Chapter 1.2, Problem 33PPS

To determine

To find: the additive and multiplicative inverse of the given number.

Answer to Problem 33PPS

The additive and multiplicative inverse of the given number is $-\frac{6}{13}$ and $\frac{13}{6}$ respectively.

Explanation of Solution

Given information:

A real number is given as

  $\frac{6}{13}$

Concept used:

For any real numbers $a,b$ , and $c$ real numbers properties are tabulated below:

Property	Addition	Multiplication
Commutative	$a+b=b+a$	$a\cdot b=b\cdot a$
Associative	$\left(a+b\right)+c=a+\left(b+c\right)$	$\left(a\cdot b\right)\cdot c=a\cdot \left(b\cdot c\right)$
Identity	$a+0=0+a=a$ 0 is identity element for addition	$a\cdot 1=1\cdot a=a$ 1 is identity element for Multiplication
Inverse	$a+\left(-a\right)=\left(-a\right)+a=0$$\left(-a\right)$ is inverse element for addition	$a\cdot \frac{1}{a}=\frac{1}{a}\cdot a=1$$\frac{1}{a}$ is inverse element for Multiplication
Closure	$a+b$ is real number	$a\cdot b$ is real number
Distributive	$\begin{array}{l}a\left(b+c\right)=ab+bc\\ \left(b+c\right)a=ba+ca\end{array}$	$\begin{array}{l}a\left(b+c\right)=ab+bc\\ \left(b+c\right)a=ba+ca\end{array}$

Calculation:

Consider the given number.

  $\frac{6}{13}$

Let the additive inverse of the number be $a$.

  $\begin{array}{c}\frac{6}{13}+a=0\\ \frac{6}{13}+a-\frac{6}{13}=0-\frac{6}{13}\\ a=-\frac{6}{13}\end{array}$

Hence, the additive inverse of the number is $-\frac{6}{13}$.

Now, let the multiplicative inverse of the number be $b$.

  $\begin{array}{c}\frac{6}{13}\cdot b=1\\ \frac{\frac{6}{13}\cdot b}{\frac{6}{13}}=\frac{1}{\frac{6}{13}}\\ b=\frac{13}{6}\end{array}$

Hence, the multiplicative inverse of the number is $\left(\frac{13}{6}\right)$.