Book Name |
: Ganit Prakash |

Subject |
: Mathematics (Maths) |

Class |
: 9 (Madhyamik/WB) |

Publisher |
: Prof. Nabanita Chatterjee |

Chapter Name |
: Real Numbers (1st Chapter) |

**Let us work out – 1.1**

**Question 1**

**Let us write the definition of rational numbers and also write 4 rational numbers.**

**Solution:**

**Rational numbers:** The numbers which can be expressed in the form of P/q where p and q are integers and q ≠ 0 are called Rational Numbers.

The 4 rational numbers are \frac{2}{3}, \frac{3}{4}, \frac{-3}{5} and \frac{6}{7}

**Question 2**

**Is 0 a rational number? Let us express 0 in the form of P/q [where p & q are integers and q ≠ 0 and P & q have no common factors]**

**Solution:**

No,

0 = \frac{0}{1} **(Ans)**

**Question 3**

**Let me place the following rational numbers on the Number line: **

**(i) 7 **

**(ii) -4 **

**(iii) \frac{3}{5} **

**(iv)\frac{9}{2} **

**(v) \frac{2}{9} **

**(vi) \frac{11}{5} **

**(vii) \frac{-13}{4}**

**Solution:**

**Question 4**

**Let me write one rational number lying between two numbers given below and place them on the Number Line. **

**(i) 4 & 5 **

**(ii) 1 & 2 **

**(iii) \frac{1}{4} & \frac{1}{2}**

** (iv) -1 & \frac{1}{2} **

**(v) \frac{1}{4} & \frac{1}{3} **

**(vi) -2 & -1**

**Solution:**

(i) 4 & 5

One rational number = \frac{4+5}{2} = \frac{9}{2}

(ii) 1 & 2

One rational number = \frac{2+1}{2} = \frac{3}{2}

(iii) \frac{1}{4} & \frac{1}{2}

One rational number = \frac{\frac{1}{2}+ \frac{1}{4}}{2} = \frac{ \frac{2+1}{4} }{2} = \frac{ \frac{3}{4} }{2} = \frac{3}{8}

(iv) -1 & \frac{1}{2}

One rational number = \frac{-1+\frac{1}{2} }{2} = \frac{\frac{-2+1}{4}}{2} = \frac{-\frac{1}{2}}{2} = \frac{-1}{4}

(v) \frac{1}{4} & \frac{1}{3}

One rational number = \frac{\frac{1}{4} + \frac{1}{3}}{2} = \frac{\frac{3 + 4}{12}}{2} = \frac{\frac{7}{12}}{2} = \frac{7}{24}

(vi) -2 & -1

One rational number = \frac{(-2) + (-1)}{2} = \frac{-2-1}{2} = \frac{-3}{2}

**Question 5**

**Let me write 3 rational numbers lying between 4 & 5 and place them on Number Line.**

**Solution:**

4 + \frac{1}{2} = \frac{9}{2},

4 + \frac{1}{3} = \frac{13}{3},

4 +\frac{1}{4} = \frac{17}{4}**Question 6**

**Let me write 6 rational numbers lying between 1 & 2 and place them on the Number Line. **

**Solution:**

**We have,**

1 + \frac{1}{2} = \frac{3}{2},

1 + \frac{1}{3} = \frac{4}{3},

1 +\frac{1}{4} = \frac{5}{4},

1 + \frac{1}{5} = \frac{6}{5},

1 + \frac{1}{6} = \frac{7}{6},

1 + \frac{1}{7} = \frac{8}{7}**Question 7**

**Let me write 3 rational numbers lying between \frac{1}{5} & \frac{1}{4}**

**Solution:**

1st. rational numbers :

= \frac{\frac{1}{5}+\frac{1}{4} }{2}

= \frac{\frac{4+5}{20} }{2}

= \frac{\frac{9}{20}}{2}

= \frac{9}{40} (Ans.)

2nd. rational numbers :

= \frac{\frac{1}{5}+\frac{9}{40} }{2}

= \frac{\frac{8+9}{40}}{2}

= \frac{\frac{17}{40}}{2}

= \frac{17}{80} (Ans.)

