Approximating irrational numbers
- I have six numbers here and you see that five of them are irrational. They involve the square root of a non-perfect square. Our goal in this video is, without a calculator, see if we can sort these numbers from least to greatest. And like always, pause this video and see if you can do that. So I'll give you a hint. The hint is-- it's very hard without a calculator. Square root of two is gonna be one point something, something. Square root of three is gonna be one point something, something. How do we do this? We just have to realize that, if I have some number, let's say I have some number a that is greater than 0. And if we know that a is less than b, then a squared is going to be less than b squared. If one positive number is less than another positive number, then the square of this positive number is going to be less than the square of that number. So one thing that we could do when we are comparing all of these irrational numbers that involve square roots of non-perfect squares, let's compare their squares. Because their squares are not going to be irrational numbers. It's going to be much easier to compare, and then we can order them. Because if we order the squares, then they'll tell us what happens if we order their square roots. What am I talking about? Well, I'm just gonna square each of these. So if I take this to the second power, this is going to be four square roots of two, times four square roots of two. You can change the order of multiplication. That's four times four times the square root of two times the square root of two. Now, four times four is 16. Square root of two times square root of two, well, that's just going to be two. So it's gonna be 16 times two which is equal to 32. Now what about two square roots of three? Well, same idea. Let's square it, let's square it. And i'll do this one a little bit faster. So if we square two square roots of three, this is going to be two squared times square root of three squared. So it's going to be two squared times the square root of three squared. Well, two squared is going to be four. Square root of three squared is going to be three. So this is going to be equal to 12. That's this thing squared. If this step seems a little bit confusing, if you have the product of two things raised to a power, that's the same thing as raising each of them to that power, and then taking the product. And you can actually see, I worked it out here, why that actually makes sense. Notice when I just changed the order of multiplication you had four times four, or four squared, times square root of two squared, which is going to be two. So let's keep doing that. So what is this value squared? It's gonna be three squared, which is nine, times square root of two squared, which is two. Nine times two is 18. What's the square root of 17 squared? That's just going to be seventeen. Do that in blue. This is just going to be 17. What is three square roots of three squared? It's gonna be three squared, which is nine, times square root of three squared. The square root of three times the square root of three is three. So it's gonna be nine times three, or 27. And what is five squared? This is pretty straightforward. That's going to be 25. So let's order them from least to greatest. Which of them, when I square it, gives me the smallest value? Compare 32 to 12 to 18 to 17 to 27 to 25. 12 is the smallest value. So if their square is the smallest, and these are all positive numbers, then this is going to be the smallest value out of all of them. Let me write that first. Two square roots of three. So I've covered that one. Now what's next? Well, now I have this value. 17 is the next smallest square, so its square root is going to be the next square root. So it's going to be two square roots of three, then square root of 17. That is this one here. Then we go to 18. So if we look at its square root, with the numbers we were originally trying to sort, that would be three square roots of two, three square roots of two, We got that one covered. Then the next one is gonna be 25, when we look at the squares. So the next value out of our original set, the next largest one, is going to be five. So then we get to five, we've covered that one. Then the next one, let's see. We have 27 and 32 left. 27 is the next largest square, so the next largest number out of the ones we care about, is three square roots of three. So three square roots of three, we covered that one. And then we finish with-- This is the largest value, four square roots of two. Four square roots of two. And we're done! That was pretty neat. Without a calculator, we were able to sort these irrational numbers, well, not all of them are irrational, but ones that involve the square root of something that is not a perfect square.