EXERCISE 1.2                                                                              PAGE:8

Question 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 formNCERT solution for class 9 Maths Chapter-1 Number Systems/image001.png, 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.

We know that irrational numbers are the numbers that cannot be converted in the formNCERT solution for class 9 Maths Chapter-1 Number Systems/image002.png, where p and q are integers andNCERT solution for class 9 Maths Chapter-1 Number Systems/image003.png.

We know that a real number is the collection of rational numbers and irrational numbers.

Therefore, we conclude that, yes every irrational number is a real number.

 

(ii) Consider a number line. We know that on a number line, we can represent negative as well as positive numbers.

We know that we cannot get a negative number after taking square root of any number.

Therefore, we conclude that not every number point on the number line is of the formNCERT solution for class 9 Maths Chapter-1 Number Systems/image001.png, where m is a natural number.

 

(iii) Consider the irrational numbers and the real numbers separately.

We know that irrational numbers are the numbers that cannot be converted in the formNCERT solution for class 9 Maths Chapter-1 Number Systems/image002.png, where p and q are integers andNCERT solution for class 9 Maths Chapter-1 Number Systems/image003.png.

We know that a real number is the collection of rational numbers and irrational numbers.

So, we can conclude that every irrational number is a real number. But every real number is not an irrational number.

Therefore, we conclude that, every real number is not a rational number.

 

Question 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 : We know that square root of every positive integer will not yield an integer.

We know thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image004.png is 2, which is an integer. But,NCERT solution for class 9 Maths Chapter-1 Number Systems/image005.pngwill give an irrational number.

Therefore, we conclude that square root of every positive integer is not an irrational number.

 

Question 3. Show how √5 can be represented on the number line
Solution :
Draw a number line and take point O and A on it such that OA = 1 unit. Draw BA ⊥ OA as BA = 1 unit. Join OB = √2 units.
Now draw BB1 ⊥ OB such that BB1 =1 unit. Join OB1 = √3 units.
Next, draw B1B2⊥ OB1such that B1B2 = 1 unit.
Join OB2 = units.
Again draw B2B3 ⊥OB2 such that B2B3 = 1 unit.
Join OB3 = √5 units.
NCERT Solutions for Class 9 Maths Chapter 1 Number Systems
Take O as centre and OB3 as radius, draw an arc which cuts the number line at D.
Point D
represents √5 on the number line.

Question 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 :

Ncert solution class 9 chapter 1-2

Constructing this manner, you can get the line segment Pn-1Pn by square root spiral drawing a line segment of unit length perpendicular to OPn-1. In this manner, you will have created the points P2, P3,….,Pn,… ., and joined them to create a beautiful spiral depicting √2, √3, √4, …

Solution:

Ncert solution class 9 chapter 1-3

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….