Question

Identify the additive inverse and multiplicative inverse for each number

11. $\frac{1}{3}$

Short answer

Additive inverse of $\frac{1}{3}$ is $-\frac{1}{3}$ and multiplicative inverse of $\frac{1}{3}$ is 3

Step-by-step solution

- Define additive inverse

Let ‘a’ be a real number such that

$a+\left(-a\right)=0=\left(-a\right)+a$

Then $‘-a’$ is the additive inverse of ‘a’

- Additive inverse of 13

Substitute $\frac{1}{3}$for $a$

$a=\frac{1}{3}$

Multiply both sides by $-1$

$\begin{array}{c}\left(-1\right)\times a=\left(-1\right)\times \left(\frac{1}{3}\right)\\ -a=-\frac{1}{3}\end{array}$

Thus additive inverse of $\frac{1}{3}$is $-\frac{1}{3}$

- Define multiplicative inverse

Let ‘a’ be a real number with $a\cancel{=}0$such that

$a·\frac{1}{a}=1=\frac{1}{a}·a$

Then '$\frac{1}{a}$' is the multiplicative inverse of ‘a’

- Multiplicative inverse of 13

Substitute $\frac{1}{3}$for $a$

$a=\frac{1}{3}$

Reciprocal on both sides

$\begin{array}{c}\frac{1}{a}=\frac{1}{\frac{1}{3}}\\ \frac{1}{a}=3\end{array}$

Thus multiplicative inverse of $\frac{1}{3}$is 3