Approximating irrational numbers
- [Voiceover] What I want to do in this video is get a little bit of experience, see a few examples of trying to roughly estimate the square root of non-perfect squares. So let's say that I had, if I wanted to estimate the square root of 32. And in particular, I'm just curious, between what two integers will this square root lie? Well one way to think about it is 32 is in between what perfect squares? We see 32 is, actually let me make sure I have some space for future examples. So 32, what's the perfect square below 32? So the greatest perfect square below 32 is 25. 32 is greater than 25. That's five squared. So maybe I should write it this way. So five squared is less than 32 and then 32, what's the next perfect square after 32? Well 32 is less than 36. So we could say 32 is less than six squared. So if you were to take the square root of all of these sides right over here, we could say that instead of here we have all of the values squared, but instead, if we took the square root, we could say five is going to be less than the square root of 32, which is less than, which is less than six. Notice, to go from here to here, to go from here to here, and here to here, all we did is we squared things, we raised everything to the second power. But the inequality should still hold. So the square root of 32 should be between five and six. It's going to be five point something. Let's do another example. Let's say we wanted to estimate, we want to say between what two integers is the square root of 55? Well we can do the same idea. Let's square it. So if we square the square root of 55, we're just gonna get to 55. We're just going to get, let me do that in the same color, 55. So okay, 55 is between which two perfect squares? So the perfect square that is below 55, or I could say the greatest perfect square that is less than 55. Let's see, six squared is 36 and seven squared is 49, eight squared is 64. So it would be 49. I could write that as seven squared. Let me write that, that is the same thing as seven squared. And what's the next perfect square above it? Well we just figured it out. Seven squared is 49, eight squared is larger than 55, it's 64. So this is going to be less than 64, which is eight squared. And of course 55, just to make it clear what's going on. 55 is the square root of 55 squared. That's kind of by definition, it's going to be the square root of 55 squared. And so the square root of 55 is going to be between what? It's going to be between seven and eight. So seven is less than the square root of 55, which is less than eight. So once again, this is just an interesting way to think about, what would you, if someone said the square root of 55 and at first you're like, "Oh, uh, I don't know what that is. "I don't have a calculator," et cetera et cetera. You're like, "Oh wait, wait, that's going to be between "49 and 64, so it's going to be seven point something." It's going to be seven point something. And you can even get a rough estimate of seven point what based on how far away it is from 49 and 64. You can begin to approximate things. Let's do one more example. Let's say we wanted to figure out where does the square root of 123 lie? And like always, I encourage you to pause the video and try to think about it yourself. Between what two integers does this lie? Well, if we were to square it, you get to 123. And what's the perfect square that is the greatest perfect square less than 123? Let's see, 10 squared is 100. 11 squared is 121. 12 squared is 144. So 11 squared. So 123, so we could write 121 is less than 123, which is less than 144, that's 12 squared. So if we take the square roots we could write that 11 is less than the square root of 123, which is less than 144. So once again, what's the square root of 123? It's going to be 11 point something. And in fact, it's going to be closer to 11 than it's going to be to 12. 123 is a lot closer to 121 than it is to 144. So it might be, I don't know, 11.1, something like that. I don't know if that's exactly right, we would have to check that on the calculator. But hopefully this gives you, oops I, that actually will be less than 144. But if we want to think about what consecutive integers is that be between, it's going to be a 12 right over there. Almost made a... Well anyway, you get the idea. Hopefully you enjoyed that.