Fractions as division
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We've already seen that a fraction like 2/9 can be interpreted as 2 divided by 9. Let me do the 2 in the same color. That if we have a fraction, it could be interpreted as the numerator divided by the denominator. And this leads to all sorts of interesting conclusions, some of which we've already seen, and some of which are a little bit new. So, for example, if I had the fraction 7/7, this can be interpreted now as 7 divided by 7, as our numerator divided by our denominator. And 7 divided by 7 is of course equal to 1. And this is consistent with what we've already seen. 7/7 would get us to a whole, and a whole is the exact same thing as one. But we could do things a little bit more interesting as well. We could take something like 18/6 and realize, wow. This is the same thing as 18 divided by 6, which we know is equal to 3. And we should do a little reality check. Does this make sense, that 18/6 should be equal to 3? Well, we could rewrite it. We could rewrite 18/6. Let me make the sixths that same orange color. That's going to be the same thing. 18 is 6 plus 6 plus 6. And then all of that over 6, and then that's the same thing. That's the same thing as 6/6 plus 6/6 plus 6/6. And I could make this right over here in orange. And we've already seen, or we've seen many, many videos ago, that 6/6, just like 7/7, these are each equal to a whole. These are each equal to 1. And we can now view this as 6 divided by 6, which is the same thing as 1. So this is 1 plus 1 plus 1, which is, of course, equal to 3. But this starts to raise an interesting question. This all worked out just fine because 18 is a multiple of 6. 6 divides evenly into 18. But what happens if we start having fractions where the denominator does not divide evenly into the numerator? Let's say we have a fraction like 23 over 6. Well, we know that we can interpret this as 23 divided by 6. And if we actually divide 23 by 6-- let's do that. So we divide 6 into 23. We know 6 goes into 23 three times. 3 times 6 is 18. And then when you subtract, you end up with a remainder of 5. So we might say, hey, 23 divided by 6 is equal to 5 remainder 3. But that's not that satisfying. What do I do with this remainder? This really isn't a number here. This is just saying that we're going five times, and then we have a little bit left over. What we can do now is manipulate this a little bit so that we can realize that this is a number, and in particular, a mixed number. So for example, we could start with a 23 over 6, and we could divide it into-- or we could decompose the numerator into one part that is divisible by 6, evenly divisible by 6, and the remainder. So for example, 23 over 6 we can rewrite as 18 plus 5 over 6. Notice I decomposed the 23 into one part that is a multiple of 6, and it's the largest multiple that essentially fits into 23, or that is less than or equal to 23, and then the remainder. When you divide 6 into 23, you get a remainder of 5. You could view it as, I divided it into the remainder and everything else. And the reason why this is interesting is because we know that this is going to be equal to 18 over 6 plus 5 over 6. Well, we already know that 18/6 is the same thing as 18 divided by 6, or 3. So this is the same thing as 3. So we know that 23 over 6, which is the same thing as 18 plus 5 over 6, is the same thing as 3 plus 5/6. Or, if we want to write it as a mixed number, we could write it as 3 and 5/6.