Lesson 15: Infinite decimal expansions
In this video, I want to talk about how we can convert repeating decimals into fractions. So let's give ourselves a repeating decimal. So let's say I had the repeating decimal 0.7. And sometimes it'll be written like that, which just means that the 7 keeps on repeating. So this is the same thing as 0.7777 and I could just keep going on and on and on forever with those 7s. So the trick to converting these things into fractions is to essentially set this equal to a variable. And we'll just show it, do it step-by-step. So let me set this equal to a variable. Let me call this x. So x is equal to 0.7, and then the 7 repeats on and on forever. Now what would 10x be? Well, let's think about this. 10x. 10x would just be 10 times this. And we could even think of it right over here. It would be, if we multiplied this times 10, you'd be moving the decimal 1 over to the right, it would be 7.777, on and on and on and on forever. Or you could say it is 7.7 repeating. Now this is the trick here. So let me make these equal to each other. So we know what x is. x is this, just 0.777 repeating forever. 10x is this. And this is another repeating thing. Now the way that we can get rid of the repeating decimals is if we subtract x from 10x. Right? Because x has all these 0.7777. If you subtract that from 7.77777, then you're just going to be left with 7. So let's do that. So let me rewrite it here just so it's a little bit neater. 10x is equal to 7.7 repeating, which is equal to 7.777 on and on forever. And we established earlier that x is equal to 0.7 repeating, which is equal to 0.777 on and on and on forever. Now what happens if you subtract x from 10x? So we're going to subtract the yellow from the green. Well, 10 of something minus 1 of something is just going to be 9 of that something. And then that's going to be equal to, what's 7.7777 repeating minus 0.77777 going on and on forever repeating? Well it's just going to be 7. These parts are going to cancel out. You're just left with 7. Or you could say these two parts cancel out. You're just left with 7. And so you get 9x is equal to 7. To solve for x, you just divide both sides by 9. Let's divide both sides by nine. I could do all three sides, although these are really saying the same thing. And you get x is equal to 7/9. Let's do another one. I'll leave this one here so you can refer to it. So let's say I have the number 1.2 repeating. So this is the same thing as 1.2222 on and on and on. Whatever the bar is on top of, that's the part that repeats on and on forever. So just like we did over here, let's set this equal to x. And then let's say 10x. Let's multiply this by 10. So 10x is equal to, it would be 12.2 repeating, which is the same thing as 12.222 on and on and on and on. And then we can subtract x from 10x. And you don't have to rewrite it, but I'll rewrite it here just so we don't get confused. So we have x is equal to 1.2 repeating. And so if we subtract x from 10x, what do we get? On the left-hand side, we get 10x minus x is 9x. And this is going to be equal to, well, the 2 repeating parts cancel out. This cancels with that. If 2 repeating minus 2 repeating, that's just a bunch of 0. So it's 12 minus 1 is 11. And you have 9x is equal to 11. Divide both sides by 9. You get x is equal to 11/9.