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Worked example: classifying numbers

How do repeating decimals fit into the number sets? We'll explore if a number like 3.4028 repeating is a real number, a rational number, or both. We'll use multiplication and subtraction, plus a bit of algebra, to convince ourselves of whether or not there is a fraction representation of the number. Created by Sal Khan and Monterey Institute for Technology and Education.

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Video transcript

What number sets does the number 3.4028 repeating belong to? And before even answering the question, let's just think about what this represents. And especially what this line on top means. So this line on top means that the 28 just keep repeating forever. So I could express this number as 3.4028, but the 28 just keep repeating. Just keep repeating on and on and on forever. I could just keep writing them forever and ever. And obviously, it's just easier to write this line over the 28 to say that it repeats forever. Now let's think about what number sets it belongs to. Well, the broadest number set we've dealt with so far is the real numbers. And this definitely belongs to the real numbers. The real numbers is essentially the entire number line that we're used to using. And 3.4028 repeating sits someplace over here. If this is negative 1, this is 0, 1, 2, 3, 4. 3.4028 is a little bit more than 3.4, a little bit less than 3.41. It would sit right over there. So it definitely sits on the number line. It's a real number. So it definitely is real. It definitely is a real number. But the not so obvious question is whether it is a rational number. Remember, a rational number is one that can be expressed as a rational expression or as a fraction. If I were to tell you that p is rational, that means that p can be expressed as the ratio of two integers. That means that p can be expressed as the ratio of two integers, m/n. So the question is, can I express this as the ratio of two integers? Or another way to think of it, can I express this as a fraction? And to do that, let's actually express it as a fraction. Let's define x as being equal to this number. So x is equal to 3.4028 repeating. Let's think about what 10,000x is. And the only reason why I want 10,000x is because I want to move the decimal point all the way to the right over here. So 10,000x. What is that going to be equal to? Well every time you multiply by a power of 10, you shift the decimal one to the right. 10,000 is 10 to the fourth power. So it's like shifting the decimal over to the right four spaces. 1, 2, 3, 4. So it'll be 34,028. But these 28's just keep repeating. So you'll still have the 28's go on and on, and on and on, and on after that. They just all got shifted to the left of the decimal point by five spaces. You can view it that way. That makes sense. It's nearly 3 and 1/2. If you multiply by 10,000, you get almost 35,000. So that's 10,000x. Now, let's also think about 100x. And my whole exercise here is I want to get two numbers that, when I subtract them and they're in terms of x, the repeating part disappears. And then we can just treat them as traditional numbers. So let's think about what 100x is. 100x. That moves this decimal point. Remember, the decimal point was here originally. It moves it over to the right two spaces. So 100x would be 300-- Let me write it like this. It would be 340.28 repeating. We could have put the 28 repeating here, but it wouldn't have made as much sense. You always want to write it after the decimal point. So we have to write 28 again to show that it is repeating. Now something interesting is going on. These two numbers, they're just multiples of x. And if I subtract the bottom one from the top one, what's going to happen? Well the repeating part is going to disappear. So let's do that. Let's do that on both sides of this equation. Let's do it. So on the left-hand side of this equation, 10,000x minus 100x is going to be 9,900x. And on the right-hand side, let's see-- The decimal part will cancel out. And we just have to figure out what 34,028 minus 340 is. So let's just figure this out. 8 is larger than 0, so we won't have to do any regrouping there. 2 is less than 4. So we will have to do some regrouping, but we can't borrow yet because we have a 0 over there. And 0 is less than 3, so we have to do some regrouping there or some borrowing. So let's borrow from the 4 first. So if we borrow from the 4, this becomes a 3 and then this becomes a 10. And then the 2 can now borrow from the 10. This becomes a 9 and this becomes a 12. And now we can do the subtraction. 8 minus 0 is 8. 12 minus 4 is 8. 9 minus 3 is 6. 3 minus nothing is 3. 3 minus nothing is 3. So 9,900x is equal to 33,688. We just subtracted 340 from this up here. So we get 33,688. Now, if we want to solve for x, we just divide both sides by 9,900. Divide the left by 9,900. Divide the right by 9,900. And then, what are we left with? We're left with x is equal to 33,688 over 9,900. Now what's the big deal about this? Well, x was this number. x was this number that we started off with, this number that just kept on repeating. And by doing a little bit of algebraic manipulation and subtracting one multiple of it from another, we're able to express that same exact x as a fraction. Now this isn't in simplest terms. I mean they're both definitely divisible by 2 and it looks like by 4. So you could put this in lowest common form, but we don't care about that. All we care about is the fact that we were able to represent x, we were able to represent this number, as a fraction. As the ratio of two integers. So the number is also rational. It is also rational. And this technique we did, it doesn't only apply to this number. Any time you have a number that has repeating digits, you could do this. So in general, repeating digits are rational. The ones that are irrational are the ones that never, ever, ever repeat, like pi. And so the other things, I think it's pretty obvious, this isn't an integer. The integers are the whole numbers that we're dealing with. So this is someplace in between the integers. It's not a natural number or a whole number, which depending on the context are viewed as subsets of integers. So it's definitely none of those. So it is real and it is rational. That's all we can say about it.