Question

Robin writes the properties of rational numbers as–

Property 1: For any two rational number $x$ and $y$ , $x + y = y + x$

Property 2: For any two rational number $x$ and $y, x - y = y - x$

Property 3: For any two rational number $x$ and $y$ , $x / y = y / x$

Property 4: For any two rational number $x$ and $y, x \times y = y \times x$

Which properties written by Robin are incorrect? Why are they incorrect?

Accepted Answer

The properties 2 and 3 written by Rohit are incorrect.

As we know that for rational numbers, addition and multiplication are commutative while subtraction and division are not commutative.

To understand this, let an example of two rational numbers: $\frac{3}{2}, \frac{2}{3}$ .

Property 1: $\frac{3}{2} + \frac{2}{3} = \frac{9 + 4}{6} = \frac{13}{6}$ and $\frac{2}{3} + \frac{3}{2} = \frac{4 + 9}{6} = \frac{13}{6}$ .

So, $\frac{3}{2} + \frac{2}{3} = \frac{2}{3} + \frac{3}{2}$ .

Property 2: $\frac{3}{2} - \frac{2}{3} = \frac{9 - 4}{6} = \frac{5}{6}$ and $\frac{2}{3} - \frac{3}{2} = \frac{4 - 9}{6} = \frac{- 5}{6}$

So, $\frac{3}{2} - \frac{2}{3} \neq \frac{2}{3} - \frac{3}{2}$ .

Property 3: $\frac{\frac{3}{2}}{\frac{2}{3}} = \frac{3}{2} \times \frac{3}{2} = \frac{9}{4}$ and $\frac{\frac{2}{3}}{\frac{3}{2}} = \frac{2}{3} \times \frac{2}{3} = \frac{4}{9}$ .

So, $\frac{\frac{3}{2}}{\frac{2}{3}} \neq \frac{\frac{2}{3}}{\frac{3}{2}}$

Property 4: $\frac{3}{2} \times \frac{2}{3} = 1$ and $\frac{2}{3} \times \frac{3}{2} = 1$ .

Hence, The properties 2 and 3 written by Rohit are incorrect.

$(i)$	Rational Numbers are commutative under addition and multiplication
$(ii)$	Rational Numbers are associative under addition and multiplication
$(iii)$	Rational Numbers are distributive under multiplication over addition
$(iv)$	Rational Numbers are closed under addition, subtraction and multiplication

Important Questions on Rational Numbers

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