Rational Numbers — Extending the Number System
"Integers are great for counting, but what about half a litre of milk? Or 0.75 metres of cloth? We need RATIONAL numbers to measure the world around us."
1. Need for Rational Numbers
Natural Numbers {1, 2, 3, ...} — used for counting. Whole Numbers {0, 1, 2, 3, ...} — added zero. Integers {...−3, −2, −1, 0, 1, 2, 3, ...} — added negatives.
Yet none of these can represent HALF (1/2), a QUARTER (1/4), or NEGATIVE two-thirds (−2/3). 'We need fractions that can also be negative — that is why RATIONAL numbers exist.'
Real-world need: 'Temperature can be 5.5°C. You can eat 3/4 of a pizza. A car's fuel efficiency could be 15.6 km/L. All these are rational numbers.'
2. What Is a Rational Number?
Definition: A number that can be expressed in the form p/q where p and q are INTEGERS and q ≠ 0.
The set of rational numbers is denoted by Q.
'Every integer is also a rational number. Why? Because 5 = 5/1, −3 = −3/1, 0 = 0/1. Integers are a SUBSET of rational numbers.'
Relationship Between Number Sets
Natural numbers ⊂ Whole numbers ⊂ Integers ⊂ Rational numbers.
| Number | As p/q | Rational? | Reason |
|---|---|---|---|
| 7 | 7/1 | YES | Integer format |
| −4 | −4/1 | YES | Negative integer |
| 0 | 0/1 | YES | Zero is rational |
| 3/5 | 3/5 | YES | Already in p/q form |
| 0.25 | 1/4 | YES | Terminating decimal |
| 0.333... | 1/3 | YES | Repeating decimal |
| √2 | — | NO | Cannot be expressed as p/q |
3. Positive and Negative Rational Numbers
| Type | Condition | Examples |
|---|---|---|
| Positive | p and q have SAME sign | 3/4, −5/−7 = 5/7, 2/3 |
| Negative | p and q have OPPOSITE signs | −5/8, 9/−11 = −9/11 |
| Zero | p = 0, q ≠ 0 | 0/5 = 0 |
Standard Form of a Rational Number
'A rational number is in STANDARD form when the denominator is POSITIVE and p and q have NO common factor other than 1.'
| Given | Standard Form | Method |
|---|---|---|
| −8/12 | −2/3 | Divide numerator and denominator by 4 |
| 15/−25 | −3/5 | Make denominator positive: −15/25 = −3/5 |
| 20/30 | 2/3 | Divide by 10 (HCF) |
4. Representation on Number Line
'Just like integers, rational numbers exist on the number line — but NOW there are INFINITELY many numbers BETWEEN any two integers.'
Example: Represent 3/5 on a number line. Divide the segment 0 to 1 into 5 EQUAL parts. The 3rd mark from 0 is 3/5.
Example: Represent −7/4 on a number line. −7/4 = −1³/₄. Start at −1. Divide the segment from −1 to −2 into 4 equal parts. Go 3 parts from −1 towards −2.
5. Comparison of Rational Numbers
Method 1: Same Denominator
'If denominators are the SAME, compare numerators directly.' −3/7 < 2/7 (because −3 < 2). 5/11 > −8/11.
Method 2: Same Numerator
'If numerators are the SAME, the fraction with the SMALLER denominator is LARGER (for positive numbers).' 5/7 > 5/9 (dividing same amount into fewer parts = bigger parts).
Method 3: Cross-Multiplication (for any pair)
'Compare 3/7 and 4/9. Cross multiply: 3 × 9 = 27, 7 × 4 = 28. Since 27 < 28, 3/7 < 4/9.'
Algorithm: For a/b and c/d (b, d > 0): If a × d > b × c then a/b > c/d.
Method 4: Convert to Decimals
Divide numerator by denominator. Compare the decimal values. 3/7 ≈ 0.4286, 4/9 ≈ 0.4444 → 3/7 < 4/9.
Ordering Rational Numbers
'When ordering rational numbers, convert all to a COMMON denominator, then compare numerators.'
6. Operations on Rational Numbers
Addition and Subtraction
Rule: 'Convert to a COMMON denominator (LCM), then add/subtract numerators. Keep the common denominator.'
| Operation | Example | Steps |
|---|---|---|
| Same denominator | 3/7 + 2/7 = 5/7 | Add numerators directly |
| Different denominator | 2/3 + 3/5 | LCM of 3 and 5 = 15. (10 + 9)/15 = 19/15 |
| With negatives | (−3/8) + (5/12) | LCM = 24. (−9 + 10)/24 = 1/24 |
| Subtraction | 5/6 − 3/4 | LCM = 12. (10 − 9)/12 = 1/12 |
Multiplication
Rule: 'Multiply numerators together, multiply denominators together. Simplify if possible.'
(a/b) × (c/d) = (a × c)/(b × d).
