NCERT Solutions Class 9 Maths Chapter 1 Number Systems Ex 1.2

NCERT Solutions Class 9 Maths 
Chapter – 1 (Number Systems ) 

The NCERT Solutions in English Language for Class 9 Mathematics Chapter – 1 Number Systems Exercise 1.2 has been provided here to help the students in solving the questions from this exercise. 

Chapter 1: Number Systems

• NCERT Solution Class 9 Maths Ex – 1.1
• NCERT Solution Class 9 Maths Ex – 1.3
• NCERT Solution Class 9 Maths Ex – 1.4
• NCERT Solution Class 9 Maths Ex – 1.5
• NCERT Solution Class 9 Maths Ex – 1.6

Exercise – 1.2 

1. 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 √m where m is a natural number.
(iii) Every real number is an irrational number.

Answer –
(i) Every irrational number is a real number. (True)
∵ Real numbers = Rational numbers + Irrational numbers.

(ii) Every point on the number line is of the form √m where m is a natural number. (False)
∵ no negative number can be the square root of any natural number.

(iii) Every real number is an irrational number. (False)
∵ rational numbers are also present in the set of real numbers.

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.
Answer – 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. 

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

Step 1 – Let line OQ be of 2 unit on a number line.
Step 2 – At Q, draw a perpendicular line QP of length 1 unit.
Step 3 – Join OP
Step 4 – Now, OPQ is a right angled triangle. Applying Pythagoras theorem,
OQ² + QP² = OP²
2² + 1² = OP²
OP² = 5
⇒ OP = √5 .
Thus, OP is a line of length √5 unit.
Now, take O as centre OP = √5 as radius, draw an arc, which intersects the line at point R.
Hence, the point R represents √5.

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 OP₁ of unit length (see Fig. 1.9). Now draw a line segment P₂P₃ perpendicular to OP₂. Then draw a line segment P₃P₄ perpendicular to OP₃. Continuing in Fig. 1.9 :

Constructing this manner, you can get the line segment P_n-1P_n 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 P₂, P₃,….,P_n,… ., and joined them to create a beautiful spiral depicting √2, √3, √4, …
Answer –

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….