3rd. rational numbers :

= \frac{\frac{9}{40}+\frac{1}{4} }{2}

= \frac{\frac{9+10}{40}}{2}

= \frac{\frac{19}{40} }{2}

= \frac{19}{80} (Ans.)

**Question 8**

**Let me put (T) if the statement is true and write (F) if the statement is false.**

- By adding and multiplying two integers, we get integers.
- By dividing two integers, we get an integer.

**Solution:**

(i) By adding and multiplying two integers, we get integers. (True)

(ii) By dividing two integers, we get an integer. (False)

**Question 9**

**Let me see and write what I will get by adding, subtracting, multiplying and dividing (divisor is non-zero) two rational numbers.**

**Solution:**

Rational Number.

Table of Contents

Toggle**Let us work out – 1.2**

**Question 1**

**Let us write the true or false statement from the following:**

- The sum of two rational numbers will always be rational
- The sum of two irrational numbers will always be irrational
- The product of two rational numbers will always be rational.
- The product of two irrational numbers will always be rational.
- Every rational number must be real.
- Every real number must be irrational.

**Solution:**

- True
- False
- True
- False
- True
- False

**Question 2**

**What is meant by an irrational number? Let me understand Let me write 4 irrational numbers.**

**Solution:**

Irrational numbers: The numbers that cannot be expressed as p/q where p and q are integers and q ≠ 0 are called Irrational Numbers.

The four irrational numbers are √2, √5, √з & \pi (Ans)

**Question 3**

**Let us write rational and irrational numbers from the following:**

- √9
- √225
- √7
- √ 50
- √100
- -√81
- √42
- √29
- – √1000

**Solution:**

- √9 = √3 × 3 = 3 It is a rational number
- √225 = √15 × 15 = 5 It is a rational number
- √7 It is an irrational number
- √50 = √5 × 5 × 2 = 5√2 It is an irrational number
- √100 = √10 × 10 = 10 It is a rational number
- -√81 =-√9 × 9 = -9 It is a rational number
- √42 It is an irrational number
- √29 It is an irrational number
- -√1000 = √10 × 10 × 10 -10√10. It is an irrational number.

**Question 4**

**Let me place √5 on number line.**

**Solution:**

By the Pythagoras theorem, we get,

OB = \sqrt{2^2 + 1^2} = \sqrt{4+1} = \sqrt{5} [ … we take AB = 1 unit and AB ⊥ OA]

Assuming O is the center and Radius of as a OB; an arc is drawn which intersects the number line at ‘P’,

Then, OP = √5 unit.

**Question 5**

**Let me place √3 on Number Line.**

**Solution:**

OD = √OB^{2} + BD^{2} = √(√2)^{2} + 1^{2} units = √3 units.

Taking center at the point O and radius OD.

an arc is drawn which intersects number line at Q.

∴ OQ = √3 unit

**Question 6**

**Let me place √5, √6, √7, -√6, -√8, -√11 on same Number Line.**

**Solution:**

**Let us work out –**** 1.****3**

**Question 1**

**Without division let’s find from the following which numbers have terminating decimals and write them.**

- \frac{17}{80},
- \frac{13}{24},
- \frac{17}{12},
- \frac{16}{125},
- \frac{4}{35}

**Solution:**

**Fractions Terminating decimal/non-terminating decimal**

- \frac{17}{80} : Terminating decimals
- \frac{13}{24} : Non-Terminating Decimals
- \frac{17}{12} : Non-Terminating Decimals
- \frac{16}{125} : Terminating decimals
- \frac{4}{35} : Non-Terminating Decimals

**Question 2**

**Let us expand each number given below into decimals and write the type of decimal expansions**

- \frac{1}{11},
- \frac{5}{8},
- \frac{3}{13},
- 3\frac{1}{8},
- \frac{2}{11},
- \frac{7}{25}

**Solution:**

**Question 3**

**Let us express each of the following numbers in the form of P/q where P and q are integers and q ≠ 0**

**Solution:**

(i) 0.\dot{3}

\frac{3}{9} = \frac{1}{3} (Ans.)

(ii) 1.\dot{3}

\frac{13-1}{9} = \frac{12}{9} = \frac{4}{3} (Ans.)

