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MATHS 4 Mark Question

Rational Numbers · Gujarat Board (GSEB) STD 8 (GSEB - English Medium). 3 questions.

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Question 14 Marks
Complete the following table
NumbersAssociative for
AdditionSubstractionMultiplicationDivision
Rational numbers---No
Integers--Yes-
Whole numbersYes---
Natural numbers-No--
Answer
Associative for
Numbers -

1) Rational numbers
Addition - $ \begin{array}{l} \text { Yes, e.g. } \\ \frac{-1}{2}+\left[\frac{3}{7}+\left(\frac{-4}{3}\right)\right] \\ =\left[\frac{-1}{2}+\frac{3}{7}\right]+\left(\frac{-4}{3}\right) \\ \Rightarrow \frac{-1}{2}+\left(\frac{9-28}{21}\right) \\ =\left(\frac{-7+6}{14}\right)-\frac{4}{3} \\ \Rightarrow-\frac{1}{2}-\frac{19}{21}=\frac{-1}{14}-\frac{4}{3} \\ \Rightarrow \frac{-59}{42}=\frac{-59}{42} \end{array} $
which is true.
Substraction - No, e.g. $ \begin{array}{l} \frac{-2}{3}-\left(\frac{-4}{5}-\frac{1}{2}\right) \\ \Rightarrow\left[\frac{-2}{3}-\left(\frac{-1}{5}\right)\right]-\frac{1}{2} \\ \Rightarrow \frac{-2}{3}-\left(\frac{-8-5}{10}\right) \\ \neq\left[\frac{-10+12}{15}\right]-\frac{1}{2} \\ \Rightarrow-\frac{2}{3}+\frac{13}{10} \neq \frac{2}{15}-\frac{1}{2} \\ \Rightarrow \frac{-20+39}{30} \neq \frac{4-15}{30} \\ \Rightarrow \frac{19}{30} \neq \frac{-11}{30} \end{array} $
So, not associative for subtraction.
Multiplication - $ \begin{array}{l} \text { Yes, e.g. } \frac{2}{3} \times\left(\frac{-6}{7} \times \frac{4}{5}\right) \\ =\left[\frac{2}{3} \times\left(\frac{-6}{7}\right)\right] \times \frac{4}{5} \\ \Rightarrow \frac{2}{3} \times\left(\frac{-24}{35}\right) \\ =\frac{-12}{21} \times \frac{4}{5} \\ \Rightarrow \frac{-16}{35}=\frac{-16}{35} \end{array} $
which is true.
Division - No, e.g
$ \begin{array}{l} \frac{1}{2}+\left[\frac{-1}{3}+\frac{2}{5}\right]+\left[\frac{1}{2}+\left(\frac{-1}{3}\right)\right]+\frac{2}{5} \\ \Rightarrow \frac{1}{2}+\left[-\frac{1}{3} \times \frac{5}{2}\right] \\ \quad \neq\left[\frac{1}{2} \times\left(-\frac{3}{1}\right)\right]+\frac{2}{5} \\ \Rightarrow \frac{1}{2} \times\left(-\frac{6}{5}\right) \neq\left[\frac{-3}{2}\right] \times \frac{5}{2} \\ \Rightarrow \frac{-3}{5} \neq \frac{-15}{4} \end{array} $
So, not associative for division.

2) Integers
Addition - Yes, e.g.
$ \begin{array}{l} (-2)+[3+(-4)] \\ \quad=[(-2)+3]+(-4) \\ \Rightarrow(-2)+[-1]=[1]+(-4) \\ \Rightarrow-3=-3 \end{array} $
which is true.
Substraction - No, e.g.
$ \begin{array}{l} 5-(7-3) \neq (5-7)-3 \\ \Rightarrow 5-4 \neq (-2)-3 \\ \Rightarrow \quad 1 \neq-5 \end{array} $
So, not associative for subtraction.
Multiplication - $ \begin{array}{l} \text { Yes, e.g. } 5 \times[(-7) \times(-8)] \\ =[5 \times(-7)] \times(-8) \\ \Rightarrow 5 \times(56)=(-35) \times(-8) \\ \Rightarrow 280=280 \end{array} $
which is true.
Division - $ \begin{array}{l} \text { No, e.g } \\ {[-10+2]+(-5)} \\ \quad \neq(-10)+[2+(-5)] \\ \Rightarrow\left[\frac{-10}{2}\right]+(-5) \neq(-10)+\left(\frac{2}{-5}\right) \\ \Rightarrow(-5) \times\left(\frac{-1}{5}\right) \neq(-10) \times\left(\frac{-5}{2}\right) \\ \Rightarrow 1 \neq 25 \end{array} $
So, not associative for division.

