, the standard way to interpret this is that you would do the multiplication first. You would do three times 56, and then add seven, which is going to give you a different value, and you could try it out, than if you were to add the 56 and the seven first. So, to make sure that you do the 56 and the seven first, you want to put this parentheses around it. So let's keep going. The sum of three times 56 and seven. So we're gonna take the sum of two things. The first thing that we're gonna take the sum of is three times 56. So, three times 56, and seven. Let me do that in a different color. And seven. So this right over here is the sum of three times 56, and seven. Now it's always good to write the parentheses. It makes it a little bit cleaner, a little bit more obvious. Look, I'm gonna take the three times 56, I'm gonna do that first, and then I'm gonna add seven, but based on what I just told you, the standard way, if someone were to just write three times 56 plus seven, this actually can still be interpreted as the sum of three times 56, and seven, because as I just said, the standard, the convention, so to speak, is to do your multiplication first. Order of operations, which you may or may not, if you're not familiar, you will be familiar with it soon, is to do the multiplication first, and then add the seven, or then do the addition. But just to make it clear, it doesn't hurt to put the parentheses there. Three times 56, plus seven. Now we have 43 minus the sum of 16 and 11. So, 43 minus, so we're gonna have 43 minus, minus the sum of 16 and 11. So, minus the sum of 16 and 11. So, from 43, we're gonna take the sum of 16 and 11, and so, once again, the parentheses make it clear that we're going to take the sum of 16 and 11, and we're gonna take that from 43. The parentheses are very, very, very important here, because if we just did 43 minus 16 plus 11, the standard way of interpreting this would be 43 minus 16, and then adding 11, which would give you a different value than 43 minus the sum of 16 and 11. So once again, the parentheses are very, very, very important here to make it clear that you're gonna add the 16 and 11 first, and then subtract that sum from 43. This is fun, let's keep going. 10 times the quotient of 104 and eight. So, we're gonna do 10 times something. 10 times the quotient of 104 and eight, and so the quotient of 104 and 8 we could write like this, 104 divided by eight, or, based on what we told you a little earlier, you could write this as 10 times the quotient of 104 and eight, or 104 divided by eight. Now let's just do this last one. Four times as large as the expression 175 minus 58. So I'm gonna do four times as large as something, so I'm gonna multiply something times four. It's four times as large as the expression 175 minus 58. And once again, I could write it as four times as large as the expression, let me do that in that purple color, as the expression 175 minus 58. Either way, and once again, if you were to do it like this, if you didn't write the parentheses, then, it wouldn't be the same thing, 'cause if the parentheses weren't here, then you would want to do the four times 175 first, and then subtract the 58, which isn't what this statement is telling us. And this last one, I think, brings up an interesting thing for us to think about, because if someone were to walk up to you on the street, and they were to show you-- Whoops, what's going on with my computer? And they were to show you two different expressions. Well, the first expression said two-- Let's write it this way, actually, I'm not gonna even speak 'em out. I'm just gonna write it down. I'm just gonna write some crazy number here. Some crazy numbers here. So that's one expression that someone were to write, and let's say another one is this one, and I'm intentionally-- What, I put the commas in the wrong place. Let me make sure I get this right. Alright, that's 183,576. This is 37,399. So that's one expression, and then another expression is this. And I'm intentionally not reading it out. Well, I'll read it out a little bit, 37,399. And someone said, "Quick! "Which expression is larger?" And you might be tempted, or you might not be tempted, but you might be tempted, "Oh, let me calculate this thing. "Gee, I'm gonna have to write this thing down, "or use a calculator or something, "or whatever else to add "183,576 "plus 37,399, "and then I'm gonna have to multiply that by two, "and figure out what that number is equal to, "and then I would have to take "183,576 plus 37,399 "and figure out what that is, "multiply that by seven, "and figure out what that's going to be. "That's hard! "That's gonna take--" Not hard, it's just gonna take you some time, you might make some careless mistakes. But the big realization to say, "Well, which one is larger? "Well I don't have to even calculate these things!" 'Cause this is two times this craziness right over here, this thing that's gonna be 200 something thousand, and this is seven times that thing that is going to be 200 and something thousand. So seven times that thing is going to be larger than two times that thing, and so, one way to-- Before you dive deep, and start computing things, it's always good to take a step back and say, "Hey, look, can I look at how the expressions are formed, "the structure of these expressions?" And say, "Look, this is two times this thing, "and this is seven times this thing." Well, the seven is going to be, this one right over here is going to be a larger expression. Anyway, hopefully you enjoyed that as much as I did.