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Studying for a test? Prepare with these 4 lessons on Module 7: Introduction to irrational numbers using geometry.
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- [Voiceover] My question to you is, "Which of these two values is greater?" 22.9%? Or, the square root of 0.45? And I encourage you to first try to see if you can think about this without a calculator. And then use a calculator to see which one is larger. Let's first try to do it without a calculator. So one thing that you could try to say is-- Well look-- You could say that 22.9% that's going to be less than-- I'm just gonna pick an arbitrary number here. Let's say that's less than 30%. So that's going to be less than-- and I picked 30% because it's easy to calculate 30%, or the square of 30%. So that's less than 30%. Or, it's another way of saying that's less than 0.3. 0.3 And then, we can try to compare 0.3 to this thing here. And if 0.3 is less than this, well, then 22.9% is going to be less than this 'cause it's less than 0.3. So why don't I change the problem to that? Let's compare 0.3-- 0.3 to square root-- to the square root of-- let me do that thing pink color-- to the square root of 0.45. And now I can use the squaring technique. What happens if I square each of these quantities? If I square this, 0.3 times 0.3, let's see, three times three is nine. But you're multiplying two things that each have one digit to the right, so you're gonna have two digits to the right. So it's going to be .09. And if you were to square this over here-- I'll just keep using the white-- if you were to square this right over here, what's that going to be? Well that's just going to be 0.45. So .09 is clearly less than 0.45. Or, 0.3 squared is clearly less than square root of-- 0.3 squared is less than the square root of 0.45 squared. And so we know that 0.3 is going to be less than, is going to be less than the square root of 0.45. And so now we can say that 22.9%, if it's less than this, and this is less than this, well 22.9% must be less than that. Now another way you can do it, you could take out a calculator-- you could take out a calculator. I'll do that. Just for kicks. So let me get the calculator out. So, 22.9%, that's the same thing as-- that's the same thing as 0.229. So we really just want to compare this quantity to the square root of 0.45. And there are two ways we can do it. We could do it the way I started. You could square this and see if it's greater than 0.45. So let's do that. You could just square it. And you see "no." This is 0.05. Which is clearly less than 0.45. And so that would validate this. Or, you could do it the other way around. You could just use your calculator to calculate the square root of this. So you could say 0.45 and then take the square root. The square root of this value right over here is approximately-- is approximately 0.67. So this thing right over here is approximately 0.67. Which is approximately 67%, which is clearly greater than 22.9%. So a bunch of ways that you could approach it. It is nice to be able to estimate things. Think in your head. So that if you didn't have access to a calculator you could get a general sense of, "Hey, would you rather have a 22.9% off the price of something?" Or, maybe some type of new store could say, "Hey, sales! Square root of 0.45 off of all goods." I don't know. (chuckles) That can be an interesting thing. Could be very confusing for customers. But, anyway. Hopefully this was helpful.