(iii) 0.5\dot{4}

\frac{54-5}{90} = \frac{49}{90} (Ans.)

(iv) 0.\dot{3} \dot{4}

\frac{34}{99} (Ans.)

(v) 3.\dot{1}\dot{4}

\frac{314-3}{99} = \frac{311}{99}

(vi) 0.1\dot{7}

\frac{17-1}{90} = \frac{16}{90}= \frac{8}{45}

(vii) 0.4\dot{7}

\frac{47-4}{90} = \frac{43}{90}

(viii) 0.\dot{5}\dot{4}

\frac{54}{99} = \frac{6}{11}

(ix) 0.\dot{0}0\dot{1}

\frac{1}{999}

(x) 0.\dot{1}6\dot{3}

\frac{163}{999}

**Question 4**

**Let’s write four numbers whose decimal expansions are non-terminating and non-recurring.**

**Solution:**

- 0.4504500450004………….
- 0.8080080008………….
- 0.85855855585555………….
- 091911911191111………….

**Question 5**

**Let us write 3 different irrational numbers lying between \frac{5}{7} & \frac{9}{7}**

**Solution:**

\frac{5}{7} = 0.714285,

\frac{9}{7} = 1.285714

The 3 different irrational numbers lying between \frac{5}{7} and \frac{9}{7} are

- 0.7142850714285000……………..
- 1.49049004900049………………….
- 1.2857140285714000………………….

**Question 6**

**Let us write 2 different irrational numbers lying between \frac{3}{7} & \frac{1}{11}**

**Solution:**

The two different irrational numbers lying between \frac{3}{7} and \frac{1}{11} are

- 0.428570 4285700 42857……………….
- 0.09 009 0009………………………

**Question 7**

**Let us write the rational and irrational numbers for the following: **

- \sqrt{47}
- \sqrt{625}
- 6.5757…..
- 1.1010010001……..

**Solution:**

- \sqrt{47} (Irrational Numbers)
- \sqrt{625} (Rational)
- 6.5757….. (Rational)
- 1.1010010001…….. (Irrational Numbers)

**Question 8**

**Let us place the following numbers on number line: **

**5.762****2.321****1.052****4.178**

**Solution:**

(i) 5.762

(ii) 2.321

(iii) 1.052

(iv) 4.178

**Question 9**

**Let us place the two numbers 2.\dot{2} \dot{6} & 5.5\dot{4} upto 4 decimal places on number line.**

**Solution:**

(i) 2.\dot{2} \dot{6} = 2.2626 ………….

(ii) 5.5\dot{4} = 5.54444……….

**Question 10**

**Let us write two rational numbers lying between 0.2323232323232 and 0.2121212121212121….**

**Solution:**

Let us suppose a = 0.23 233 2333 233332….. and b = 0.212112111211112.. a & b non-terminating and a non-recurring decimal number.

Both a and b have the number 2 in the first decimal place after the decimal point, but the number in the second decimal place of a is 3 and that of b is 1 so a > b.

Let c = 0.21 and d = 0.2121

Hence c & d are rational.

So, the two rationals lying between a & b are 0.21 & 0.2121. The decimal expansion of all real numbers is possible.

**Question 11**

**Let us write the two rational numbers lying between 0.2101 & 0.2222…. or 0.\dot{2}**

**Solution:**

The two rational numbers lying between 0.2101 & 0.2222 or 0.2 are 0.219 & 0.22 (Ans.)

**Question 12**

**Let us write 10 right and 10 wrong statements about natural numbers, whole numbers, integers, rational & irrational numbers and real numbers.**

**Solution:**

**Right statement: **

- If we take any two other real numbers and add, subtract, multiply and divide, we will always get a real number.
- If we take any two other natural numbers and add, subtract, and multiply, we will always get a natural number.
- The addition & Subtraction of two irrational numbers are not always irrational numbers.
- Multiplication & division of two irrational numbers is not always an irrational number.
- Addition, Subtraction and Multiplication of two integers are not integers.

