Exercise 1.3 - Chapter 1 Numbers 8th Maths Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Updated By SaraNextGen

On April 24, 2024, 11:35 AM

Question $1 .$
Verify the closure property for addition and multiplication for the rational numbers $\frac{-5}{7}$ and $\frac{8}{9}$.
Answer:
closure property for addition
Let $\mathrm{a}=\frac{-5}{7}$ and $\mathrm{b}=\frac{8}{9}$
$\begin{aligned}
a+b &=\frac{-5}{7}+\frac{8}{9} \\
&=\frac{(-5 \times 9)+(8 \times 7)}{7 \times 9} \\
&=\frac{-45+56}{63}=\frac{11}{63} \text { is in } \mathrm{Q}
\end{aligned}$
i.e $a+b=\frac{-5}{7}+\frac{8}{9}=\frac{11}{63}$ is in Q.
$\therefore$ Closure property is true for addition of rational numbers.

Question 2 .
Verify the commutative property for addition and multiplication for the rational numbers $\frac{-10}{11}$ and $\left(\frac{-8}{33}\right)$.
Answer:
Let $\mathrm{a}=\frac{-10}{11}$ and $\left(\frac{-8}{33}\right)$ be the given rational numbers.
Now $a+b=\frac{-10}{11}+\left(\frac{-8}{33}\right)=\frac{(-10 \times 3)+(-8 \times 1)}{33}=\frac{-30+(-8)}{33}$ $a+b=\frac{-38}{33}$
$b+a=\frac{-8}{33}+\left(\frac{-10}{11}\right)=\frac{(-8 \times 1)+((-10) \times 3)}{33}=\frac{-8+(-30)}{33}$
$b+a=\frac{-38}{33}$

$\begin{aligned}
&b+a=\frac{-8}{33}+\left(\frac{-10}{11}\right)=\frac{(-8 \times 1)+((-10) \times 3)}{33}=\frac{-8+(-30)}{33} \\
&b+a=\frac{-38}{33}
\end{aligned}$
From (1) and (2)
$\mathrm{a}+\mathrm{b}=\mathrm{b}+\mathrm{a}$ and hence additionis commutative for rational numbers
Further $a \times b=\frac{-10}{11} \times\left(\frac{-8}{33}\right)=\frac{80}{363}$
$\begin{aligned}
a \times b &=\frac{80}{363} \\
b \times a &=\frac{-8}{33} \times\left(\frac{-10}{11}\right)=\frac{80}{363} \\
b \times a &=\frac{80}{363}
\end{aligned}$
From (3) and (4) $a \times b=b \times a$
Hence multiplication is commutative for rational numbers.

Question $3 .$
Verify the associative property for addition and multiplication for the rational numbers $\frac{-7}{9}, \frac{5}{6}$ and $\frac{-4}{3}$.
Answer:
Let $a=\frac{-7}{9}, b=\frac{5}{6}, c=\frac{-4}{3}$ be the given rational numbers.
$(a+b)+c=\left(\frac{-7}{9}+\frac{5}{6}\right)+\left(\frac{-4}{3}\right)=\left(\frac{-7 \times 2+5 \times 3}{18}\right)+\left(\frac{-4}{3}\right)$
$=\left(\frac{-14+15}{18}\right)+\left(\frac{-4}{3}\right)=\frac{1}{18}+\left(\frac{-4}{3}\right)$

$\begin{aligned}
&a+(b+c)=-\frac{7}{9}+\left(\frac{5}{6}+\frac{(-4)}{3}\right)=\frac{-7}{9}+\left(\frac{5+(-4) 2}{6}\right)\\
&=\frac{-7}{9}+\left(\frac{5+(-8)}{6}\right)=-\frac{7}{9}+\left(\frac{-3}{6}\right)=-\frac{7}{9}+\left(\frac{-1}{2}\right)\\
&=\frac{-7 \times 2+(-1) \times 9}{18}=\frac{-14+(-9)}{18}=\frac{-23}{18}\\
&\text { From (1) and (2), }(a+b)+c=a+(b+c) \text { is true for rational numbers. }\\
&\text { Given the rational number } a=\frac{-1}{2} ; b=\frac{2}{3} \text { and } c=\frac{-5}{6}\\
&a \times(b+c)=\frac{-1}{2} \times\left(\frac{2}{3}+\left(\frac{-5}{6}\right)\right)=\frac{-1}{2} \times\left(\frac{(2 \times 2)+(-5 \times}{6}\right.\\
&=\frac{-1}{2} \times\left(\frac{4+(-5)}{6}\right)=\frac{-1}{2} \times\left(\frac{-1}{6}\right)\\
&a \times(b+c)=\frac{1}{12}\\
&(a \times b)+(a \times c)=\left(\frac{-1}{2} \times \frac{2}{3}\right)+\left(\frac{-1}{2} \times\left(\frac{-5}{6}\right)\right)\\
&=\frac{-2}{6}+\frac{5}{12}=\frac{(-2 \times 2)+5 \times 1}{12}=\frac{-4+5}{12}\\
&(a \times b)+(a \times c)=\frac{1}{12}
\end{aligned}$

From $(1)$ and $(2)(a \times b) \times c=(a \times b) \times c$ is true for rational numbers.
Thus associative property.

