yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Meaning of the reciprocal


3m read
·Nov 10, 2024

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.

More Articles

View All
A Belief in Black Magic | Uncensored with Michael Ware
NARRATOR: Finally, the [inaudible] leader agrees to talk. [non-english speech]. He is speaking on behalf of everyone. He says, “Yeah, I’m a pastor. I’m a leader. I’m a Christian. I’m a father. But this belief is part of my life. And I can’t just take it o…
Estimating limits from tables | Limits and continuity | AP Calculus AB | Khan Academy
The function g is defined over the real numbers. This table gives select values of g. What is a reasonable estimate for the limit as x approaches 5 of g of x? So pause this video, look at this table. It gives us the x values as we approach five from value…
15 Reasons Persuasive People Always Get What They Want
No matter how hard you work at something, if you don’t know how to persuade people, you’re never going to get what you want. Hard work falls flat without the driving force of persuasion. Good persuasion skills beat hard work any day. That’s why a charisma…
How We’re Fooled By Statistics
Which is most effective for helping people learn: punishment or reward? Well, consider the case of Israeli fighter pilot training, because instructors there found that negative feedback was far superior to positive feedback. If a cadet performed a particu…
THE FEDERAL RESERVE JUST FLIPPED | Major Changes Explained
What’s up guys, it’s Graham here. So this is big. After more than a year of patiently waiting, the Federal Reserve has just officially paused their rate hikes for the first time since March of 2022, marking the beginning of a brand new market cycle that’s…
The actual reason why you procrastinate and how to fix it
It’s 6:00 p.m. You just got back home, and you’ve got a task that has been lingering in your mind, waiting to be checked off your list. It could be a project for work or school, house chores that can’t be ignored any longer, or maybe it’s about spending q…