**Exercise 1.1**

**Question 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 1:**

**Question 2: Find five rational numbers between 1 and 2.**

**Solution 2:**

**Question 3:**

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

**Solution 3:**

**Question 4:**

**Find five rational number between 3/5 and 4/5**

**Solution 4:**

**Question 5:**

**Are the following statements true or false? Give reason for your answer.**

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

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

**(iii) Every rational number is an integer.**

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

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

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

**Solution 5:**

**Exercise 1.2**

**Question 1:**

**Express the following rational number as decimals:**

**(i) 42/100 (ii) 327/500 (iii) 15/4**

**Solution 1:**

**Question 2:**

**Express the following rational number as decimals:**

**(i) 2/3 (ii) - 4/9 (iii) - 2/15 (iv) - 22/13 (v) 437/999 (vi) 33/26**

**Solution 2:**

**Question 3: Look at several examples of rational number 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. Can you guess what property q must satisfy?**

**Solution 3:**

**Exercise 1.3**

**Question 1: Express each of the following decimals in the form p/q:**

**(i) 0.39 (ii) 0.750 (iii) 2.15 (iv) 7.010 (v) 9.90 (vi) 1.0001**

**Solution 1:**

**Question 2: Express each of the following decimals in the form p/q:**

**Solution 2:**

**Exercise 1.4**

**Question 1: Define an irrational number.**

**Solution 1:**

**Question 2: Explain how irrational numbers differ from rational numbers?**

**Solution 2:**

**Question 3: Examine, whether the following number are rational or irrational:**

**(i) √7 (ii) √4 (iii) 2 + √3 (iv) √3 + √2 (v) √3 + √5 (vi) (√2 – 2)**

^{2 }(vii) (2 – √2) (2 +√2) (viii) (√2 – √3)^{2}(ix) √5-2 (x) √23 (xi) √225 (xii) 0.3796 (xiii) 7.478478… (xiv) 1.101001000100001…..**Solution 3:**

^{2}

^{2}– 2ab + b

^{2}

^{2}– 2(√2) (2) + (2)

^{2}

^{2}– b

^{2}= (a+ b) (a- b)

^{2}– (√2)

^{2}

^{2}

^{2}= a

^{2}+ 2 ab + b

^{2}

^{2}+ 2(√3) (√2) + (√2)

^{2}

^{2}is an irrational number.

**Question 4. Identify the following as rational or irrational numbers. Give the decimal representation of rational number.**

**(i) √4 (ii) 3√(18 ) (iii) √1.44 (iv) √(9/27) (v) - √64 (vi) √100**

**Solution 4:**

**Question 5: In the following equation, find which variables x, y, z etc. represent rational or irrational numbers:**

**(i) x^{2 }= 5**

**(ii) y^{2} = 9**

**(iii) z^{2 }= 0.04**

**(iv) u^{2} = 17/4**

**(v) v^{2 }= 3**

**(vi) w^{2 }= 27**

**(vii) t^{2} = 0.4**

**Solution 5:**

^{2}= 5

^{2}= 5 is an irrational number.

^{2}= 9

^{2}= 9 is a rational number.

^{2}= 0.04

^{2}= 0.04

^{2}= 004/100

^{2}= 17/4

^{2}= 17/4

^{2}= 17/4 is also an irrational number.

^{2}= 3

^{2}= 3 is also an irrational number.

^{2}= 27

^{2}= 27 is also an irrational number.

^{2}= 0.4

^{2}= 4/10

^{2}= 0.4 is also an irrational number.

**Question 6: Give two rational number lying between 0.232332333233332…. and 0.212112111211112....**

**Solution 6:**

**Question 7: Give two rational number lying between 0.515115111511115…. and 0.5353353335…..... .**

**Solution 7:**

**Question 8: Find one irrational number between 0.2101 and 0.2222…… = 0.2.**

**Solution 8:**

**Question 9: Find a rational number and also an irrational number lying between the numbers 0.3030030003…… and 0.3010010001……….. .**

**Solution 9:**

**Question 10: Find three different irrational number between the rational numbers 5/7 and 9/11 .**

**Solution 10:**

**Question 11: Give an example of each, of two irrational number whose:**

**(i) Difference is a rational number.**

**(i) Difference is an irrational number.**

**(ii) Difference in an irrational number.**

**(iii) Sum in a rational number.**

**(iv) Sum is an irrational number.**

**(v) Product in a rational number.**

**(vi) Product in an irrational number.**

**(vii) Quotient in a rational number.**

**(viii) Quotient in an irrational number.**

**Solution 11:**

**Question 12: Find two irrational numbers between 0.5 and 0.55 .**

**Solution 12:**

**Question 13: Find two irrational number lying between 0.1 and 0.12 .**

**Solution 13:**

**Question 14: Prove that √3+√5 is an irrational number.**

**Solution 14:**

^{2}= (√3+√5)

^{2}

^{2}= x

^{2}+ 2xy + y

^{2}

^{2}= (√3+√5)

^{2}

^{2}= (√3)

^{2}+ 2√3×√5 + (√5)

^{2}

^{2}= 3 + 2√15 + 5

^{2}= 8 + 2√15

^{2}– 8 = 2√15

^{2}-8)/2 = √15

^{2}-8)/2 = √15

^{2}-8)/2 is rational number. But we know that √15 is an irrational number. Hence our assumption is wrong √3+√5 is an irrational number.

