What Is a Reciprocal?

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 0\hspace{0.2em} 0 \hspace{0.2em}) is 1\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 4/9\hspace{0.2em} 4/9 \hspace{0.2em} is 9/4\hspace{0.2em} 9/4 \hspace{0.2em}. That of 1/2\hspace{0.2em} 1/2 \hspace{0.2em} is 2/1\hspace{0.2em} 2/1 \hspace{0.2em} or simply 2\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 n\hspace{0.2em} n \hspace{0.2em}, the reciprocal would be 1/n\hspace{0.2em} 1/n \hspace{0.2em}.

For example, the reciprocal of 2\hspace{0.2em} 2 \hspace{0.2em} is 1/2\hspace{0.2em} 1/2 \hspace{0.2em}. And that of 7\hspace{0.2em} -7 \hspace{0.2em} would be 1/7\hspace{0.2em} - 1/7 \hspace{0.2em} (or 17\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, n\hspace{0.2em} n \hspace{0.2em} is 1/n\hspace{0.2em} 1/n \hspace{0.2em}. But as I mentioned, this isn't true for 0\hspace{0.2em} 0 \hspace{0.2em}. Here's why.

Division by 0\hspace{0.2em} 0 \hspace{0.2em} is not defined. That means 1/0\hspace{0.2em} 1/0 \hspace{0.2em} is not defined. And hence, 0\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 215\hspace{0.2em} 2 \frac{1}{5} \hspace{0.2em}.

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

215=5×2+15=115\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 215\hspace{0.2em} 2 \frac{1}{5} \hspace{0.2em} would be 5/11\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 1\hspace{0.2em} 1 \hspace{0.2em}.

And for obvious reasons.

n1n=1\cancel{n} \cdot \frac{1}{\cancel{n}} = 1

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

12÷58=12×85\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.

an=(1a)na^{-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.