Reciprocals

The reciprocal of a non-zero number allows us to express division by a number as multiplication by its reciprocal.

For a non-zero number $x,$ the reciprocal of $x$ is $\frac{1}{x}$. $_\square$

Using the reciprocal, dividing a number $y$ by a number $x$ can be expressed as the product of $y$ with the reciprocal of $x$:

$$\frac{y}{x} = y \left( \frac{1}{x} \right).$$

Here are a few properties of reciprocals:

1. Since $x \cdot \frac{1}{x} = 1$ for every non-zero number $x$, this shows multiplying any number by its reciprocal gives the value $1$.
2. Since $$\frac{1}{\left( \frac{1}{x} \right)} = x$$ for every non-zero number $x$, this shows the reciprocal of the reciprocal of a number gives the number itself.

What is the reciprocal of $x = 5$?

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The reciprocal of $x=5$ is $\frac{1}{x} = \frac{1}{5}.$ $_\square$

What is the reciprocal of $x = \frac{7}{9}?$

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The reciprocal of $x = \frac{7}{9}$ is

$$\frac{1}{x} = \frac{1}{ \left( \frac{7}{9}\right)} = \frac{9}{7}. \ _\square$$

What is the product of $\pi$ and the reciprocal of $\pi$?

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By Property (1) above, $x \cdot \frac{1}{x} = 1$ for every non-zero number $x$, so

$$\pi \cdot \frac{1}{\pi} = 1.$$

Since multiplying a number by its reciprocal always gives the value $1$, the reciprocal of a number is also called the multiplicative inverse. $_\square$

What is the reciprocal of $\displaystyle \sum_{n=1}^{10} n$?

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We have

$$\begin{aligned} \displaystyle \sum_{n=1}^{10} n & = \dfrac {10 × 11}{2}\\ & = 55. \end{aligned}$$

So, the reciprocal is $\dfrac {1}{55}.$ $_\square$