Question

Find the multiplicative inverse of $-5-\dfrac{1}{-5}$ .

Answer 1

Hint: Think of a number which, when multiplied by $-5-\dfrac{1}{-5}$, give the product 1. Make sure that you convert your answer to the simplest form. You might have to use the identity $\dfrac{1}{-a}=-\dfrac{1}{a}$ .

Complete step by step answer:
Before moving to the question, let us talk about the multiplicative inverse.
The multiplicative inverse of a number is defined as the reciprocal of the number itself. In other words, an expression is said to be the multiplicative inverse of the other if on multiplying, their product is 1.
For example: if $xy=1$ , then, x and y are the multiplicative inverses of each other.
Now, moving to the solution to the question mentioned above.
Let the given expression be ‘m.’
$\therefore m=-5-\dfrac{1}{-5}$
Let the multiplicative inverse of m be ‘n.’
Now, according to the definition of the multiplicative inverse:
$mn=1$
$\Rightarrow n=\dfrac{1}{m}$
$\Rightarrow n=\dfrac{1}{-5-\dfrac{1}{-5}}$
Now we know that $\dfrac{1}{-a}=-\dfrac{1}{a}$ .
$n=\dfrac{1}{-5-\left( -\dfrac{1}{5} \right)}$
$\Rightarrow n=\dfrac{1}{-5+\dfrac{1}{5}}$
Now let us take the LCM of the required part to be 5. On doing so, we get
$n=\dfrac{1}{\dfrac{-25+1}{5}}$
$\Rightarrow n=\dfrac{5}{-24}$
Again, using $\dfrac{1}{-a}=-\dfrac{1}{a}$ give us:
$n=\dfrac{-5}{24}$
So, the multiplicative inverse of the complex number $\left( -5-\dfrac{1}{-5} \right)$ is $\dfrac{-5}{24}$ .

Note: The inverse, in general, refers to multiplicative inverse, but there is a possibility of additive inverse being asked. An expression is said to be the additive inverse of other if both expressions add up, giving 0 as the sum. For example: if $x+y=0$ , then, x and y are the additive inverses of each other. Also, multiplicative inverse and reciprocal are the same in case of numbers.