Meaning of the reciprocal

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 two-thirds, the reciprocal of two-thirds, if I swap the numerator and the denominator, is three-halves. If I had the fraction five-sixths, the reciprocal of that is going to be six-fifths.

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.

So, two-thirds times three-halves equals one, or five-sixths times six-fifths equals one. Another way to think about reciprocals is 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 two-thirds, one interpretation of this is saying how many two-thirds fit into one. If I take one divided by five-sixths, an interpretation of this is how many five-sixths fit into one. And we'll see that three-halves of a two-thirds fit into one, and we'll see that in a second, or that six-fifths of a five-sixths fit into one.

So, let's start with a very straightforward example. Let's say that I have the fraction one-half. So, if I have one-half, if that whole rectangle is a whole, this is one-half here.

So, if I were to ask how many one-halves fit into one, so one divided by one-half, how many one-halves fit into one? Well, I have one one-half right over here, and then I would have another one-half right over there. So, we have two one-halves, 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 one-half is indeed two over one, or if you take two over one, and if you have two one-halves, that is indeed going to be equal to one.

But now, let's work on two-thirds, things that are a little bit more nuanced. So, two-thirds, here I can shade that in. That's one-third, and then two-thirds. So, this right over here is two-thirds. Now, how many of these fit into one?

If we were to say what's one divided by two-thirds, well, we can clearly get a whole two-thirds into one, and then we can get another third, which is half of a two-thirds. So, we can have a whole two-thirds, and then half of a two-thirds, or one-and-a-half two-thirds.

So, we could say one divided by two-thirds is equal to one-and-a-half. Well, one-and-a-half is the exact same thing as three-halves. So, once again, you can see that three-halves times two-thirds is equal to one, or that three-halves of a two-thirds fit into one.

Let's do another example. If we were to think about three-halves, so three-halves would be, let's see, that's a half, that's two halves, and then this is three-halves right over here. So, let me mark all of that.

So, this whole thing right over here is three-halves. Now, how many three-halves fit into a whole? Well, you can see that you can't even fit a whole three-halves into a whole. You can only fit two of the three-halves.

So, one one-half and two halves of the three-halves. So, what you can see here is that this is two-thirds of the three-halves. So, if you say one divided by three-halves, how many three-halves can fit into one? Well, you can always fit two-thirds of a three-halves into one.

And this is interesting because the reciprocal of two-thirds is three-halves, and the reciprocal of three-halves is two-thirds.