3) Whole numbers
Addition - Yes, e.g.
$ \begin{array}{l} 2+(0+5) \\ =(2+0)+5 \end{array} $ $ \Rightarrow 7=7 $
which is true.
Substraction - No, e.g.
$ \begin{array}{l} 2-(0-5) \neq(2-0)-5 \\ \Rightarrow \quad 7 \end{array} $
So, not associative for subtraction.
Multiplication - $ \begin{array}{l} \text { Yes, e.g. } 2 \times[3 \times(-5)] \\ =[2 \times 3] \times(-5) \\ \Rightarrow 2 \times(-15)=(6) \times(-5) \\ \Rightarrow-30=-30 \end{array} $
which is true.
Division - No, e.g.
$ \begin{array}{l} (2+3)+5 \neq 2+(3+5) \\ \Rightarrow\left(\frac{2}{3}\right)+5 \neq 2+\left(\frac{3}{5}\right) \\ \Rightarrow \frac{2}{3} \times \frac{1}{5} \neq 2 \times \frac{5}{3} \\ \Rightarrow \frac{2}{15} \neq \frac{10}{3} \end{array} $
So, not associative for division.

4) Natural numbers
Addition - Yes, e.g.
$ \begin{array}{l} 2+(4+5) \\ =(2+4)+5 \\ \Rightarrow 11=11 \end{array} $
which is not true
Substraction - No, e.g.
$ \begin{array}{l} 4-(5-1) \neq(4-5)-1 \\ \Rightarrow 0 \neq-2 \end{array} $
So, not associative for subtraction.
Multiplication - Yes, e.g.
$ \begin{array}{l} 3 \times(4 \times 5)=(3 \times 4) \times 5 \\ \Rightarrow 60=60 \end{array} $
which is true.
Division - No, e.g.
$ \begin{array}{l} (4+2)+5 \neq 4+(2+5) \\ \Rightarrow 2+5 \neq 4+\frac{2}{5} \\ \Rightarrow 2 \times \frac{1}{5} \neq 4 \times \frac{5}{2} \\ \Rightarrow \quad \frac{2}{5} \neq \frac{20}{2} \end{array} $
So, not associative for division.
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Question 24 Marks
Complete the following table
NumbersCommutative for
AdditionSubstractionMultiplicationDivision
Rational numbersYes
IntegersNo
Whole numbersYes
Natural numbersNo
Answer
Associative for
Numbers -

1) Rational numbers
Addition - $ \begin{array}{l} \text { Yes, e.g } \\ \left(\frac{2}{3}+\frac{5}{7}=\frac{5}{7}+\frac{2}{3}\right) \\ \Rightarrow \frac{14+15}{21}=\frac{15+14}{21} \\ \Rightarrow \frac{29}{21}=\frac{29}{21} \end{array} $
which is true.
Substraction - $ \begin{array}{l} \text { No, e. } 1 \\ \left(\frac{1}{2}-\frac{3}{5} * \frac{3}{5}-\frac{1}{2}\right) \\ \Rightarrow \frac{5-6}{10} \neq \frac{6-5}{10} \\ \Rightarrow \frac{-1}{10} \neq \frac{1}{10} \end{array} $
So, not commutative for subtraction
Multiplication - Yes, e.p $ \begin{array}{l} \frac{7}{3} \times \frac{6}{5}+\frac{6}{5} \times\left(\frac{-7}{3}\right) \\ \Rightarrow \frac{-42}{15}=\frac{-42}{15} \end{array} $
which is true.
Division - No, e.g $ \begin{array}{l} \frac{-5}{4}+\frac{3}{7} \times \frac{3}{7}+\left(\frac{-5}{4}\right) \\ \Rightarrow \frac{-5}{4} \times \frac{7}{3} \neq \frac{3}{7} \times \frac{4}{-5} \\ \Rightarrow-\frac{35}{12} \times \frac{12}{-35} \end{array} $
So, not commutative for division.

