# Question #c7863

Oct 18, 2016

To find the multiplicative inverse, convert the number to a fraction and flip it over.

#### Explanation:

Make fraction

$0.1 = \frac{1}{10}$

Flip

$\frac{1}{10} \rightarrow \frac{10}{1}$

Simplify

$\frac{10}{1} = 10$

Oct 18, 2016

$10$

#### Explanation:

First, note that $0.1$ just has a $1$ in the ${10}^{\text{th}} s$ position. As a fraction, then, we can represent it as $0.1 = \frac{1}{10}$

Next, remember that the multiplicative inverse of any nonzero number is the number whose product with the original number is $1$. So, for any nonzero number $x$, the multiplicative inverse of $x$ is $\frac{1}{x}$. This is because $x \times \frac{1}{x} = \frac{x}{x} = 1$

So, if we want the multiplicative inverse of $0.1$, we look at $\frac{1}{0.1}$ and get

$\frac{1}{0.1} = \frac{1}{\frac{1}{10}}$

$= \frac{1}{\frac{1}{10}} \times \frac{10}{10}$

(multiplying the numerator and denominator by the same number doesn't change our value)

$= \frac{10}{\left(\frac{1}{10}\right) \times 10}$

$= \frac{10}{1}$

$= 10$

So the multiplicative inverse of $0.1$ is $10$.

As a side note, it's a useful trick to remember that $\frac{1}{\frac{1}{x}} = x$ for any nonzero $x$. Multiplying by $\frac{x}{x} = 1$ shows it to be true, as done above.