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Class 8 Maths Notes Rational Numbers

🔢 Chapter 1: Rational Numbers (Class 8 Maths) 🧮

📌 What are Rational Numbers?

Rational Number: A number which can be written in the form pq, where p and q are integers and q ≠ 0.

Parimey Sankhya: Ek number jise p/q form mein likh sakte hain, jahan p aur q integers hain aur q zero nahi hai.

Examples: 12, -34, 5 (means 5/1), 0 (means 0/1), -2 (means -2/1).

All fractions, integers, whole numbers, and natural numbers are rational numbers.

Sabhi fractions, integers, whole numbers, natural numbers rational hote hain.

⚙️ Properties of Rational Numbers (Gundharm)

✅ 1. Closure Property (Samvrit)

Check if performing an operation (like +, -, ×, ÷) on two rational numbers gives a rational number.

Check karte hain ki kya do rational numbers par koi operation karne se rational number hi milta hai.
  • Addition: YES. ab + cd is rational. 👍
  • Subtraction: YES. ab - cd is rational. 👍
  • Multiplication: YES. ab × cd is rational. 👍
  • Division: NO. Division by zero is not defined (ab ÷ 0 not allowed). Otherwise, yes. 👎 (Except division by 0)
Closure Examples

(See previous version’s examples if needed, presented vertically if complex)

🔄 2. Commutativity Property (Kramvinimeyata)

Check if changing the order changes the result (a op b = b op a).

Check karte hain ki order badalne se result badalta hai ya nahi.
  • Addition: YES. ab + cd = cd + ab. ✅
  • Subtraction: NO. ab - cd ≠ cd - ab. ❌
  • Multiplication: YES. ab × cd = cd × ab. ✅
  • Division: NO. ab ÷ cd ≠ cd ÷ ab. ❌
Commutativity Example (Subtraction)

Is 12 - 14 = 14 - 12?

LHS = 24 - 14 = 14

RHS = 14 - 24 = -14

LHS ≠ RHS. Not commutative. ❌

🔗 3. Associativity Property (Sahacharyata)

Check if changing grouping changes the result ((a op b) op c = a op (b op c)).

Check karte hain ki grouping badalne se result badalta hai ya nahi.
  • Addition: YES. (ab + cd) + ef = ab + (cd + ef). ✅
  • Subtraction: NO. ❌
  • Multiplication: YES. (ab × cd) × ef = ab × (cd × ef). ✅
  • Division: NO. ❌
Associativity Example (Multiplication)

Check (-23 × 35) × 54 = -23 × (35 × 54).

LHS = -615 × 54 = -25 × 54 = -1020 = -12

RHS = -23 × (1520) = -23 × 34 = -612 = -12

LHS = RHS. Associative. ✅

0️⃣ 4. Role of Zero (Additive Identity)

Zero (0) is the additive identity because adding 0 to any rational number doesn’t change it.

Zero (0) additive identity hai kyunki kisi bhi rational number mein 0 jodne se woh badalta nahi hai.

ab + 0 = ab

1️⃣ 5. Role of One (Multiplicative Identity)

One (1) is the multiplicative identity because multiplying any rational number by 1 doesn’t change it.

One (1) multiplicative identity hai kyunki kisi bhi rational number ko 1 se guna karne par woh badalta nahi hai.

ab × 1 = ab

➖➕ 6. Negative of a Number (Additive Inverse / Yojya Pratilom)

The additive inverse of ab is -ab such that their sum is 0.

a/b ka additive inverse -a/b hota hai, jisse unka jod 0 ho. Bas sign badal do.

ab + (-ab) = 0

Additive Inverse Examples

Additive inverse of 28 is -28.

Additive inverse of -59 is 59.

➗ 7. Reciprocal (Multiplicative Inverse / Gunatmak Pratilom)

The reciprocal of a non-zero rational number ab is ba such that their product is 1.

Kisi non-zero rational number a/b ka reciprocal b/a hota hai, jisse unka product 1 ho. Fraction ko ulta kar do.

ab × ba = 1   (where a, b ≠ 0)

Zero (0) has no reciprocal! (Zero ka reciprocal nahi hota!).

Reciprocal Examples

Reciprocal of 131 (or 13) is 113.

Reciprocal of -58 is 8-5 (or -85).

➕✖️ 8. Distributivity (Vitarakta)

Multiplication distributes over addition and subtraction.

