Rational Numbers
1. Introduction and Definition
A RATIONAL NUMBER is any number that can be written in the form p/q where p and q are INTEGERS and q ≠ 0.
Examples: 3/4, -5/2, 7 (= 7/1), 0.5 (= 1/2), 0 (= 0/1), -2 1/3 (= -7/3).
Set notation: Q = {p/q | p, q ∈ Z, q ≠ 0}
Why Do We Need Rational Numbers?
- Integers are NOT enough for measurement (e.g., half a metre).
- Fractions represent PARTS of a whole.
- Negative fractions are needed for temperature below zero, debt, etc.
Important Observations
- Every INTEGER is a rational number (because n = n/1).
- Every FRACTION is a rational number (but NOT every rational number is a fraction — fractions have positive denominators).
- DECIMALS that TERMINATE or RECUR are rational.
2. Standard Form (Simplest Form)
A rational number is in STANDARD FORM when:
- The denominator is POSITIVE.
- p and q have NO COMMON FACTOR other than 1.
Converting to Standard Form
Example: Convert -15/20 to standard form.
- Step 1: Make denominator positive: -15/20 (already positive).
- Step 2: Find HCF of 15 and 20 = 5.
- Step 3: Divide numerator and denominator by 5: (-15÷5)/(20÷5) = -3/4.
Example 2: Convert 36/(-48) to standard form.
- Step 1: Make denominator positive: 36/(-48) = -36/48.
- Step 2: HCF of 36 and 48 = 12.
- Step 3: Divide: (-36÷12)/(48÷12) = -3/4.
Common Mistake
'Do NOT stop at -36/48 — you must reduce to LOWEST terms.'
3. Equivalent Rational Numbers
Two rational numbers are EQUIVALENT if they represent the SAME value.
How to Find Equivalent Rationals
Multiply (or divide) BOTH numerator and denominator by the SAME NON-ZERO integer.
Example: Write 5 equivalent rational numbers for 2/3.
- 2/3 = 4/6 = 6/9 = 8/12 = 10/15 = 12/18
- All these represent the same value: 0.666...
Checking Equivalence
Cross-multiply: a/b = c/d if a × d = b × c. Example: Is 4/6 equivalent to 6/9? 4 × 9 = 36, 6 × 6 = 36. YES, they are equivalent.
4. Comparison of Rational Numbers
Method
- Convert to LIKE fractions (same POSITIVE denominator).
- Compare numerators — the larger numerator means the larger rational number.
Comparison Rules
- If both are POSITIVE: Larger value means larger rational.
- If both are NEGATIVE: The one with the SMALLER absolute value is LARGER.
- POSITIVE rational > 0 > NEGATIVE rational.
Worked Example (ICSE 2024, 2 marks)
Arrange in ascending order: -3/5, -2/3, 1/2, -4/15.
Solution: LCM of 5, 3, 2, 15 = 30. Convert: -3/5 = -18/30, -2/3 = -20/30, 1/2 = 15/30, -4/15 = -8/30. Ascending: -20/30 < -18/30 < -8/30 < 15/30. Answer: -2/3 < -3/5 < -4/15 < 1/2.
5. Operations on Rational Numbers
Addition
- Find LCM of denominators.
- Convert each to equivalent fraction with LCM as denominator.
- Add numerators. Keep denominator same.
- Reduce to standard form.
Example: -2/5 + 7/10 = -4/10 + 7/10 = 3/10.
Subtraction
Same as addition, but subtract numerators. Example: 3/4 - (-2/3) = 9/12 + 8/12 = 17/12 = 1 5/12.
Multiplication
(a/b) × (c/d) = (a × c) / (b × d) Example: (-2/3) × (9/4) = -18/12 = -3/2.
Division
(a/b) ÷ (c/d) = (a/b) × (d/c) = (a × d) / (b × c) Example: (-5/6) ÷ (2/3) = (-5/6) × (3/2) = -15/12 = -5/4.
6. Rational Numbers Between Two Rationals
There are INFINITELY many rational numbers between any two distinct rationals.
Method 1: Average Method
Take the AVERAGE: (a + b) / 2. Example: Find one rational between 1/3 and 1/2. Average = (1/3 + 1/2) / 2 = (2/6 + 3/6) / 2 = (5/6) / 2 = 5/12.
Method 2: Equivalent Fraction Expansion
Write both with a large common denominator. Example: Find 3 rational numbers between 1/5 and 3/5. 1/5 = 4/20, 3/5 = 12/20. Numbers: 5/20 = 1/4, 7/20, 11/20.
7. ICSE Exam Focus
| Topic | Marks | Frequency |
|---|---|---|
| Standard form and equivalence | 2 marks | High |
| Comparison and ordering | 2-3 marks | Very High |
| Addition and subtraction | 2-3 marks | Very High |
| Multiplication and division | 2-3 marks | High |
| Rationals between two numbers | 2 marks | Medium |
Common Mistakes
- Forgetting to make denominator POSITIVE before standard form.
- Adding/subtracting only numerators without finding LCM.
- Sign errors when subtracting with negative rationals.
- Forgetting to take RECIPROCAL when dividing.
Self-Test (5 Questions)
Q1. Write in standard form: 45/(-63). (2 marks)
Q2. Arrange in descending order: -7/10, -5/8, 3/4, -2/5. (3 marks)
Q3. Find the value of: (-3/4) + (5/6) - (-1/3). (2 marks)
- A) 13/12
- B) 11/12
- C) 1/2
- D) 5/4
Q4. Divide: (-7/9) ÷ (14/27). (2 marks)
- A) -3/2
- B) 3/2
- C) -2/3
- D) 2/3
Q5. Find two rational numbers between 2/5 and 3/5. (2 marks)
Answers
A1. -5/7. (45/(-63) = -45/63. HCF = 9. (-45÷9)/(63÷9) = -5/7.) A2. 3/4 > -2/5 > -5/8 > -7/10. (LCM=40: 30/40, -16/40, -25/40, -28/40) A3. A) 13/12. (LCM=12: -9/12 + 10/12 + 4/12 = 5/12 + 4/12 = 13/12.) A4. A) -3/2. ((-7/9) × (27/14) = -189/126 = -3/2.) A5. 2/5 = 10/25, 3/5 = 15/25. Numbers: 11/25, 12/25. (Any two valid.)
