Exercise 1.1 - Complete Explanation Guide
Question 1: Identify Rational or Irrational Numbers
Concept Explanation:
Rational Numbers: Numbers that can be written as p/q where q ≠ 0
Terminating decimals (e.g., 0.5, 2.75)
Repeating decimals (e.g., 0.333…, 1.252525…)
All integers and fractions
Irrational Numbers: Numbers that cannot be written as p/q
Non-terminating, non-repeating decimals
Square roots of non-perfect squares
π (pi), e (Euler’s number)
(i) 2.353535
Type: ✅ Rational Number
Explanation:
Yeh number repeating decimal hai (353535 repeat ho raha hai)
Jo bhi number repeat hota hai, wo rational hota hai
Isay p/q ki form mein likha ja sakta hai
Iski rational form: 233/99 (approx)
(ii) 0.6
Type: ✅ Rational Number
Explanation:
Yeh terminating decimal hai (khatam hone wala decimal)
0.6 = 6/10 = 3/5
Jo bhi terminating decimal hai, wo rational hota hai
p/q form mein likha ja sakta hai (q ≠ 0)
(iii) 2.236067…
Type: ❌ Irrational Number
Explanation:
Yeh non-terminating, non-repeating decimal hai
Ye dots (…) show kar rahe hain ke ye number kabhi khatam nahi hota
Yeh asal mein √5 hai
√5 irrational hai kyunke 5 perfect square nahi hai
Isay p/q form mein nahi likha ja sakta
(iv) √7
Type: ❌ Irrational Number
Explanation:
√7 means “7 ka square root”
7 perfect square nahi hai (1²=1, 2²=4, 3²=9)
Jo number perfect square nahi hai, uska square root irrational hota hai
Iski decimal value: 2.64575131… (kabhi khatam nahi hoti)
(v) e
Type: ❌ Irrational Number
Explanation:
e (Euler’s number) ≈ 2.718281828…
Yeh ek famous irrational number hai
Non-terminating, non-repeating decimal hai
Mathematics mein natural logarithms mein use hota hai
(vi) π (pi)
Type: ❌ Irrational Number
Explanation:
π (pi) ≈ 3.1415926535…
Yeh sab se famous irrational number hai
Circle ke perimeter aur diameter ka ratio hai
Iski decimal value kabhi khatam nahi hoti aur koi pattern nahi repeat karta
(vii) 5 + √1
Type: ✅ Rational Number
Explanation:
√1 = 1 (kyunke 1 × 1 = 1)
5 + 1 = 6
6 ek integer hai aur sab integers rational hote hain
Isay 6/1 likha ja sakta hai
(viii) √3 + √13
Type: ❌ Irrational Number
Explanation:
√3 irrational hai (3 perfect square nahi)
√13 irrational hai (13 perfect square nahi)
Do irrational numbers ka sum bhi irrational hota hai
Isay p/q form mein nahi likha ja sakta
(ix) (2 – √2)(2 + √2)
Type: ✅ Rational Number
Explanation:
Yeh (a – b)(a + b) = a² – b² ka formula use kar raha hai
(2 – √2)(2 + √2) = 2² – (√2)²
= 4 – 2 = 2
2 ek integer hai, jo rational hai
Isay 2/1 likha ja sakta hai
(x) 15/4
Type: ✅ Rational Number
Explanation:
Yeh already p/q ki form mein hai
p = 15, q = 4 (q ≠ 0)
Isliye yeh rational number hai
Question 2: Represent Numbers on Number Line
Concept Explanation:
Number line ek straight line hoti hai jis par numbers ko show kiya jata hai. Zero center mein hota hai, right side positive numbers, left side negative numbers.
(i) √2 (Square root of 2)
How to represent:
Step 1: Ek number line banayein aur 0 aur 1 mark karein
Step 2: Point 1 se perpendicular line 1 unit upward draw karein
Step 3: 0 se perpendicular line ke end tak line draw karein (ye √2 hai)
Step 4: Compass se 0 se √2 tak ka distance measure karein
Step 5: Number line par 0 se right side ye distance mark karein
Explanation:
√2 ≈ 1.414 (1 aur 2 ke darmiyan)
Pythagoras theorem use hoti hai (1² + 1² = 2)
(ii) √3 (Square root of 3)
How to represent:
Step 1: Pehle √2 locate karein (upar wali method se)
Step 2: Point √2 se perpendicular line 1 unit upward draw karein
Step 3: 0 se perpendicular line ke end tak line draw karein (ye √3 hai)
Step 4: Compass se 0 se √3 tak ka distance measure karein
Step 5: Number line par 0 se right side ye distance mark karein
Explanation:
√3 ≈ 1.732 (1 aur 2 ke darmiyan)
√2 aur √3 dono irrational hain
(iii) 100/5
How to represent:
Step 1: Pehle simplify karein: 100 ÷ 5 = 20
Step 2: Number line par 0 aur 20 ke darmiyan distance measure karein
Step 3: 0 se right side 20 mark karein
Explanation:
100/5 = 20 (integer hai)
Sab integers number line par show ho sakte hain
(iv) -2 (Negative 2)
How to represent:
Step 1: Number line par 0 mark karein
Step 2: 0 se left side 2 units count karein
Step 3: Us point ko -2 mark karein
Explanation:
Negative numbers 0 ke left side show hote hain
-2 ka matlab hai 0 se 2 units left
(v) √8 (Square root of 8)
How to represent:
Step 1: Simplify karein: √8 = √(4 × 2) = 2√2
Step 2: Pehle √2 locate karein
Step 3: 0 se √2 tak ka distance measure karein aur 2 times karein
Step 4: Number line par 0 se right side ye distance mark karein
Explanation:
√8 ≈ 2.828 (2 aur 3 ke darmiyan)
Isay 2√2 likha ja sakta hai
Question 3: Express as Rational Number (p/q)
Concept Explanation:
Kisi bhi number ko p/q ki form mein likhna hai, jahan q ≠ 0.
