Question

How do you find the additive inverse and multiplicative inverse of $ - \dfrac{8}{3}$ ?

Answer 1

Hint: In the given question, we have to find the additive inverse of the given number. To find the additive inverse of a number, we must know the meaning of the term additive inverse. The additive inverse of a number $x$ is a number which when added to $x$ gives zero as result. Similarly, the multiplicative inverse of $x$ is a number which when multiplied to $x$ gives unity as a result. So, to find the additive inverse and multiplicative inverses, we form the required mathematical equations and solve using a method of transposition.

Complete step by step answer:
The number given to us in the problem is $ - \dfrac{8}{3}$. Let us assume the additive inverse of the number to be a variable, let's say $x$. Then, the sum of $ - \dfrac{8}{3}$ and $x$ should be equal to zero. So, we get,
$x + \left( { - \dfrac{8}{3}} \right) = 0$

Now, we use the transposition method to find the value of variable x in the equation above. So, shifting the constant to the right side of the equation, we get,
$ \Rightarrow x = - \left( { - \dfrac{8}{3}} \right)$
Now, opening the brackets, we know that multiplication of two negative signs yields a positive sign. Hence, we get,
$ \Rightarrow x = \dfrac{8}{3}$
Hence, the additive inverse of the number $\left( { - \dfrac{8}{3}} \right)$ is $\dfrac{8}{3}$.

Now, let us consider the multiplicative inverse of the number as $y$. Then the product of $ - \dfrac{8}{3}$ and y should be equal to unity. So, we get,
$ \Rightarrow \left( { - \dfrac{8}{3}} \right) \times y = 1$
Isolating y and shifting other terms to the right side of the equation. So, dividing both sides by $\left( { - \dfrac{8}{3}} \right)$, we get,
$ \Rightarrow y = \dfrac{1}{{\left( { - \dfrac{8}{3}} \right)}}$
Simplifying for the value of y,
$ \therefore y = - \dfrac{3}{8}$

Therefore, the multiplicative inverse of the number given to us in the problem itself, $ - \dfrac{8}{3}$ is obtained as $ - \dfrac{3}{8}$.

Note: We can see that the additive inverse of the number is opposite in sign when compared to the original number. Thus, we can notice that the additive inverse of a number is actually the same as the negation of the number. Similarly, we can also notice that the multiplicative inverse of a number is reciprocal of the number. We can find additive and multiplicative inverses through these techniques.