NCERT Solutions for Chapter 1 Number System Class 9 Maths

Chapter Name

NCERT Solutions for Chapter 1 Number System

Class

Class 9

Topics Covered

  • Rational and Irrational Numbers
  • Decimal Expansion of Rational Numbers and Irrational Numbers
  • Finding Rational Numbers between two rationals
  • Real Numbers

Related Study Materials

  • NCERT Solutions for Class 9 Maths
  • NCERT Solutions for Class 9
  • Revision Notes for Chapter 1 Number System Class 9 Maths
  • Important Questions for Chapter 1 Number System Class 9 Maths
  • MCQ for for Chapter 1 Number System Class 9 Maths

Short Revision for Ch 1 Number system Class 9 Maths 

  1. The collection of real numbers is made up by all the rationals and irrationals. 
  2. Every real number is either a rational or an irrational number.
  3. Every rational number can be written in the form p/q , where p and q are integers with q ≠ 0. 
  4. No irrational number can be written in the form p/q , where p and q are integers with q ≠ 0. 
  5. The decimal expansion of a rational number is either terminating or non - terminating repeating (recurring).
  6. The decimal expansion of an irrational is non-terminating non-repeating (recurring). 
  7. 0, 1 and 22/7 are rational numbers. 
  8. 2.833..... or 2.83 is rational numbers.
  9. 22/7 is not an exact value of π. 
  10. π is an irrational number. 
  11. 0.4040040004..... is an irrational number. 
  12. Every integer or a fraction made up by them is a rational number. 
  13. Square root of every prime number is an irrational number. 
  14. There are infinitely many rational numbers between any two distinct rational numbers. 
  15. All the rational and irrational numbers lie on the number line. 
  16. Every real number is represented by a unique point on the number line. 
  17. The sum or difference of a rational and an irrational number is an irrational number. 
  18. The product or quotient of a non - zero rational and an irrational number is an irrational number.

Exercise 1.1


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



2. Find six rational numbers between 3 and 4.


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

Solution


4. State whether the following statements are true or false. Give reasons for your answers.
(i) Every natural number is a whole number.
(ii) Every integer is a whole number.
(iii) Every rational number is a whole number.



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.


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.


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


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, …

Exercise 1.3 


1. Write the following in decimal form and say what kind of decimal expansion each has :
(i)36/100
(ii)1/11


(iv) 3/13
(v) 2/11
(vi) 329/400

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?

[Hint: Study the remainders while finding the value of 1/7 carefully.]


3. Express the following in the form p/q, where p and q are integers and q 0.
(i) 0. 
6
(ii) 0.47
(iii) 0.001


4. Express 0.99999…. in the form p/q . Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.


5. What can the maximum number of digits be in the repeating block of digits in the decimal expansion of 1/17 ? Perform the division to check your answer.

Thus, 1/17 = 0.0588235294117647
Hence, the required number of digits in the repeating block is 16.

6. Look at several examples of rational numbers in the form p/q (q ≠ 0), 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?


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


8. Find three different irrational numbers between the rational numbers 5/7 and 9/11.


9. Classify the following numbers as rational or irrational according to their type:
(i) √23 
(ii) 
√225
(iii) 0.3796
(iv) 7.478478.....
(v) 1.101001000100001......


Exercise 1.4


1. Visualise 3.765 on the number line, using successive magnification.



2. Visualise 4.26 on the number line, up to 4 decimal places. 

Point O in step 5 represents the number 4.26 on the number line.

Exercise 1.5


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) 2π


2. Simplify each of the following expressions : 
(i) (3+ √3)(2+√2)
(ii) (3+ √3)(3 - √3)
(iii) (√5 + √2)2 
(iv) (√5 - √2)(√5 + √2)


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?

Solution

On measuring c with any device, we get only approximate measurement. Therefore, π is an irrational.


4. Represent (√9.3) on the number line.


5. Rationalize the denominators of the following:
(i) 1/√7 
(ii) 1/(√7 -√6)
(iii) 1/(√5+√2)
(iv) 1/(√7-2)


Exercise 1.6


1. Find:
(i) 641/2
(ii) 321/5
(iii) 1251/3

2. Find : 
(i) 93/2
(ii) 322/5
(iii) 163/4
(iv) 125-1/3


3. Simplify :
(i) 22/3 ∙ 21/5
(ii) (1/33 )7
(iii) 111/2/111/4
(iv) 71/2 ∙ 81/2


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