(i) 0.4
Solution:
0.4 = 4/10
Simplify: 4/10 ÷ 2/2 = 2/5
✅ p = 2, q = 5
(ii) 0.37
Solution:
0.37 = 37/100
Already simplified (37 prime hai)
✅ p = 37, q = 100
(iii) 0.21
Solution:
0.21 = 21/100
Already simplified (21 aur 100 ka common factor 1 hai)
✅ p = 21, q = 100
Question 4: Name the Property Used
Concept Explanation:
Properties of real numbers ko pehchanana hai.
(i) (a + 4) + b = a + (4 + b)
Answer: ✅ Associative Property of Addition
Explanation:
Numbers ko different groups mein add karne se result same aata hai
Formula: (a + b) + c = a + (b + c)
(ii) √2 + √3 = √3 + √2
Answer: ✅ Commutative Property of Addition
Explanation:
Numbers ka order change karne se sum same aata hai
Formula: a + b = b + a
(iii) x – x = 0
Answer: ✅ Additive Inverse Property
Explanation:
Kisi bhi number mein uski negative value add karne se 0 aata hai
Formula: a + (-a) = 0
(iv) a(b + c) = ab + ac
Answer: ✅ Distributive Property
Explanation:
Multiplication addition par distribute hoti hai
Formula: a × (b + c) = a × b + a × c
(v) 16 + 0 = 16
Answer: ✅ Additive Identity Property
Explanation:
Kisi bhi number mein 0 add karne se wahi number aata hai
Formula: a + 0 = a
(vi) 100 × 1 = 100
Answer: ✅ Multiplicative Identity Property
Explanation:
Kisi bhi number mein 1 multiply karne se wahi number aata hai
Formula: a × 1 = a
(vii) 4 × (5 × 8) = (4 × 5) × 8
Answer: ✅ Associative Property of Multiplication
Explanation:
Numbers ko different groups mein multiply karne se result same aata hai
Formula: (a × b) × c = a × (b × c)
(viii) ab = ba
Answer: ✅ Commutative Property of Multiplication
Explanation:
Numbers ka order change karne se product same aata hai
Formula: a × b = b × a
Question 5: Name the Property Used
(i) -3 < -10 < 2
Answer: ✅ Transitive Property of Inequality
Explanation:
Agar a < b aur b < c, toh a < c
Yahan: -3 < -10? Check karein: -3 > -10 (ye galat hai!)
Asal mein: -10 < -3 < 2 hona chahiye tha
Correction: Yeh statement incorrect hai kyunke -3, -10 se bara hai.
(ii) If a < b then 1/a > 1/b
Answer: ✅ Reciprocal Property of Inequality
Explanation:
Agar a < b (positive numbers), toh 1/a > 1/b
Example: 2 < 5, toh 1/2 > 1/5 (0.5 > 0.2)
(iii) If a < b then a + c < b + c
Answer: ✅ Addition Property of Inequality
Explanation:
Dono sides mein same number add karne se inequality same rehti hai
Formula: If a < b, then a + c < b + c
(iv) If ac < bc and c > 0 then a < b
Answer: ✅ Multiplication Property of Inequality (Positive)
Explanation:
Positive number se divide karne se inequality same rehti hai
c > 0 hai, isliye a < b
(v) If ac < bc and c < 0 then a > b
Answer: ✅ Multiplication Property of Inequality (Negative)
Explanation:
Negative number se divide karne se inequality reverse ho jati hai
c < 0 hai, isliye a > b (inequality flip ho gayi)
(vi) Either a > b or a = b or a < b
Answer: ✅ Trichotomy Property
Explanation:
Do numbers ke beech mein 3 possibilities hain
Ya toh a bara hai, ya equal hai, ya chota hai
Koi aur option nahi hai
Question 6: Insert Two Rational Numbers Between
Concept Explanation:
Do numbers ke darmiyan rational numbers find karne hain.
(i) Between 1/3 and 4/5
Step 1: LCM find karein (3, 5) = 15
Step 2: Denominator same karein
1/3 = 5/15
4/5 = 12/15
Step 3: 5/15 aur 12/15 ke darmiyan numbers find karein
6/15, 7/15, 8/15, 9/15, 10/15, 11/15
Step 4: Simplest form mein likhein
6/15 = 2/5
7/15
✅ Answer: 2/5 and 7/15 (among many)
(ii) Between 3 and 4
Step 1: 3 aur 4 ke darmiyan numbers find karein
Step 2: Common denominator use karein (e.g., 2)
3 = 6/2
4 = 8/2
Step 3: 6/2 aur 8/2 ke darmiyan
7/2
Step 4: Another number find karein (denominator 3)
3 = 9/3, 4 = 12/3
10/3, 11/3
✅ Answer: 7/2 and 10/3 (among many)
(iii) Between 1/4 and 5/4
Step 1: Denominator same hai (4)
Step 2: 1/4 aur 5/4 ke darmiyan numbers
2/4 = 1/2
3/4
✅ Answer: 1/2 and 3/4 (among many)