The reciprocal of any number (except $\hspace{0.2em} 0 \hspace{0.2em}$) is $\hspace{0.2em} 1 \hspace{0.2em}$ divided by that number.

Most often, when we talk of reciprocals, it's in the context of fractions. But reciprocals are not limited to fractions.

Yeah, that's all there is to it. That said, let's take a closer look.

The reciprocal of a fraction is the fraction turned upside down (i.e. its top and bottom numbers interchanged.)

For example, the reciprocal of $\hspace{0.2em} 4/9 \hspace{0.2em}$ is $\hspace{0.2em} 9/4 \hspace{0.2em}$. That of $\hspace{0.2em} 1/2 \hspace{0.2em}$ is $\hspace{0.2em} 2/1 \hspace{0.2em}$ or simply $\hspace{0.2em} 2 \hspace{0.2em}$.

That said, the concept of reciprocals is not limited to fractions.

The reciprocal of a whole number or integer is simply one over that number. So for $\hspace{0.2em} n \hspace{0.2em}$, the reciprocal would be $\hspace{0.2em} 1/n \hspace{0.2em}$.

For example, the reciprocal of $\hspace{0.2em} 2 \hspace{0.2em}$ is $\hspace{0.2em} 1/2 \hspace{0.2em}$. And that of $\hspace{0.2em} -7 \hspace{0.2em}$ would be $\hspace{0.2em} - 1/7 \hspace{0.2em}$ (or $\hspace{0.2em} \frac{1}{-7} \hspace{0.2em}$, the same thing).

We just learned that the reciprocal of a whole number, $\hspace{0.2em} n \hspace{0.2em}$ is $\hspace{0.2em} 1/n \hspace{0.2em}$. But as I mentioned, this isn't true for $\hspace{0.2em} 0 \hspace{0.2em}$. Here's why.

Division by $\hspace{0.2em} 0 \hspace{0.2em}$ is not defined. That means $\hspace{0.2em} 1/0 \hspace{0.2em}$ is not defined. And hence, $\hspace{0.2em} 0 \hspace{0.2em}$ doesn't have a reciprocal.

To get the reciprocal of a mixed number, convert the mixed number into a simple fraction and then take the reciprocal of the fraction.

As an example, let's get the reciprocal of $\hspace{0.2em} 2 \frac{1}{5} \hspace{0.2em}$.

To start with, we turn it into a simple fraction. So —

$\begin{align*} 2 \frac{1}{5} \hspace{0.25em} &= \hspace{0.25em} \frac{5 \times 2 + 1}{5} \\[1em] &= \hspace{0.25em} \frac{11}{5} \end{align*}$

And now we turn the fraction upside down. So the reciprocal of $\hspace{0.2em} 2 \frac{1}{5} \hspace{0.2em}$ would be $\hspace{0.2em} 5/11 \hspace{0.2em}$.

Reciprocal of a number is its multiplicative inverse. In other words, if you multiply a number with its reciprocal, you end up with $\hspace{0.2em} 1 \hspace{0.2em}$.

And for obvious reasons.

$\cancel{n} \cdot \frac{1}{\cancel{n}} = 1$

That is why to divide by a fraction, we multiply by it reciprocal. For example,

$\frac{1}{2} \div \frac{ {\color{Red} 5} }{ {\color{Teal} 8} } \hspace{0.25em} = \hspace{0.25em} \frac{1}{2} \times \frac{ {\color{Teal} 8} }{ {\color{Red} 5} }$

Another related idea is that we can turn a negative exponent into positive (and the other way round) by taking the reciprocal of the base.

$a^{-n} \hspace{0.25em} = \hspace{0.25em} \left ( \frac{1}{a} \right )^n$

I hope that gives you some idea of why reciprocals are important.

And with that, we come to the end of this elementary tutorial on absolute value. Until next time.

We use cookies to provide and improve our services. By using the site you agree to our use of cookies. Learn more