**Wrong statement:**

- Zero is not an integer.
- Fraction is possible in natural numbers.
- The division of two natural numbers is always natural numbers.
- The division of two integers is always an integer.
- Addition, Subtraction, Multiplication & Divide of two irrational numbers are always irrational numbers.

**Question 13**

**Let us find, how much rupees will be required to determine the values of the following expressions of Rs.2 for one multiplication and Rs.1 for one addition is required and let us also see which law can be applied to find out the value of the expression with the least amount of money.**

- 3
*x*^{2}+ 2*x*+ 1, when*x*= 5 - 2
*x*^{3}+ 3*x*^{2}+ 2*x*+ 3, when*x*= 7

**Solution:**

(i) 3*x*^{2}+ 2*x* + 1, when *x* = 5

3*x*^{2} + 2*x* + 1 = 3 × 5 × 5 + 2 × 5+ 1, here numbers of multiplications and additions are 3 & 2 respectively, So Rs.8 is required.

But if we write 3*x*^{2} + 2*x* + 1 = *x* (3*x *+ 2) + 1 by applying the law, then for two multiplications and two additions Rs.6 is required.

(ii) 2*x*^{3} + 3*x*^{2} + 2*x* + 3, when *x* = 7

= 2 × 7^{3} + 3 × 7^{2} + 2 × 7 + 3

= 2 × 7 × 7 × 7 + 3 × 7 × 7 + 2 × 7 + 3

Here numbers of multiplications and additions are 6 and 3. Respectively, So, Rs.15 is required.

But if we write 2*x*^{3} + 3*x*^{3} + 2*x* + 3 = *x*(2*x*^{2} + 3*x *+ 2) + 3 by applying distributive law, then for three multiplications and three additions Rs.9 is required.

**Question 14**

**Multiple Choice Questions (M.C.Q) :**

**Question 14 (i)**

** The decimal expansion of √5 is**

**a terminating decimal,****a terminating or recurring decimal****a non-terminating and non-recurring decimal.****None of them.**

**Solution:**

(a) A terminating decimal.

**Question 14 (ii)**

**The product of two irrational numbers is**

**always irrational number****always rational number****always an integer****rational or irrational number**

**Solution:**

(a) always an irrational number.

**Question 14 (iii)**

*\pi *and \frac{22}{7} are

**(a) always rational number**

**(b) always irrational number**

**(c) \pi is rational and \frac{22}{7} irrational**

**(d) \pi is irrational and \frac{22}{7} irrational**

**Solution:**

(d) *\pi * is irrational and \frac{22}{7} irrational

**Question 14 (iv)**

**Between two rational numbers, there exists**

**a rational number****only one rational number****infinite number of rational numbers****no irrational number**

**Solution:**

(c) infinite number of rational numbers.

**Question 14 (v)**

**Between two irrational numbers, there exists**

**no rational number****only one rational number****infinite number of rational numbers****no irrational number.**

**Solution:**

(c) infinite number of irrational numbers.

**Question 14 (vi)**

**The number 0 is**

**whole number but not an integer****integer but not rational****rational but not a real number****whole number, integer, rational and real number but not irrational**

**Solution:**

(d) whole number, integer, rational and real number but not irrational

**Question 15**

**Short answer type question:**

**Question 15 (i) **

**Let us write a number where the sum of two is irrational rational.**

**Solution:**

(2 + √3) + (2 – √3) = 4

**Question 15 (ii) **

**Let us write a number where the difference of two irrationals is a rational number:**

**Solution:**

(√3 + 2) – (√3 – 2) = 4

**Question 15 (iii) **

**Let us write a rational number lying between \frac{1}{7} & \frac{2}{7}**

**Solution:**

**Question 15 (iv) **

**Let us write an irrational number lying between \frac{1}{7} & \frac{2}{7}**

**Solution:**

0.\dot{2}8571\dot{4}

The irrational number lying between

\frac{1}{7} & \frac{2}{7} = 2857140 28571400 285714……..

**Question 15 (v) **

**Let us write a common fraction of the recurring decimal 0.012\dot{3}**

**Solution:**

0.012\dot{3}

= \frac{123-12}{999}

= \frac{111}{999}

= \frac{1}{9} (Ans.)