Rational Numbers — Mental Maths

Fill in the blanks:

(i) The product of two rational numbers is a ..... .

(ii) Subtraction of rational numbers is ..... commutative.

(iii) The rational number $\dfrac{-7}{4}$ lies ..... of zero on the number line.

(iv) Division of rational numbers is ..... associative.

(v) Negative of a rational number is called its ..... .

(vi) Multiplicative identity of rational numbers is ..... .

(vii) Multiplication of rational numbers is ..... over addition.

(viii) Division of a rational number by a non-zero rational number is a ..... .

Answer

(i) Rational numbers are closed under multiplication, i.e. if $\dfrac{p}{q} \text{ and } \dfrac{r}{s}$ are two rational numbers, then $\dfrac{p}{q} \times \dfrac{r}{s}$ is also a rational number.

Hence, the product of two rational numbers is a rational number.

(ii) For any two rational numbers $\dfrac{p}{q} \text{ and } \dfrac{r}{s}$, we have $\dfrac{p}{q} - \dfrac{r}{s} \neq \dfrac{r}{s} - \dfrac{p}{q}$

Hence, subtraction of rational numbers is not commutative.

(iii) As $\dfrac{-7}{4}$ is a negative rational number, it is less than zero.

Every negative rational number lies to the left of zero on the number line.

Hence, the rational number $\mathbf{\dfrac{-7}{4}}$ lies to the left of zero on the number line.

(iv) For any three non-zero rational numbers, we have $\dfrac{p}{q} \div \left(\dfrac{r}{s} \div \dfrac{t}{u}\right) \neq \left(\dfrac{p}{q} \div \dfrac{r}{s}\right) \div \dfrac{t}{u}$

Hence, division of rational numbers is not associative.

(v) The negative of a rational number is obtained by changing its sign, and the sum of a rational number and its negative is 0.

Hence, negative of a rational number is called its additive inverse.

(vi) For every rational number $\dfrac{p}{q}$, we have $\dfrac{p}{q} \times 1 = \dfrac{p}{q} = 1 \times \dfrac{p}{q}$

Hence, multiplicative identity of rational numbers is 1.

(vii) For any three rational numbers, we have $\dfrac{p}{q} \times \left(\dfrac{r}{s} + \dfrac{t}{u}\right) = \dfrac{p}{q} \times \dfrac{r}{s} + \dfrac{p}{q} \times \dfrac{t}{u}$

Hence, multiplication of rational numbers is distributive over addition.

(viii) If we exclude 0, the remaining rational numbers are closed under division, i.e. if $\dfrac{p}{q} \text{ and } \dfrac{r}{s}$ are two non-zero rational numbers, then $\dfrac{p}{q} \div \dfrac{r}{s}$ is also a rational number.

Hence, division of a rational number by a non-zero rational number is a rational number.

State whether the following statements are true (T) or false (F):

(i) $\dfrac{-5}{9} \text{ is the additive inverse of } \dfrac{5}{9}$

(ii) Every integer is a rational number.

(iii) Zero has its multiplicative inverse.

(iv) Every rational number is an integer.

(v) Division of two rational numbers is always closed.

(vi) All terminating decimal numbers are rational numbers.

(vii) 0 is the multiplicative identity of rational numbers.

(viii) For rational numbers x and y, if x < y, then x - y is a positive rational number.

(ix) Subtraction of rational numbers is not associative.

(x) Reciprocal of 1 is 1.

(xi) Multiplicative inverse is also called a reciprocal.

(xii) Between two different rational numbers, there are infinitely many number of rational numbers.

Answer

(i) The sum of a rational number and its additive inverse is 0.

$\Rightarrow \dfrac{5}{9} + \left(\dfrac{-5}{9}\right) \\[1em] \Rightarrow \dfrac{5 + (-5)}{9} = \dfrac{0}{9} = 0$

Hence, the given statement is true.

(ii) Every integer $p$ can be written as $\dfrac{p}{1}$, which is of the form $\dfrac{p}{q}$ where $q \neq 0$. For example, $3 = \dfrac{3}{1} \text{ and } -3 = \dfrac{-3}{1}$

Hence, the given statement is true.

(iii) There is no rational number which when multiplied by 0 gives 1, so 0 has no multiplicative inverse.

Hence, the given statement is false.

(iv) Rational numbers of the form $\dfrac{p}{q}$ where $q \neq 1$ are generally not integers. For example, $\dfrac{-3}{5}$ is a rational number but it is not an integer.

Hence, the given statement is false.

(v) Division of a rational number by zero is not defined, so rational numbers are not closed under division. For example, $\dfrac{4}{5} \div 0$ is not defined.

Hence, the given statement is false.

(vi) Every terminating decimal number can be written in the form $\dfrac{p}{q}$ where $q \neq 0$. For example, $0.8 = \dfrac{4}{5} \text{ and } 0.28 = \dfrac{7}{25}$

Hence, the given statement is true.

(vii) For every rational number $\dfrac{p}{q}$, we have $\dfrac{p}{q} \times 1 = \dfrac{p}{q}$, so 1 is the multiplicative identity and 0 is the additive identity.

Hence, the given statement is false.

(viii) If $x$ < $y$, then $x$ lies to the left of $y$ on the number line, so $x - y$ is always negative. For example, taking $x = \dfrac{1}{2} \text{ and } y = \dfrac{3}{2}$, we get $x - y = -1$

Hence, the given statement is false.

(ix) For any three rational numbers, we have $\left(\dfrac{p}{q} - \dfrac{r}{s}\right) - \dfrac{t}{u} \neq \dfrac{p}{q} - \left(\dfrac{r}{s} - \dfrac{t}{u}\right)$

Hence, the given statement is true.

(x) As $1 \times 1 = 1$, the number 1 is its own reciprocal.

Hence, the given statement is true.

(xi) For every non-zero rational number $\dfrac{p}{q}$, the rational number $\dfrac{q}{p}$ is called its multiplicative inverse or reciprocal.

Hence, the given statement is true.

(xii) Between any two different rational numbers, we can always insert their mean, and this process can be repeated as many times as we want.

Hence, the given statement is true.