Meaning of the reciprocal

0:00- [Instructor] Let's talk a little bit about reciprocals. Now, when you first learn reciprocals some folks will immediately tell you, Hey, just swap the numerator and the denominator. So for example, if I have the fraction 2/3, the reciprocal of 2/3, if I swap the numerator and the denominator, is 3/2.

0:17If I had the fraction 5/6, the reciprocal of that is going to be six over five. And that's all well and good, but what does this actually mean? Well, one interpretation of a reciprocal is it's the number that when you multiply it by the original number, you get one.

0:36So 2/3 times 3/2 we'll see is equal to one or 5/6 times 6/5 is equal to one. Another way to think about reciprocals are, how many of that number or how many of that fraction fit into one? So if I were to take one and I divide it by 2/3, one interpretation of this is saying how many 2/3 fit into one.

1:00If I take one divided by 5/6, an interpretation of this is how many 5/6 fit into one. And we'll see that 3/2 of a 2/3 fit into one. And we'll see that in a second. Or that 6/5 of a 5/6 fit into one. So let's start with a very straightforward example. Let's say that I have the fraction 1/2.

1:27So if I have 1/2. If that whole rectangle is a whole, this is 1/2 here. So if I were to ask how many one halves fit into one, so one divided by 1/2. How many one halves fit into one? Well, I have one 1/2 right it over here. And then I would have another 1/2 right over there.

1:47So we have two 1/2. So this is equal to two. Now you might be saying, wait, two doesn't look like I just swapped the numerator and the denominator but you have to realize that two is the same thing as two wholes. So the reciprocal of 1/2 is indeed two over one.

2:05Or if you take two over one, or if you have two 1/2, that is indeed going to be equal to one. But now let's work on 2/3, things that are a little bit more nuanced. So 2/3 here, I can shade that in. That's 1/3 and then 2/3. So this right over here is 2/3. Now how many of these fit into one?

2:31If we were to say what's one divided by 2/3? Well, we can clearly get a whole 2/3 into one and then we can get another third, which is half of a 2/3. So we can have a whole 2/3 and then half of a 2/3, or one and a half 2/3. So we could say one divided by 2/3 is equal to one and a half.

2:55Well, one and a half is the exact same thing as three halves. So once again, you can see that 3/2 times 2/3 is equal to one or that 3/2 of a 2/3 fit into one. Let's do another example. If we were to think about 3/2. So 3/2 would be, let's see that's a half, that's two halves, and then this is three halves right over here.

3:26So let me mark all of that. So this whole thing right over here is 3/2. Now how many 3/2 fit into a whole? Well, you can see that you can't even fit a whole 3/2 into a whole. You can only fit two of the three halves. So one half, two halves of the three halves.

3:49So what you can see here is that this is 2/3 of the 3/2. So if you say one divided by 3/2, how many 3/2 can fit into one? Well, you can only fit 2/3 of a 3/2 into one. And this is interesting because the reciprocal of 2/3 is 3/2, and the reciprocal of 3/2 is 2/3.