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

\[\begin{align} \displaystyle \sum_{n=1}^{10} n & = \dfrac {10 × 11}{2}\\ & = 55. \end{align}\]

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