**Exercise 1.5**

**Question 1: Complete the following sentences:**

**(i) Every point on the number line corresponds to a _________ number which many be either ___________ or __________.**

**(ii) The decimal form of an irrational number is neither __________ nor ___________.**

**(iii) The decimal representation of a rational number is either ___________ or ___________.**

**(iv) Every real number is either ___________ number or ___________ number.**

**Solution 1:**

(i) Every point on the number line corresponds to a** real **number which many be either **rational **or** irrational **.

(ii) The decimal form of an irrational number is neither** terminating **nor** repeating **.

(iii) The decimal representation of a rational number is either** terminating, non-terminating** or** recurring **.

(iv) Every real number is either** a rational **number or** an irrational **number.

**Question 2: Represent √6, √7, √8 on the number line.**

**Solution 2:**

**Representation of √6 on number line:-**

**Steps of making √6 on number line:-**

^{2}= OB

^{2}+ BC

^{2}

^{2}= (2)

^{2}+ (1)

^{2}

^{2}= 4 + 1

^{2}= 5

^{2}= OD

^{2}+ DE

^{2}

^{2}= (√5)

^{2}+ (1)

^{2}

^{2}= 5 + 1

^{2}= 6

**(ii) Representation of √7 on number line:-**

^{2}= OF

^{2}+ FG

^{2}

^{2}= (√6)

^{2}+ (1)

^{2}

^{2}= 6 + 1

^{2}= 7

**(iii) Representation of √8 on number line:-**

^{2}= OA

^{2}+ AB

^{2}

^{2}= (2)

^{2}+ (2)

^{2}

^{2}= 4 + 4

^{2}= 8

**Question 3: Represent√3.5, √9.4, √10.5 on the real number line.**

**Solution 3:**

**(i) Representation of √(3.5) on number line:-**

^{2}= OD

^{2}– OB

^{2}

^{2}= OD

^{2}– (OC – BC)

^{2}

^{2}= OD

^{2}– (OC – BC)

^{2}

^{2}= OD

^{2}– {(OC)

^{2}– 2(OC)(BC) + (BC)

^{2}}

^{2}= OD

^{2}– OC

^{2}+ 2(OC)(BC) – BC

^{2 }[we know that OC = OD]

^{2}= 2(OC)(BC) – BC

^{2}

^{2}= 2(2.25)(1) – (1)

^{2}

**(ii) Representation of √(9.4) on number line:-**

^{2}= OD

^{2}– OB

^{2}

^{2}= OD

^{2}– (OC – BC)

^{2}

^{2}= OD

^{2}– (OC – BC)

^{2}

^{2}= OD

^{2}– {(OC)

^{2}– 2(OC)(BC) + (BC)

^{2}}

^{2}= OD

^{2}– OC

^{2}+ 2(OC)(BC) – BC

^{2}[we know that OC = OD]

^{2}= 2(OC)(BC) – BC

^{2}

^{2}= 2(5.2)(1) – (1)

^{2}

**(iii) Representation of √(10.5) on number line:-**

^{2}= OD

^{2}– OB

^{2}

^{2}= OD

^{2}– (OC – BC)

^{2}

^{2}= OD

^{2}– (OC – BC)

^{2}

^{2}= OD

^{2}– {(OC)

^{2}– 2(OC)(BC) + (BC)

^{2}}

^{2}= OD

^{2}– OC

^{2}+ 2(OC)(BC) – BC

^{2 }[we know that OC = OD]

^{2}= 2(OC)(BC) – BC

^{2}

^{2}= 2(5.75)(1) – (1)

^{2}

**Question 4: Find whether the following statements are true or false.**

**(i) Every real number is either rational or irrational.**

**(ii) π is an irrational number.**

**(iii) Irrational numbers cannot be represented by point on the number line.**

**Solution 4:**

**Exercise 1.6**

**Question 1: Visualise 2.665 on the number line, using successive magnification.**

**Solution 1:**

**Question 2: Visualise the representation of 5.37̅ on the number line upto 5 decimal places, that is upto 5.37777.**

**Solution 2:**