Guna jod aur ghatav par distribute hota hai.

a × (b + c) = (a × b) + (a × c)

a × (b - c) = (a × b) - (a × c)

Distributivity Example

Use Distributivity: 75 × (-312) + 75 × 512

Solution:

= 75 × [ (-312) + 512 ]   (Take 7/5 common)

= 75 × [ -3+512 ]

= 75 × 212

= 75 × 16   (Simplify 2/12)

= 7 × 15 × 6 = 730

↔️ Representation on Number Line (Number Line Par Dikhana)

Rational numbers ko number line par locate kar sakte hain.

Rational numbers ko number line par dhoondha ja sakta hai.
Steps to Represent pq:
  1. Draw number line, mark integers (0, 1, -1 etc).
  2. Divide the segment between consecutive integers (like 0 and 1 or -1 and -2) into q equal parts (q = denominator).
    3. Do integers ke beech ke hisse ko ‘q’ barabar parts mein baanto.
  3. Starting from the integer part, move p steps (p = numerator) right (if positive) or left (if negative). Mark the point.
    4. Integer se shuru karke ‘p’ steps aage (positive) ya peeche (negative) jao.
Represent 56 on Number Line

1. It’s between 0 and 1.

2. Divide 0 to 1 into 6 equal parts.

3. Mark the 5th part from 0.

 0 |---|---|---|---|---|---|
    1/6              5/6  1
Represent -74 on Number Line

1. -7/4 = -134. It’s between -1 and -2.

2. Divide -1 to -2 into 4 equal parts.

3. Mark the 3rd part from -1 towards left.

 ...|---|---|---|---|
    -2     -7/4(-1 3/4) -3/2     -1

🧐 Rational Numbers between Two Rational Numbers

You can find unlimited (infinite) rational numbers between any two given rational numbers.

Aap kisi bhi do diye gaye rational numbers ke beech anant numbers dhoondh sakte hain.

Method 1: Average / Mean Method (Ek number nikalne ke liye)

Find the average: (a + b) ÷ 2. This average will be between a and b.

Dono numbers ko jodo aur 2 se divide karo. Jawaab dono ke beech mein hoga.
Find a rational number between 14 and 12

Avg = ( 14 + 12 ) ÷ 2

= ( 14 + 24 ) ÷ 2

= ( 34 ) ÷ 2

= 34 × 12 = 38

Result: 38 lies between them.

Method 2: Making Denominators Same (Zyada numbers nikalne ke liye)

Steps:
  1. Make the denominators of the two rational numbers equal using LCM.
    2. LCM use karke dono ke denominators barabar karo.
  2. If needed, multiply numerator & denominator of *both* fractions by 10 (or suitable number) to create a wider gap between numerators.
    3. Agar zaroorat ho, toh dono fractions ke upar-neeche 10 (ya koi aur number) se multiply karke numerators ke beech zyada gap banao.
  3. Write down the rational numbers with the new numerators lying between them.
    4. Naye numerators ke beech wale numbers ko common denominator ke saath likho.
Find 5 rational numbers between -25 and 12

1. LCM(5, 2) = 10.

2. Convert: -25 = -410; 12 = 510.

3. Numbers between -4 and 5 are: -3, -2, -1, 0, 1, 2, 3, 4.

4. Pick any 5.

Result: -310, -210, -110, 010, 110

Find 10 rational numbers between 14 and 12

1. Convert to denominator 8: 14 = 28; 12 = 48. (Gap is too small: only 3/8).

2. Multiply by 10: 2080 and 4080.

3. Numbers between 20 and 40 are 21, 22,…, 39.

Result: Ten numbers are 2180, 2280, 2380, 2480, 2580, 2680, 2780, 2880, 2980, 3080.

❓ Sawal Jawab (Questions & Answers)

🤏 Very Short Answer Questions

1. What is the standard form of a rational number?

A rational number pq is in standard form if q is positive and p, q have no common factor other than 1.

2. What is the additive identity?

0 (Zero).

3. What is the multiplicative identity?

1 (One).

4. Find the additive inverse of -4-5.

First simplify: -4-5 = 45. Additive inverse is -45.

5. Find the multiplicative inverse of -1 17.

Convert to improper fraction: -117 = -87. Reciprocal is -78.

6. Are whole numbers closed under division?

No (e.g., 5 ÷ 2 = 2.5, which is not a whole number).

7. Which operation is commutative for rational numbers?

Addition and Multiplication.

8. Which operation is associative for rational numbers?

Addition and Multiplication.

9. What is the reciprocal of x?

1/x (if x ≠ 0).

10. Is 30 a rational number?

No, because the denominator cannot be zero.

11. What is 0 × 57?

0 (Property of Zero).

12. What is 95 ÷ 1?

9/5 (Role of 1 – not applicable here directly, but division by 1 doesn’t change).

