NCERT Solutions for Maths Chapter 1 Exercise 1.2 Class 9 - Number Systems

Exercise (1.2)

1. State whether the following statements are true or false. Justify your answers.

(i) Every irrational number is a real number. 

Ans: Write the irrational numbers and the real numbers in a separate manner.

• The irrational numbers are the numbers that cannot be represented in the form $\dfrac{p}{q},$ where $p$ and $q$ are integers and $q\ne 0.$

For example, $\sqrt{2},3\pi ,\text{ }.011011011...$ are all irrational numbers.

• The real number is the collection of both rational numbers and irrational numbers.

For example, $0,\,\pm \dfrac{1}{2},\,\pm \sqrt{2}\,,\pm \pi ,...$ are all real numbers.

Thus, it is concluded that every irrational number is a real number.

Hence, the given statement is true.

(ii) Every point on the number line is of the form $\sqrt{m}$, where m is a natural number. 

Ans: Consider points on a number line to represent negative as well as positive numbers.

Observe that, positive numbers on the number line can be expressed as $\sqrt{1,}\sqrt{1.1,}\sqrt{1.2},\sqrt{1.3},\,...$, but any negative number on the number line cannot be expressed as $\sqrt{-1},\sqrt{-1.1},\sqrt{-1.2},\sqrt{-1.3},...$, because these are not real numbers.

Therefore, it is concluded from here that every number point on the number line is not of the form $\sqrt{m}$, where $m$ is a natural number. 

Hence, the given statement is false.

(iii) Every real number is an irrational number. 

Ans: Write the irrational numbers and the real numbers in a separate manner.

• The irrational numbers are the numbers that cannot be represented in the form $\dfrac{p}{q},$ where $p$ and $q$ are integers and $q\ne 0.$

For example, $\sqrt{2},3\pi ,\text{ }.011011011...$ are all irrational numbers.

• Real numbers are the collection of rational numbers (Ex: $\dfrac{1}{2},\dfrac{2}{3},\dfrac{3}{5},\dfrac{5}{7},$……) and the irrational numbers (Ex: $\sqrt{2},3\pi ,\text{ }.011011011...$).

Therefore, it can be concluded that every irrational number is a real number, but every real number cannot be an irrational number.

Hence, the given statement is false. 

2. Are the square roots of all positive integer numbers irrational? If not, provide an example of the square root of a number that is not an irrational number.

Ans: Square root of every positive integer does not give an integer. 

For example: $\sqrt{2},\sqrt{3,}\sqrt{5},\sqrt{6},...$ are not integers, and hence these are irrational numbers. But $\sqrt{4}$ gives $\pm 2$ , these are integers and so, $\sqrt{4}$ is not an irrational number.

Therefore, it is concluded that the square root of every positive integer is not an irrational number.

3. Show how $\sqrt{5}$ can be represented on the number line.

Ans: Follow the procedures to get $\sqrt{5}$ on the number line.

• Firstly, Draw a line segment $AB$ of $2$ unit on the number line.

• Secondly, draw a perpendicular line segment $BC$ at $B$ of $1$ units.

• Thirdly, join the points $C$ and $A$, to form a line segment $AC$. 

• Fourthly, apply the Pythagoras Theorem as 

$A{{C}^{2}}=A{{B}^{2}}+B{{C}^{2}} $ 

$A{{C}^{2}}={{2}^{2}}+{{1}^{2}}$

$A{{C}^{2}}=4+1=5 $ 

$AC=\sqrt{5} $

• Finally, draw the arc $ACD$, to find the number $\sqrt{5}$ on the number line as given in the diagram below.

4. Classroom activity (Constructing the ‘square root spiral’) : Take a large sheet of paper and construct the ‘square root spiral’ in the following fashion. Start with a point O and draw a line segment $OP_{1}$ of unit length. Draw a line segment $P_{1}$ $P_{2}$ perpendicular to $OP_{1}$ of unit length (see Fig. 1.9). Now draw a line segment $P_{2}$ $P_{3}$ perpendicular to $OP_{2}$ . Then draw a line segment $P_{3}$ $P_{4}$ perpendicular to $OP_{3}$ . Continuing in this manner, you can get the line segment $P_{n–1}$Pn by drawing a line segment of unit length perpendicular to OPn–1. In this manner, you will have created the points $P_{2}$ ,$ P_{3}$ ,...., $P_{n}$ ,... ., and joined them to create a beautiful spiral depicting $\sqrt{2}, \sqrt{ 3}, \sqrt{4}, …$

Ans:

Step 1: Mark a point O

Choose a point O on your paper. This will be the centre of your square root spiral.

Step 2: Draw line OA of 1 cm horizontally

From point O, draw a straight line OA horizontally to the right. The length should be 1 cm.

Step 3: Draw perpendicular line AB of 1 cm

From point A (the end of OA), draw a line AB vertically upwards. The length of AB should also be 1 cm.

Step 4: Join OB (length √2)

Join point O to point B (the end of AB). The length of OB should be √2 cm.

Step 5: Draw perpendicular line from B and mark C

From point B, draw another line perpendicular to OB (going upwards) of 1 cm. Mark the endpoint of this line as C.

Step 6: Join OC (length √3)

Join point O to point C. The length of OC should be √3 cm.

Step 7: Repeat the process

Repeat steps 5 and 6 to continue the spiral:

From point C, draw a perpendicular line of 1 cm upwards and mark the endpoint as D.

Join O to D. The length OD should be √4 cm.

Continue this process, each time increasing the length of the perpendicular line by 1 cm and joining the new point to O to form the next segment of the spiral, where the length of each segment from O increases by 1 each time (resulting in √2, √3, √4, etc.).

Conclusion

In NCERT Solutions for Class 9 Chapter 1 Exercise 1.2 on the Number System by Vedantu, students delve into the basics of numbers. Key points include understanding the classification of numbers into natural, whole, integers, rational, and irrational. Focus on mastering operations like addition, subtraction, multiplication, and division with these numbers. Previous year question papers typically feature 5–7 questions, testing the application of these concepts. It's crucial to grasp the properties of different number types and how they interact in mathematical operations. By mastering these fundamentals, students build a solid foundation for more advanced mathematical concepts.

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