Examples: (−2/3) × (5/7) = −10/21. (4/9) × (3/8) = 12/72 = 1/6.
Division
Rule: 'Multiply the first rational number by the RECIPROCAL of the second.'
(a/b) ÷ (c/d) = (a/b) × (d/c).
Examples: (3/5) ÷ (2/7) = (3/5) × (7/2) = 21/10. (−4/9) ÷ (2/3) = (−4/9) × (3/2) = −12/18 = −2/3.
7. Finding Rational Numbers Between Two Rationals
'Between any two rational numbers, there are INFINITELY many rational numbers.'
Method 1: Mean Method
'If a and b are two rational numbers, then (a + b)/2 lies BETWEEN them.' Between 2 and 3: (2+3)/2 = 2.5. Between 2 and 2.5: (2+2.5)/2 = 2.25. Continue forever.
Example: Find three rational numbers between 1/3 and 1/2. First: (1/3 + 1/2)/2 = (2/6 + 3/6)/2 = (5/6)/2 = 5/12. Second: (1/3 + 5/12)/2 = (4/12 + 5/12)/2 = 9/24 = 3/8. Third: (5/12 + 1/2)/2 = (5/12 + 6/12)/2 = 11/24.
Method 2: Equivalent Fractions
'Write the rational numbers as EQUIVALENT fractions with LARGER denominators.'
Between 1/3 and 1/2: Convert to denominator 30. 1/3 = 10/30, 1/2 = 15/30. Numbers: 11/30, 12/30 (= 2/5), 13/30, 14/30 (= 7/15).
8. Properties of Rational Numbers
| Property | Addition | Multiplication |
|---|---|---|
| Closure | Sum of two rationals is rational ✓ | Product of two rationals is rational ✓ |
| Commutative | a + b = b + a ✓ | a × b = b × a ✓ |
| Associative | (a+b)+c = a+(b+c) ✓ | (a×b)×c = a×(b×c) ✓ |
| Identity | a + 0 = a (0 is identity) ✓ | a × 1 = a (1 is identity) ✓ |
| Inverse | a + (−a) = 0 (additive inverse) ✓ | a × (1/a) = 1 (multiplicative inverse, a≠0) ✓ |
| Distributive | — | a(b+c) = ab + ac ✓ |
'Rational numbers satisfy ALL these properties — unlike integers which lack multiplicative inverses within the set.'
9. Common Mistakes and Fixes
| Mistake | Why It Is Wrong | Correct Approach |
|---|---|---|
| −3/5 is not rational | It IS rational — just a negative fraction | −3/5 = −0.6 = −3/5 ∈ Q |
| 0/0 is rational | Division by zero is undefined | q ≠ 0 is mandatory |
| 2/3 < 3/5 | Cross-multiplying incorrectly | 2×5 = 10, 3×3 = 9. Since 10 > 9, 2/3 > 3/5 |
| 3/8 − 1/4 = 2/4 | Subtracted both numerator and denominator | Common denominator: 3/8 − 2/8 = 1/8 |
| There are FINITE rationals between 1 and 2 | Infinitely many exist | Mean method gives a NEW rational every time |
10. AP SSC Exam Focus
| Topic | Marks | Question Type |
|---|---|---|
| Identifying rational numbers | 1-2 | True/False |
| Standard form | 2-3 | Simplify to standard |
| Comparison | 2-3 | Compare using methods |
| Operations | 3-4 | Add, subtract, multiply, divide |
| Finding between two rationals | 3-4 | Find given number of rationals |
Quick Self-Test
Q1. Is 0 a rational number? A1. Yes. 0 = 0/1 = 0/5 = 0/−3 (q ≠ 0). Zero is rational.
Q2. Express −24/36 in standard form. A2. Divide by 12: −2/3.
Q3. Compare 5/8 and 7/12. A3. Cross-multiply: 5 × 12 = 60, 8 × 7 = 56. Since 60 > 56, 5/8 > 7/12.
Q4. Find the sum: (−3/7) + (2/5). A4. LCM = 35. (−15 + 14)/35 = −1/35.
Q5. Find the product: (−4/9) × (3/8). A5. (−4 × 3)/(9 × 8) = −12/72 = −1/6.
Q6. Find three rational numbers between 1/4 and 1/3. A6. LCM = 12. 1/4 = 3/12, 1/3 = 4/12. Convert to denominator 24: 6/24 and 8/24. Number: 7/24. Convert to 48: 12/48 and 16/48. Numbers: 13/48, 14/48 = 7/24, 15/48 = 5/16. Three numbers: 13/48, 7/24, 5/16.
Q7. The product of two rational numbers is −8/15. One is −2/3. Find the other. A7. (−2/3) × x = −8/15 → x = (−8/15) ÷ (−2/3) = (−8/15) × (−3/2) = 24/30 = 4/5.
