Question

How do you find the additive and multiplicative inverse of $- \dfrac{{11}}{5}$?

Answer 1

We have to find the additive and multiplicative inverse of $- \dfrac{{11}}{5}$. For this, we will assume that $x$ is the multiplicative inverse of $- \dfrac{{11}}{5}$ and use the concept that if the sum of two numbers is equal to zero then these numbers are called the additive inverse. Then we will assume $y$ is the multiplicative inverse of $- \dfrac{{11}}{5}$ and use the concept that if the product of two numbers is equal to one then these numbers are called the multiplicative inverse.

Complete step by step solution:
We have to find the additive and multiplicative inverse of $- \dfrac{{11}}{5}$.
As we know, two numbers are called additive inverse when their sum is equal to $0$.
Let us assume that the additive inverse of $- \dfrac{{11}}{5}$ be $x$. Then we will get,
$\Rightarrow - \dfrac{{11}}{5} + x = 0$
Adding $\dfrac{{11}}{5}$ both the sides, we get
$\Rightarrow x = \dfrac{{11}}{5}$
Hence, we get the additive inverse of $- \dfrac{{11}}{5}$ is $\dfrac{{11}}{5}$.
Now, two numbers are called the multiplicative inverse if the product of two numbers is equal to $1$.
So, let us assume that the multiplicative inverse of $- \dfrac{{11}}{5}$ is $y$. Then we get,
$\Rightarrow - \dfrac{{11}}{5} \times y = 1$
On simplifying the above obtained equation, we get
$\Rightarrow y = - \dfrac{5}{{11}}$
Hence, we get the multiplicative inverse of $- \dfrac{{11}}{5}$ is $- \dfrac{5}{{11}}$.
Therefore, the additive and multiplicative inverse of $- \dfrac{{11}}{5}$ is $\dfrac{{11}}{5}$ and $- \dfrac{5}{{11}}$ respectively.

Note:
As an additive inverse the sum of two numbers is equal to zero, so $0$ is known as the additive identity. Similarly, in the multiplicative inverse the product of two numbers is equal to one, so $1$ is known as the multiplicative identity. Multiplicative inverse of a number is also called the reciprocal of that number.