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Rational Numbers — MATHS STD 8 — Question

Gujarat BoardEnglish MediumSTD 8MATHSRational Numbers4 Marks
Question
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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