** **The **NCERT Solutions for Class 9 Maths Chapter 1** Number System are created by the expert faculty of Study Circle. The **NCERT Solutions for Class 9 Maths Chapter 1** aims to provide the students with detailed and step-wise explanations for the answers to all the questions given in the exercises of this chapter.

**NCERT solutions for Class 9 Maths Chapter 1** help students to solve problems efficiently and efficiently for board exams. They also focus on preparing maths solutions in such a way that it is easy for students to understand and all students score very well all students score more than 90% in the exam

## ** **** Exercise 1.1**

** (****Class – 9 Maths NCERT Chapter -1 )**

**Q 1. Is zero a rational number? Can you write it in the form p/q, where p and q are integers and q ≠ 0?**

**Solution:**

**We know that a number is said to be rational if it can be written in the form p/q , where p and q are integers and q ≠ 0.**

**Taking the case of ‘0’,**

**Zero can be written in the form 0/1, 0/2, 0/3 … as well as , 0/1, 0/2, 0/3 ..**

**Since it satisfies the necessary condition, we can conclude that 0 can be written in the p/q form, where q can either be positive or negative number.**

**Hence, 0 is a rational number.**

** **

**Q 2. Find six rational numbers between 3 and 4.**

**Solution:**

**There are infinite rational numbers between 3 and 4.**

**As we have to find 6 rational numbers between 3 and 4, we will multiply both the numbers, 3 and 4, with 6+1 = 7 (or any number greater than 6)**

**i.e., 3 × (7/7) = 21/7**

**and, 4 × (7/7) = 28/7. The numbers in between 21/7 and 28/7 will be rational and will fall between 3 and 4.**

**Hence, 22/7, 23/7, 24/7, 25/7, 26/7, 27/7 are the 6 rational numbers between 3 and 4.**

** **

**Q 3. Find five rational numbers between 3/5 and 4/5.**

**Solution:**

**There are infinite rational numbers between 3/5 and 4/5.**

**To find out 5 rational numbers between 3/5 and 4/5, we will multiply both the numbers 3/5 and 4/5**

**with 5+1=6 (or any number greater than 5)**

**i.e., (3/5) × (6/6) = 18/30**

**and, (4/5) × (6/6) = 24/30**

**The numbers in between18/30 and 24/30 will be rational and will fall between 3/5 and 4/5.**

**Hence,19/30, 20/30, 21/30, 22/30, 23/30 are the 5 rational numbers between 3/5 and 4/5**

** **

**Q 4. State whether the following statements are true or false. Give reasons for your answers.**

**(i) Every natural number is a whole number.**

** **

**Solution:**

**True**

**Natural numbers- Numbers starting from 1 to infinity (without fractions or decimals)**

**i.e., Natural numbers = 1,2,3,4…**

**Whole numbers – Numbers starting from 0 to infinity (without fractions or decimals)**

**i.e., Whole numbers = 0,1,2,3…**

**Or, we can say that whole numbers have all the elements of natural numbers and zero.**

**Every natural number is a whole number; however, every whole number is not a natural number.**

** **

**(ii) Every integer is a whole number.**

**Solution:**

**False**

**Integers- Integers are set of numbers that contain positive, negative and 0; excluding fractional and decimal numbers.**

**i.e., integers= {…-4,-3,-2,-1,0,1,2,3,4…}**

**Whole numbers- Numbers starting from 0 to infinity (without fractions or decimals)**

**i.e., Whole numbers= 0,1,2,3….**

**Hence, we can say that integers include whole numbers as well as negative numbers.**

**Every whole number is an integer; however, every integer is not a whole number.**

** **

**(iii) Every rational number is a whole number.**

**Solution:**

**False**

**Rational numbers- All numbers in the form p/q, where p and q are integers and q≠0.**

**i.e., Rational numbers = 0, 19/30 , 2, 9/-3, -12/7…**

**Whole numbers- Numbers starting from 0 to infinity (without fractions or decimals)**

**i.e., Whole numbers= 0,1,2,3….**

**Hence, we can say that integers include whole numbers as well as negative numbers.**

**All whole numbers are rational, however, all rational numbers are not whole numbers.**

### ** **** Exercise 1.2**

** (****Class – 9 Maths NCERT Chapter -1****)**

**Q 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.**

** **

**Q 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).**

** **

**Q 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,**

**AB2+BC2 = CA2**

**22+12 = CA2 = 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.**

** **

**Q 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 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 :–**

**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….**

** **

## **EXERCISE 1.3**

** (****Class – 9 Maths NCERT Chapter -1****)**

** **

**Q 1. Write the following in decimal form and say what kind of decimal expansion each has :**

**(i) We have
**

**⇒ The decimal expansion of
is terminating.**

**(ii) Dividing 1 by 11, we have:**

**
**

**Note:**

**The bar above the digits indicates the block of digits that repeats. Here, the repeating block is 09.**

**Q- 2. You know that 1/7 = 0.142857. Can you predict what the decimal expansions of 2/7, 3/7, 4/7, 5/7, 6/7 are, without actually doing the long division? If so, how?**

**Solution:–**

Q -3. Express the following in the form p/q, where p and q are integers and q ≠ 0.

**Solution;–**

(ii)

(iii)

** Q- 4** Let x = 0.99999 . . . in the form
Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

**Q- 5 **** What can the maximum number of digits be in the repeating block of digits in the decimal expansion of **

**Perform the division to check your answer.**

**Solution;–**

**Q–6. Look at several examples of rational numbers in the form where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?**

** **

**Q–7. Write three numbers whose decimal expansions are non-terminating
non-recurring.**

** **

**Solution:–**

** **

** **

**Q–8 Find three different irrational numbers between the rational numbers
**

**Solution:–**

**Q–9**

**Classify the following numbers as rational or irrational:**

**Solution:–**

**EXERCISE 1.4**

** (****Class – 9 Maths NCERT Chapter -1)**

**1. Classify the following numbers as rational or irrational:**

**(i) 2 –√5**

**(ii) (3 +√23)- √23**

**(iii) 2√7/7√7**

**(iv) 1/√2**

**(v) 2****π**

**Solution–:**

** **

**Q– 2 Simplify each of the followings:**

**(i) (3+√3)(2+√2)**

**(ii)**

**(3+√3)(3-√3 )**

**(iii) (√5+√2)2**

**(iv) (√5-√2)(√5+√2)**

** **

**Q- 3. Recall, π is defined as the ratio of the circumference (say c) of a**

**circle**

**to its diameter, (say d). That is, π =c/d. This seems to**

**contradict the fact**

**that π is irrational. How will you resolve**

**this contradiction?**

**Q-4. Represent (√9.3) on the number line.**** **

**Q-5 Rationalise the denominator of the following;–**

**EXERCISE 1.5 (Class – 9 Maths NCERT Chapter -1 )**

**Q-1 Find :–**

**(iii) 1251/3**

**Q- 2 Find:–**

**Q- 3 Simplify:-**

** **