What Is a Reciprocal?

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

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.

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

Reciprocal of a Fraction

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.

Reciprocal of a Whole Number

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).

Reciprocal of 0

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.

Reciprocal of a Mixed Number

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 $\hspace{0.2em} 5/11 \hspace{0.2em}$ .

What Makes Reciprocals Important?

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.