13. Simplify: 23 + (-23).

0 (Additive inverse).

14. Find -(-56).

5/6.

15. How do you represent rational numbers on a number line?

By dividing segments between integers into equal parts based on the denominator.

📝 Short Answer Questions

1. Verify the commutative property of addition for -23 and 57.

  • LHS = -23 + 57 = -14 + 1521 = 121.
  • RHS = 57 + (-23) = 15 - 1421 = 121.
  • Since LHS = RHS, it is verified.

2. Verify the associative property of multiplication for 12, -23, 34.

  • LHS = (12 × -23) × 34 = (-26) × 34 = (-13) × 34 = -312 = -14.
  • RHS = 12 × (-23 × 34) = 12 × (-612) = 12 × (-12) = -14.
  • Since LHS = RHS, it is verified.

3. Find the additive inverse and multiplicative inverse of -1319.

  • Additive Inverse: 1319.
  • Multiplicative Inverse: -1913.

4. Find the additive inverse and multiplicative inverse of -15.

  • Additive Inverse: 15.
  • Multiplicative Inverse: 5-1 = -5.

5. Using appropriate properties find: 25 × (-37) - 16 × 32 + 114 × 25.

  • Rearrange: [25 × (-37)] + [114 × 25] - (16 × 32)
  • Distributivity: 25 × [ (-37) + 114 ] - (16 × 32)
  • Solve bracket: 25 × [ -6 + 114 ] - 312
  • = 25 × (-514) - 14
  • = -1070 - 14 = -17 - 14
  • Solve: = -4 - 728 = -1128

6. Represent -211, -511, -911 on the number line.

  • Draw line, mark 0 and -1.
  • Divide segment 0 to -1 into 11 equal parts.
  • Mark the 2nd part left of 0 as -2/11.
  • Mark the 5th part left of 0 as -5/11.
  • Mark the 9th part left of 0 as -9/11.
 |-,-,-,-,-,-,-,-,-,-,-,-| 
-1     -9/11   -5/11   -2/11      0

7. Find one rational number between 13 and 14 using the mean method.

  • Mean = (13 + 14) ÷ 2
  • = (4+312) ÷ 2 = 712 ÷ 2
  • = 712 × 12 = 724

8. Find 5 rational numbers between -12 and 23.

  • LCM(2,3)=6. Convert: -12=-36; 23=46.
  • Numbers between -3/6 and 4/6 are: -2/6, -1/6, 0/6, 1/6, 2/6, 3/6.
  • Choose 5: e.g., -13, -16, 0, 16, 13.

9. Multiply 613 by the reciprocal of -716.

  • Reciprocal of -7/16 is -16/7.
  • Multiply: 613 × (-167)
  • = -9691.

10. Is 0 the multiplicative identity? Why or why not?

  • No.
  • The multiplicative identity is the number ‘x’ such that a × x = a.
  • For rational numbers, this number is 1 (because a/b × 1 = a/b).
  • Multiplying by 0 gives 0 (a/b × 0 = 0), not the original number.

11. Name the property that allows calculation of 23 × (67 × 12) as (23 × 67) × 12.

Associative property of multiplication.

📜 Long Answer Questions

1. Explain Closure, Commutativity, and Associativity for all four basic operations (+, -, ×, ÷) on Rational Numbers.

  • Closure: Result is rational? Yes for +, -, ×. No for ÷ (due to 0).
  • Commutativity: Order matters? No for + and × (a op b = b op a). Yes for – and ÷ (a op b ≠ b op a).
  • Associativity: Grouping matters? No for + and × ((a op b) op c = a op (b op c)). Yes for – and ÷.

2. Define Additive and Multiplicative Inverse. Give examples for 45 and -3.

  • Additive Inverse (Negative): Number that adds to give 0. Ex: For 4/5, it’s -4/5. For -3 (or -3/1), it’s 3.
  • Multiplicative Inverse (Reciprocal): Number that multiplies to give 1. Ex: For 4/5, it’s 5/4. For -3 (or -3/1), it’s 1/-3 or -1/3. (0 has no reciprocal).

3. Using appropriate properties find: 25 × (-37) - 114 - 37 × 35

= 25 × (-37) - 37 × 35 - 114 (Commutativity)

= [25 × (-37)] + [(-37) × 35] - 114

= (-37) × [25 + 35] - 114 (Distributivity)

= (-37) × [55] - 114

= (-37) × 1 - 114

= -37 - 114 = -614 - 114 = -714 = -12

4. Find ten rational numbers between -34 and 56.

  • LCM(4, 6) = 12.
  • Convert: -3/4 = -9/12; 5/6 = 10/12.
  • Integers between -9 and 10 are -8, -7, …, 0, …, 8, 9. (Plenty here).
  • Ten numbers are: -812, -712, -612, -512, -412, -312, -212, -112, 012, 112.

