Number Systems

Class 9 NCERT Maths

NCERT

1   Is zero a rational number? Can you write it in the form $\dfrac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$?

Solution :

Consider the definition of a rational number.$\\$ A rational number is the one that can be written in the form of $\dfrac{p}{q}$ where $p$ and $q$ are integers and $q\neq 0$$\\$ $\bullet$ Zero can be written as $\dfrac{0}{1},\dfrac{0}{2},\dfrac{0}{3},\dfrac{0}{4},\dfrac{0}{5}....$$\\$ $\bullet $ Zero can be written as well$\dfrac{0}{-1},\dfrac{0}{-2},\dfrac{0}{-3},\dfrac{0}{-4},\dfrac{0}{-5}....$$\\$ So, we arrive at the conclusion that 0 can be written in the form of $\dfrac{p}{q}$ Where $p$ and $q$ are integers ( q can be positive or negative integers).$\\$ Therefore, zero is a rational number.

2   Find six rational numbers between $3$ and $4.$

Solution :

We know that there are infinite rational numbers between any two numbers. As we have to find $6$ rational numbers between $3$ and $4$ So multiply and divide by $7$ (or any number greater than $6$)$\\$ We get,$3=3*\dfrac{7}{7}=\dfrac{21}{7}\\ 4=4*\dfrac{7}{7}=\dfrac{28}{7}$$\\$ Thus the $6$ rational numbers are $\\$ $\dfrac{22}{7},\dfrac{23}{7},\dfrac{24}{7},\dfrac{25}{7},\dfrac{26}{7},\dfrac{28}{7}$

3   Find five rational numbers between $\dfrac{3}{5} $ and $\dfrac{4}{5}$

Solution :

We know that there are infinite rational numbers between any two numbers.$\\$ As we have to find $5$ rational numbers between $\dfrac{3}{5}$ and $ \dfrac{4}{5}$$\\$So, multiply and divide by $6$ (Or any number greater than $5$)$\\$ $\dfrac{3}{5}=\dfrac{3}{5}*\dfrac{6}{6}=\dfrac{18}{30}$$\\$ $\dfrac{4}{5}=\dfrac{4}{5}*\dfrac{6}{6}=\dfrac{24}{30}$$\\$ Thus the $5$ rational numbers are$\\$ $\dfrac{19}{30},\dfrac{20}{30},\dfrac{21}{30},\dfrac{22}{30},\dfrac{23}{30}$

4   State whether the following statements are true or false. Give reasons for your answers.$\\$ (i) Every natural number is a whole number.$\\$ (ii) Every integer is a whole number.$\\$ (iii) Every rational number is a whole number.

Solution :

(i) Consider the whole numbers and natural numbers separately.$\\$ We know that whole number series is $0,1, 2, 3, 4,5..... .$$\\$ We know that natural number series is $1, 2, 3, 4,5..... .$$\\$ So, we can conclude that every natural number lie in the whole number series.$\\$ Diagrammatically, we can represent as follows:$\\$ Therefore, we conclude that, yes every natural number is a whole number.$\\$ (ii) Consider the integers and whole numbers separately.$\\$ We know that integers are those numbers that can be written in the form of $\dfrac{p}{q}$ where $q=1.$$\\$ Now, considering the series of integers, we have$ - 4, - 3, - 2, - 1, 0, 1, 2,3, 4..... .$$\\$ We know that whole number are $0, 1, 2, 3, 4, 5..... .$$\\$ We can conclude that whole number series lie in the series of integers. But every integer does not appear in the whole number series.$\\$ Therefore, we conclude that every integer is not a whole number.$\\$ But, clearly every whole number is an integer.$\\$ (iii) Consider the rational numbers and whole numbers separately.$\\$ We know that rational numbers are the numbers that can be written in the form $\dfrac{p}{q}$ ,where $ q\neq 0$$\\$ We know that whole numbers are $0, 1, 2, 3, 4, 5..... .$$\\$ We know that every whole number can be written in the form of $\dfrac{p}{q}$ as follows $\dfrac{0}{1},\dfrac{1}{1},\dfrac{2}{1},\dfrac{3}{1},\dfrac{4}{1}\dfrac{5}{1}........$$\\$ We conclude that every whole number is a rational number.$\\$ But, every rational number $(1/2, 1/3, 1/4,1/5,1/6.... )$ is not a whole number. Therefore, we conclude that every rational number is not a whole number.$\\$ But, clearly every whole number is a rational number.

5   State whether the following statements are true or false. Justify your answers.$\\$ (i) Every irrational number is a real number.$\\$ (ii) Every point on the number line is of the form $\sqrt{m}$ , where m is a natural number.$\\$ (iii) Every real number is an irrational number.

Solution :

(i)Consider the irrational numbers and the real numbers separately.$\\$ $\bullet $ The irrational numbers are the numbers that cannot be converted in the form $\dfrac{p}{q}$,where $p$ and $q$ are integers and $q\neq 0$. (Eg: $\sqrt{2}, 3\pi, .011011011...$)$\\$ $\bullet $ The real number is the collection of rational numbers and irrational numbers.$\\$ Therefore, we conclude that, every irrational number is a real number.$\\$ (ii)Consider a number line. on a number line, we can represent negative as well as positive numbers$\\$ $\bullet $ Positive numbers are represented in the form of $\sqrt{1}, \sqrt{1.1}, \sqrt{1.2} ... ...$$\\$ $\bullet $ But we cannot get a negative number after taking square root of any number.$\\$ (Eg: $\sqrt{-5} = 5i $ is a complex number (which you will study in higher classes))$\\$ Therefore, we conclude that every number point on the number line is not of the form $\sqrt{m}$ , where m is a natural number.$\\$ (iii) Consider the irrational numbers and the real numbers separately.$\\$ $\bullet $ Irrational numbers are the numbers that cannot be converted in the form $\dfrac{p}{q},$ where $p$ and $q$ are integers and $q\neq 0.$$\\$ $\bullet $ A real number is the collection of rational numbers (Eg:$ \dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{4},\dfrac{1}{5}...)$ and irrational numbers (Eg: $\sqrt{2}, 3\pi, .011011011...$)$\\$ So, we can conclude that every irrational number is a real number. But every real number is not an irrational number.$\\$ Therefore, every real number is not an irrational number.