NCERT Solutions Class 9 Chapter 1 Ex 1.2 Number Systems

NCERT Solutions Class 9 Chapter 1 Exercise 1.2 introduces the concept of irrational numbers and helps students clearly understand how they are different from rational numbers. NCERT Solutions are given here in a simple and explanatory way. These solutions help students identify irrational numbers, understand real numbers, and learn how to differentiate between terminating and non-terminating decimals. 

Class 9 Chapter 1 Number System Exercise 1.2 Questions and Answers

Class 9 chapter 1 number system exercise 1.2 questions are given here along with the answers for better understanding:

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

(i) Every irrational number is a real number.

Solution: True

Irrational Numbers – A number is said to be irrational, if it cannot be written in the p/q, where p and q are integers and q ≠ 0. i.e., Irrational numbers = π, e, √3, 5+√2, 6.23146…. , 0.101001001000….

Real numbers – The collection of both rational and irrational numbers are known as real numbers. i.e., Real numbers = √2, √5, , 0.102… Every irrational number is a real number, however, every real number is not an irrational number.

(ii) Every point on the number line is of the form √m where m is a natural number.

Solution: False

The statement is false since as per the rule, a negative number cannot be expressed as square roots.

E.g., √9 =3 is a natural number. But √2 = 1.414 is not a natural number. Similarly, we know that there are negative numbers on the number line, but when we take the root of a negative number it becomes a complex number and not a natural number.

E.g., √-7 = 7i, where i = √-1 The statement that every point on the number line is of the form √m, where m is a natural number is false.

(iii) Every real number is an irrational number.

Solution: False

The statement is false. Real numbers include both irrational and rational numbers. Therefore, every real number cannot be an irrational number.

Real numbers – The collection of both rational and irrational numbers are known as real numbers. i.e., Real numbers = √2, √5, , 0.102

Irrational Numbers – A number is said to be irrational, if it cannot be written in the p/q, where p and q are integers and q ≠ 0. i.e., Irrational numbers = π, e, √3, 5+√2, 6.23146…. , 0.101001001000…. Every irrational number is a real number, however, every real number is not irrational.

2. Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.

Solution: No, the square roots of all positive integers are not irrational. For example, √4 = 2 is rational. √9 = 3 is rational. Hence, the square roots of positive integers 4 and 9 are not irrational. ( 2 and 3, respectively).

3. Show how √5 can be represented on the number line.

Solution:

Step 1: Let line AB be of 2 unit on a number line.

Step 2: At B, draw a perpendicular line BC of length 1 unit.

Step 3: Join CA Step 4: Now, ABC is a right angled triangle. Applying Pythagoras theorem, AB 2 +BC 2 = CA 2 2 2 +1 2 = CA 2 = 5 ⇒ CA = √5 .Thus, CA is a line of length √5 unit.

Step 4: Taking CA as a radius and A as a center draw an arc touching the number line. The point at which number line get intersected by arc is at √5 distance from 0 because it is a radius of the circle whose center was A. Thus, √5 is represented on the number line as shown in the figure. 

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 OP1 of unit length. Draw a line segment P1P2 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 Fig. 1.9 :

Constructing this manner, you can get the line segment P n-1 Pn by square root spiral drawing a line segment of unit length perpendicular to OP n-1 . In this manner, you will have created the points P 2 , P 3 ,….,Pn,… ., and joined them to create a beautiful spiral depicting √2, √3, √4, …

Solution: Step 1: Mark a point O on the paper. Here, O will be the center of the square root spiral.

Step 2: From O, draw a straight line, OA, of 1cm horizontally.

Step 3: From A, draw a perpendicular line, AB, of 1 cm.

Step 4: Join OB. Here, OB will be of √2

Step 5: Now, from B, draw a perpendicular line of 1 cm and mark the end point C.

Step 6: Join OC. Here, OC will be of √3

Step 7: Repeat the steps to draw √4, √5, √6….

NCERT Maths Class 9 Number System Exercise 1.2 PDF

The NCERT Maths Class 9 Number System Exercise 1.2 PDF is helpful for students who want to practice this exercise anytime. The PDF includes neatly formatted, step-wise answers so you can practise comfortably and double-check your solutions whenever needed. It also explains important points to remember when identifying irrational numbers.

Number System Class 9 chapter 1.2 exercises must be practiced well to get a better understanding of the entire chapter. You can revise the definitions, examples, and solved questions easily by downloading the PDF:

NCERT Maths Class 9 Number System Exercise 1.1 PDF

Number System ONE SHOT Full Chapter Class 9th Maths Youtube Video