5. Represent 38 and -58 on the same number line.

  • Draw a number line. Mark 0, 1, -1.
  • Divide the segment from 0 to 1 into 8 equal parts. Mark the 3rd part as 3/8.
  • Divide the segment from 0 to -1 into 8 equal parts. Mark the 5th part to the left of 0 as -5/8.
|....|....|....|....0....|....|....|....|
-1(-8/8)  (-5/8)     0         (3/8)     1(8/8)

6. What is the role of 0 and 1 for rational numbers?

  • Role of 0 (Zero): It is the **Additive Identity**. Adding 0 to any rational number leaves it unchanged (a + 0 = a).
  • Role of 1 (One): It is the **Multiplicative Identity**. Multiplying any rational number by 1 leaves it unchanged (a × 1 = a).

7. Use the mean method to find three rational numbers between 1/3 and 1/2.

  • 1st number = (13 + 12) / 2 = (2+36) / 2 = (56) / 2 = 512. Now numbers are 1/3, 5/12, 1/2.
  • 2nd number (between 1/3 and 5/12) = (13 + 512) / 2 = (4+512) / 2 = (912) / 2 = 34 / 2 = 38.
  • 3rd number (between 5/12 and 1/2) = (512 + 12) / 2 = (5+612) / 2 = (1112) / 2 = 1124.
  • Three numbers: 3/8, 5/12, 11/24.

8. Explain why rational numbers are closed under Addition, Subtraction, and Multiplication but not always under Division.

  • Closure Definition:** Operation on two set members gives a result within the same set.
  • Addition/Subtraction/Multiplication:** Adding, subtracting, or multiplying any two fractions (pq, rs) where denominators are non-zero will always result in another fraction xy where y (the new denominator, like qs or lcm(q,s)) is also non-zero. Hence, closed.
  • Division:** Dividing pq by rs (where r/s ≠ 0) involves multiplying by the reciprocal: pq × sr = psqr. This is rational if qr ≠ 0.
  • Issue with Division by Zero: The rational number 0 can be written as 0/1, 0/2 etc. If we try to divide any rational number ab by 0, it’s ab ÷ 0 which is mathematically undefined. Since division by one member (0) of the rational number set does not yield a result within the set (it’s undefined), the set is not closed under division.

9. Find 37 + (-611) + (-821) + 522 using rearrangement (Commutativity and Associativity).

Group terms with compatible denominators:

= [37 + (-821)] + [(-611) + 522]

  • Solve first bracket [LCM(7,21)=21]:
    = [921 + -821] = 121
  • Solve second bracket [LCM(11,22)=22]:
    = [-1222 + 522] = -722
  • Add the results: 121 + (-722)
  • LCM(21, 22) = 21 × 22 = 462.
  • = 1×22462 + -7×21462
  • = 22 - 147462 = -125462

Result: -125462

10. Define rational numbers and give examples of rational numbers that are integers, fractions, positive, and negative.

Rational Number: Any number that can be expressed as pq, where p and q are integers, and q is not zero.

  • Examples that are Integers:
    • 5 (written as 5/1)
    • -3 (written as -3/1)
    • 0 (written as 0/1)
  • Examples that are Fractions (but not integers):
    • 12
    • 34
    • 75
  • Examples that are Positive:
    • 23
    • 7
    • 54
  • Examples that are Negative:
    • -12
    • 3-5 (which is same as -3/5)
    • -6

🤔 Check Your Understanding! (Quiz Time!)

1. Which number is a rational number?

2. Which property states that a+b = b+a?

3. The additive identity for rational numbers is:

4. The multiplicative identity for rational numbers is:

5. Additive inverse of -719?

6. Multiplicative inverse of -1?

7. Reciprocal of -38?

8. Property: a × (b+c) = a×b + a×c?

9. Rational numbers are NOT associative under:

10. How many rational numbers exist between 2 and 3?

11. To represent 47, divide segment 0 to 1 into:

12. Which number lies between 14 and 13?

13. Sum 56 + (-23)?

14. Product (-35) × (-53)?

15. Is 0 rational?

16. Which operation is NOT always closed for Rational Numbers?

17. Result of -12 - 14?

18. Reciprocal does not exist for:

19. Where is -114 located?

20. Evaluate: 12 × (13 + 14)?

21. Additive inverse of ab?

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