Reciprocidade na divisão

Turn that Frown Upside Down

A reciprocal of a number is $1$ divided by the number. Before we start complaining about having to do division, there's a really easy way to find the reciprocal.

The reciprocal of a number is just the numerator and denominator switched! That means for a fraction $\frac{a}{b}$, the reciprocal is

$$\frac{b}{a}$$

Okay, but what about numbers that don't have a denominator, like $7$. Oh please, math has thought of this. Just put the number over $1$. If we divide a number by $1$, we get the same number. That is, putting a $1$ in the denominator doesn't change the value.

$$7\rightarrow\frac71$$

Now, we flip the numerators and denominators. $$\frac71\rightarrow\frac17$$

It's worth noting that this process works for proper and improper fractions. The type of fraction doesn't make a difference here.

We can also find reciprocals of mixed numbers. To get those, we have to convert to an improper fraction first.

As for decimal numbers, we can convert these to fractions first as well. In finding the reciprocal of a decimal, the final answer may be improper or proper. However, if we find the reciprocal of a mixed number, the fraction at the end will be proper.

I think you're ready for the property. It's not too difficult to remember.

Any non-zero number multiplied by its reciprocal will always equal 1. $$\frac7{10}\times\frac{10}{7}=1$$

Division and Reciprocals

A number's reciprocal is the upside down version of that number when it's written as a fraction. Another name for reciprocal is multiplicative inverse.

Reciprocals are really helpful when it comes to dividing fractions. We can use reciprocals to turn fraction division into fraction multiplication.

So, before we try out some of these division problems, let's review how to find a number's reciprocal.

Finding the Reciprocal of a Fraction

This is easy business. All you need to do to find the reciprocal of a fraction is to flip the fraction over.

Take a look at the table below for some examples:

Notice how we just flip the fraction over to find its reciprocal. Multiply any number by its reciprocal, and you'll get $1$.

Let's practice.

Finding the Reciprocal of a Whole Number

To do this, it helps to imagine regular whole numbers (e.g., $1$, $2$, $3$, etc.) as fractions themselves.

Let's try that.

Dividing Fractions Using Reciprocals

When you're presented with a problem where you need to divide one fraction by another, using a reciprocal will come in handy.

Dividing one fraction by another is EQUIVALENT to multiplying one fraction by the other's reciprocal.

So, let's rewrite the problem.

Now, we can multiply the two fractions pretty easily, reduce the product, and then we're done!

(Video) Art of Problem Solving: What's a Reciprocal?

by Art of Problem Solving

This video by Richard Rusczyk explains why we have reciprocals of numbers.

$\large{\text{Summary}}$

In his first example, he shows how we have negative numbers in order to add something to a positive number to get $0$.

Specifically, negative numbers allow us to have the inverse property of addition. Take any positive number, and we can pick a different number, add them together, and get $0$.

Then, we want a similar thing for multiplication: for any number $x$, we want some number $y$ where $x \cdot y = 1$. We call this $y$ value $\frac{1}{x}$.

He uses $8$ as an example. So for $8$, if we want to find its reciprocal, we just do $1$ over $8$, which is simply $\frac{1}{8}$. So we know $8 \cdot \frac{1}{8} = 1$.

The ability to find reciprocals is called the inverse property of multiplication.

One important thing is that if you have any fraction $\frac{a}{b}$, then its reciprocal is $\frac{b}{a}$. We can see this with the example above since $8 = \frac{8}{1}$, so its reciprocal is $\frac{1}{8}$.

Another important thing to remember, which Richard mentions in the video, is that $0$ does not have a reciprocal. This is because $0$ times anything is $0$, so we can't have any number $x$ where $0 \cdot x = 1$. Another reason is because if we wanted the reciprocal of $0$, we would have to divide $1$ by $0$, but we cannot divide by $0$ ever, so there is no reciprocal.