Question

What is the sum of additive inverse and multiplicative inverse of $ - 7$.

Answer 1

Hint: First let us know, what are additive inverse and multiplicative inverse. So, additive inverse refers to the number on adding with which the given number becomes $0$. And, the multiplicative number refers to the number on multiplying with which the given number becomes $1$. So, to solve this question, we have to find the additive inverse and multiplicative inverse of $ - 7$ first, and then just add the two numbers in order to get the required answer.

Complete step-by-step solution:
So, we need to first find the additive inverse of $ - 7$.
We know, additive inverse refers to the number on adding with which the given number becomes $0$
So, let the additive inverse of $ - 7$ be $x$.
So, according to the definition of additive inverse,
$ - 7 + x = 0$
Now, adding $7$ on both sides of the equation, we get,
$ \Rightarrow x = 7$
Therefore, the additive inverse of $ - 7$ is $7$.
Now, we are to find the multiplicative inverse of $ - 7$.
We know, the multiplicative number refers to the number on multiplying with which the given number becomes $1$.
So, let the multiplicative inverse of $ - 7$ be $y$.
So, according to the definition of multiplicative inverse,
$ - 7 \times y = 1$
Now, dividing both sides of the equation by $ - 7$, we get,
$ \Rightarrow y = - \dfrac{1}{7}$
Therefore, the multiplicative inverse of $ - 7$ is $ - \dfrac{1}{7}$.
Therefore, we need to find the sum of the additive inverse and multiplicative inverse of $ - 7$.
Therefore, the sum of the additive inverse and multiplicative inverse of $ - 7$$ = x + y$
$ = 7 + \left( { - \dfrac{1}{7}} \right)$
$ = 7 - \dfrac{1}{7}$
Taking LCM, we get,
$ = \dfrac{{49 - 1}}{7}$
$ = \dfrac{{48}}{7}$
Therefore, the sum of the additive inverse and multiplicative inverse of $ - 7$ is $\dfrac{{48}}{7}$.

Note: There are simpler ways to find the additive inverse and multiplicative inverse of a number. The additive of a number is simply the number with the opposite symbol of that number. Like, the additive inverse of $ - 7$ is $7$. And the multiplicative inverse of a number is simply the reciprocal of that number. Like, the multiplicative inverse of $ - 7$ is $\dfrac{1}{{ - 7}}$.