Question $4 .$
Verify the distributive property $a \times(b+c)=(a \times b)+(a+c)$ for the rational numbers $\mathrm{a}=\frac{-1}{2}, \mathrm{~b}=\frac{2}{3}$ and $\mathrm{c}=\frac{-5}{6}$.
Answer:
Given the rational number $a=\frac{-1}{2} ; b=\frac{2}{3}$ and $c=\frac{-5}{6}$
$\begin{aligned}
a \times(b+c) &=\frac{-1}{2} \times\left(\frac{2}{3}+\left(\frac{-5}{6}\right)\right)=\frac{-1}{2} \times\left(\frac{(2 \times 2)+(-5 \times}{6}\right.\\
&=\frac{-1}{2} \times\left(\frac{4+(-5)}{6}\right)=\frac{-1}{2} \times\left(\frac{-1}{6}\right) \\
a \times(b+c) &=\frac{1}{12}
\end{aligned}$

$(a \times b)+(a \times c)=\left(\frac{-1}{2} \times \frac{2}{3}\right)+\left(\frac{-1}{2} \times\left(\frac{-5}{6}\right)\right)$
$\begin{aligned}
&=\frac{-2}{6}+\frac{5}{12}=\frac{(-2 \times 2)+5 \times 1}{12}=\frac{-4+5}{12} \\
(a \times b)+(a \times c) &=\frac{1}{12}
\end{aligned}$
From (1) and (2) we have $a \times(b+c)=(a \times b)+(a \times c)$ is true Hence multiplication is distributive over addition for rational numbers $Q$.

Question $5 .$
Verify the identity property for addition and multiplication for the rational numbers $\frac{15}{19}$ and $\frac{-18}{25}$.
Answer:
$\begin{aligned}
\frac{15}{19}+0 &=\frac{15}{19}+\frac{0}{19}=\frac{15+0}{19}=\frac{15}{19} \\
\frac{-18}{25}+0 &=\frac{-18}{25}+\frac{0}{25}=\frac{-18+0}{25}=\frac{-18}{25}
\end{aligned}$

$$
\frac{-18}{25}+0=\frac{-18}{25}+\frac{0}{25}=\frac{-18+0}{25}=\frac{-18}{25}
$$
Identify property for addition verified.
$$
\begin{aligned}
\frac{15}{19} \times 1 &=\frac{15 \times 1}{19}=\frac{15}{19} \\
\frac{-18}{25} \times 1 &=\frac{-18 \times 1}{25}=\frac{-18}{25}
\end{aligned}$
Identify property for multiplication verified.

Question $6 .$
Verify the additive and multiplicative inverse property for the rational numbers $\frac{-7}{17}$ and $\frac{17}{27}$.
Answer:
$\begin{aligned}
\frac{-7}{17}+\frac{7}{17} &=\frac{-7+7}{17}=\frac{0}{17}=0 \\
\frac{17}{27}+\left(-\frac{17}{27}\right) &=\frac{17+(-17)}{27}=\frac{0}{27}=0
\end{aligned}$
Additive inverse for rational numbers verified.

$\begin{aligned}
&\frac{-7}{17} \times \frac{17}{-7}=\frac{-7 \times 17}{17 \times(-7)}=1 \\
&\frac{17}{27} \times \frac{27}{17}=\frac{17 \times 27}{27 \times 17}=1
\end{aligned}$
Mulplicative inverse for rational numbers verified.

Objective Type Questions
Question 7.
Closure property is not true for division of rational numbers because of the number
(A) 1
(B) 1
(C) 0
(D) $\frac{1}{2}$
Answer:
(C) 0

Question $8 .$
$\frac{1}{2}-\left(\frac{3}{4}-\frac{5}{6}\right) \neq\left(\frac{1}{2}-\frac{3}{4}\right)-\frac{5}{6}$ illustrates that subtraction does not satisfy the property for rational numbers.
(A) commutative
(B) closure
(C) distributive
(D) associative
Answer:
(D) associative

Question $9 .$
Which of the following illustrates the inverse property for addition?
(A) $\frac{1}{8}-\frac{1}{8}=0$
(B) $\frac{1}{8}+\frac{1}{8}=\frac{1}{4}$
(C) $\frac{1}{8}+0=\frac{1}{8}$
(D) $\frac{1}{8}-0=\frac{1}{8}$
Answer:
(A) $\frac{1}{8}-\frac{1}{8}=0$

Question $10 .$
$\frac{3}{4} \times\left(\frac{1}{2}-\frac{1}{4}\right)=\frac{3}{4} \times \frac{1}{2}-\frac{3}{4} \times \frac{1}{4}$ illustrates that multiplication is distributive over
(A) addition
(B) subtraction
(C) multiplication
(D) division
Answer:
(B) subtraction