2) Integers
Addition - Yes, e.g. $5+3=3+5$ $\Rightarrow 8=8$
which is true.
Substraction - No, $ \begin{array}{l} \text { e.g }[5-(-3) \neq-3-5] \\ \Rightarrow 5+3 \neq-3-5 \\ \Rightarrow 8 \neq-8 \end{array} $
So, not commutative for subtraction.
Multiplication - Yes, $ \begin{array}{l} \text { e.g. } 3 \times(-5)=(-5) \times 3 \\ \Rightarrow-15=-15 \end{array} $
which is true.
Division - $ \begin{array}{l} \text { No, e.g. }(3+5+5+3) \\ \Rightarrow \frac{3}{5} \neq \frac{5}{3} \end{array} $
So, not cornmutative for division.

3) Whole numbers
Addition - Yes, $ \begin{array}{l} \text { e.g. }(0+7=7+0) \\ \Rightarrow 7=7 \end{array} $
which is true.
Substraction - $ \text { No, e.g. }(5-4 \neq 4-5) $
$ 1 \neq-1 $
So, not commutative for subtraction.
Multiplication - Yes, $ \begin{array}{l} \text { e.g. }(5 \times 4=4 \times 5) \\ \Rightarrow 20=20 \end{array} $
which is true.
Division - No, e.g. $(5+0 \neq 0+5)$
So. not commutative for division.

4) Natural numbers
Addition - Yes, e.g. $(2+3=3+2)$
$ \Rightarrow 5=5 $
which is true.
Substraction - No, $ \begin{array}{l} \text { No, e.g. }(2-3 \neq 3-2) \\ \Rightarrow-1 \neq 1 \end{array} $
So, not commutative for subtraction.
Multiplication - Yes, $ \begin{array}{l} \text { e.g. }(2 \times 3=3 \times 2) \\ \Rightarrow 6=6 \end{array} $
which is true.
Division - No, e.g. $(2+4 \neq 4+2)$
$ \Rightarrow \frac{2}{4} \neq \frac{4}{2} $
So, not commutative for division.
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Question 34 Marks
Fill in the blanks in the following table
Numbers Closed under 
 AdditionSubstractionMultiplicationDivision
Rational numbersYesYes No
Integers Yes No
Whole numbers  Yes 
Natural numbers No  
Answer
Associative for
Numbers -

1) Rational numbers
Addition - $\begin{array}{l} \text { Yes, e.g. }\left(\frac{3}{8}+\left(\frac{(-5)}{7}\right)\right) \\=\frac{3 \times 7+(-5) \times 8}{56} \\=\frac{21+(-40)}{56}=\frac{-19}{56},\end{array}$
Rational number
Substraction - $\begin{array}{l}\text { Yes, e.g. }\left[\left(\frac{(-5)}{7}\right)-\frac{2}{3}\right] \\ =\frac{(-5) \times 3-2 \times 7}{21} \\ =\frac{-15-14}{21}=\frac{-29}{21}\end{array}$
Rational number
Multiplication - Yes, e.g.
$\left[\frac{-4}{5} \times\left(\frac{-6}{11}\right)\right]=\frac{24}{55}$
Rational number
Division - No, e.g $\left(\frac{4}{5}+0\right)$,
Not defined

2) Integers
Addition - Yes, e.g. (-6 + 5 = -1)
integer
Substraction - Yes, e.g.
$(7-5=2)$ Integer
Multiplication - Yes, e.g. $(5 \times 8=40)$
Integer
Division - No, e.g $(5+8)=$ Integer

3) Whole numbers
Addition - Yes, e.g. (0 + 5 = 5)
Whole number
Substraction - No, e.g. $(5-7=-2)$
Not a whole number
So, not commutative for subtraction.
Multiplication - Yes, e.g. $(3 \times 7=21)$
Whole number
Division - No, e.g. $(5+8)$
Not a whole number

4) Natural numbers
Addition - Yes, e.g. (2 + 3 = 5)
Natural number
Substraction - No, e.g. $(2-3=-1)$
Not a natural number
Multiplication - Yes, e.g. $(2 \times 3=6)$
Natural number
Division - No, e.g. $(2+3)$
Not a natural number
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