Rational Numbers
1. What Are Rational Numbers?
A number that can be expressed as p/q where p and q are integers and q ≠ 0. The set of rational numbers is denoted by Q.
Examples: 2/3, -5/7, 4, 0, -3 (since 4 = 4/1, 0 = 0/1, -3 = -3/1).
'Every integer is a rational number, but not every rational number is an integer.'
2. Properties of Rational Numbers
Closure Property
| Operation | Statement | Example |
|---|---|---|
| Addition | Sum of two rationals is ALWAYS rational | 2/3 + 4/5 = 22/15 (rational) |
| Subtraction | Difference of two rationals is ALWAYS rational | 7/8 - 1/4 = 5/8 (rational) |
| Multiplication | Product of two rationals is ALWAYS rational | -2/3 × 9/4 = -3/2 (rational) |
| Division | Quotient of two rationals is rational (divisor ≠ 0) | 3/4 ÷ 2/7 = 21/8 (rational) |
Commutative Property
| Operation | Property | Example |
|---|---|---|
| Addition | a + b = b + a | 2/3 + 1/5 = 1/5 + 2/3 = 13/15 |
| Multiplication | a × b = b × a | 3/4 × 2/5 = 2/5 × 3/4 = 3/10 |
| Subtraction | NOT commutative | 3/4 - 1/2 ≠ 1/2 - 3/4 |
| Division | NOT commutative | 5/6 ÷ 2/3 ≠ 2/3 ÷ 5/6 |
Associative Property
| Operation | Property |
|---|---|
| Addition | (a + b) + c = a + (b + c) |
| Multiplication | (a × b) × c = a × (b × c) |
| Subtraction | NOT associative |
| Division | NOT associative |
Distributive Property
Multiplication distributes over addition: a × (b + c) = a × b + a × c
Example: 2/3 × (1/4 + 3/5) = 2/3 × 17/20 = 34/60 = 17/30 Check: 2/3 × 1/4 + 2/3 × 3/5 = 2/12 + 6/15 = 1/6 + 2/5 = 5/30 + 12/30 = 17/30 ✓
3. Identity and Inverse
Additive Identity
0 is the additive identity. a + 0 = 0 + a = a. Example: 3/7 + 0 = 0 + 3/7 = 3/7.
Additive Inverse
For every rational a, there exists —a such that a + (—a) = 0. Example: Additive inverse of 5/8 is —5/8.
Multiplicative Identity
1 is the multiplicative identity. a × 1 = 1 × a = a. Example: -2/3 × 1 = 1 × (-2/3) = -2/3.
Multiplicative Inverse (Reciprocal)
For every non-zero rational a, there exists 1/a such that a × (1/a) = 1. Example: Multiplicative inverse of 3/5 is 5/3.
4. Representation on the Number Line
To represent a rational number on the number line:
- If the number is positive, it lies to the RIGHT of 0.
- If the number is negative, it lies to the LEFT of 0.
- Divide the segment between 0 and 1 (or -1 and 0) into equal parts based on the denominator.
Example: Represent 3/4 on the number line. Divide the segment from 0 to 1 into 4 equal parts. The THIRD point from 0 represents 3/4.
Example: Represent -5/3 on the number line. -5/3 = -1⅔. It lies between -1 and -2. Divide the segment from -1 to -2 into 3 equal parts. The SECOND point from -1 gives -5/3.
5. Finding Rational Numbers Between Two Given Rational Numbers
'Between any two rational numbers, there are INFINITELY MANY rational numbers. Rational numbers are DENSE on the number line.'
Method 1: Using the Mean (Average)
The rational number halfway between a and b is (a + b)/2.
Worked Example: Find three rational numbers between 1/3 and 2/5.
1/3 = 5/15, 2/5 = 6/15
First: (1/3 + 2/5)/2 = (5/15 + 6/15)/2 = 11/30
Between 1/3 and 11/30: (1/3 + 11/30)/2 = (10/30 + 11/30)/2 = 21/60 = 7/20
Between 11/30 and 2/5: (11/30 + 12/30)/2 = 23/60
Therefore: 1/3, 7/20, 11/30, 23/60, 2/5 are five numbers in order.
Method 2: Equivalent Fractions
Write the rational numbers with a LARGE common denominator.
Worked Example: Find five rational numbers between 3/5 and 4/5.
3/5 = 30/50, 4/5 = 40/50
Five numbers: 31/50, 32/50 = 16/25, 33/50, 34/50 = 17/25, 35/50 = 7/10
6. Standard Form of a Rational Number
A rational number is in STANDARD FORM if its denominator is POSITIVE and the numerator and denominator have NO COMMON FACTOR other than 1.
Example: Reduce -15/25 to standard form. -15/25 = -3/5 (dividing numerator and denominator by 5).
7. Comparison of Rational Numbers
| Case | Rule |
|---|---|
| Same denominator | Compare numerators directly |
| Different denominators | Convert to equivalent fractions with LCM as denominator |
| Positive vs Negative | Positive > Negative always |
| On number line | Numbers to the RIGHT are GREATER |
Worked Example: Compare -7/12 and -5/8.
LCM of 12 and 8 is 24. -7/12 = -14/24, -5/8 = -15/24 Since -14 > -15: -7/12 > -5/8
Common Mistakes and Fixes
| Mistake | Fix |
|---|---|
| '0 is not a rational number' | 0 = 0/1, so 0 IS a rational number |
| 'Division is commutative' | 2 ÷ 3 ≠ 3 ÷ 2. Only addition and multiplication are commutative |
| 'Density property means equally spaced' | Density means BETWEEN any two rationals there is ANOTHER rational — they are NOT equally spaced |
| 'Forgetting to check q ≠ 0' | Any number with denominator 0 is UNDEFINED, not rational |
ICSE Exam Focus (5–6 marks)
- 2-mark questions: Properties of rational numbers (name the property used)
- 3-mark questions: Find rational numbers between two given numbers
- 4-mark questions: Represent on number line with verification of properties
- 6-mark questions: Application of distributive property in complex expressions
Self-Test
Q1. State whether the following statement is TRUE or FALSE: 'Subtraction is commutative for rational numbers.' A1. FALSE. Counterexample: 2/3 - 1/4 ≠ 1/4 - 2/3.
Q2. Find the additive inverse of -7/9. A2. 7/9 (since -7/9 + 7/9 = 0).
Q3. Find three rational numbers between 1/2 and 3/4. A3. 1/2 = 4/8, 3/4 = 6/8. Numbers: 9/16, 5/8, 11/16 (using mean method).
Q4. Verify the distributive property: 2/3 × (1/4 + 5/6) = (2/3 × 1/4) + (2/3 × 5/6). A4. LHS = 2/3 × (3/12 + 10/12) = 2/3 × 13/12 = 26/36 = 13/18. RHS = 2/12 + 10/18 = 1/6 + 5/9 = 3/18 + 10/18 = 13/18. Hence verified.
Q5. Express 42/-98 in standard form. A5. 42/-98 = -42/98 = -21/49 = -3/7. Standard form: -3/7.
Q6. Which is greater: -3/7 or -5/8? A6. LCM = 56. -3/7 = -24/56, -5/8 = -35/56. Since -24 > -35, -3/7 > -5/8.
