Rational Numbers — Class 8 Mathematics
1. What Are Rational Numbers?
A number that can be expressed in the form p/q, where p and q are INTEGERS and q ≠ 0, is a RATIONAL NUMBER. The word 'rational' comes from 'RATIO.' Examples: 3/4, −5/7, 8 (= 8/1), 0 (= 0/1), −2.5 (= −5/2). 'Every integer IS a rational number. Every fraction IS a rational number. But NOT every rational number is an integer.'
Are These Rational?
- 0.75 → 75/100 = 3/4 ✓ (terminating decimal → rational)
- 0.333... → 1/3 ✓ (recurring decimal → rational)
- √2 → Cannot be expressed as p/q ✗ (irrational)
- −7 → −7/1 ✓ (integer → rational)
2. Properties of Rational Numbers
Closure Property
A set of numbers is CLOSED under an operation if the result of that operation ALWAYS belongs to the same set.
| Operation | Rational + Rational | Result |
|---|---|---|
| Addition | a/b + c/d = (ad+bc)/bd | RATIONAL ✓ |
| Subtraction | a/b − c/d = (ad−bc)/bd | RATIONAL ✓ |
| Multiplication | a/b × c/d = ac/bd | RATIONAL ✓ |
| Division | a/b ÷ c/d = ad/bc | RATIONAL ✓ (if divisor ≠ 0) |
'Rational numbers are CLOSED under addition, subtraction, multiplication, and division (except division by zero). This is a powerful property — you can perform ANY of these operations and stay within rational numbers.'
Commutative Property
An operation is commutative if changing the ORDER does NOT change the result (a ○ b = b ○ a).
| Operation | Commutative? | Example |
|---|---|---|
| Addition | YES | 2/3 + 5/7 = 5/7 + 2/3 |
| Subtraction | NO | 2/3 − 5/7 ≠ 5/7 − 2/3 |
| Multiplication | YES | 2/3 × 5/7 = 5/7 × 2/3 |
| Division | NO | 2/3 ÷ 5/7 ≠ 5/7 ÷ 2/3 |
'Addition and multiplication ARE commutative. Subtraction and division are NOT. This is why the ORDER matters when you subtract or divide.'
Associative Property
An operation is associative if the GROUPING does NOT change the result: (a ○ b) ○ c = a ○ (b ○ c).
| Operation | Associative? | Example |
|---|---|---|
| Addition | YES | (1/2 + 2/3) + 3/4 = 1/2 + (2/3 + 3/4) |
| Subtraction | NO | (1/2 − 2/3) − 3/4 ≠ 1/2 − (2/3 − 3/4) |
| Multiplication | YES | (1/2 × 2/3) × 3/4 = 1/2 × (2/3 × 3/4) |
| Division | NO | (1/2 ÷ 2/3) ÷ 3/4 ≠ 1/2 ÷ (2/3 ÷ 3/4) |
Distributive Property
Multiplication DISTRIBUTES over addition: a × (b + c) = a×b + a×c. This property holds for rational numbers: 2/3 × (1/2 + 3/4) = 2/3 × 1/2 + 2/3 × 3/4.
3. Identity Elements
Additive Identity — ZERO
For any rational number 'a': a + 0 = a. 0 + a = a. Zero is the ADDITIVE IDENTITY for rational numbers. 'Adding zero changes NOTHING — the number retains its identity.'
Multiplicative Identity — ONE
For any rational number 'a': a × 1 = a. 1 × a = a. One is the MULTIPLICATIVE IDENTITY for rational numbers. 'Multiplying by one changes NOTHING — the number retains its identity.'
4. Inverse Elements
Additive Inverse (Negative)
For rational number a, its additive inverse is −a: a + (−a) = 0. Examples: 3/5 → −3/5. −7/9 → 7/9. 0 → 0 (0 is its OWN additive inverse).
Multiplicative Inverse (Reciprocal)
For rational number a (a ≠ 0), its multiplicative inverse is 1/a: a × (1/a) = 1. Examples: 3/5 → 5/3. −7/9 → −9/7. 1 → 1 (1 is its OWN multiplicative inverse). 'ZERO has NO multiplicative inverse — because 1/0 is UNDEFINED. Division by zero does not exist in mathematics.'
5. Representing Rational Numbers on the Number Line
Draw a horizontal line. Mark 0 at the centre. Positive numbers to the RIGHT. Negative numbers to the LEFT. For 3/4: divide the segment from 0 to 1 into 4 EQUAL parts. The 3rd mark from 0 is 3/4. For −5/3 = −1⅔: go left of 0 to −1. Then divide the segment from −1 to −2 into 3 equal parts. The 2nd mark is −1⅔.
6. Finding Rational Numbers Between Two Given Rationals
Method 1 — Mean Method: The AVERAGE of two rational numbers is a rational number BETWEEN them. (a+b)/2 lies exactly midway between a and b. Repeat this to get MORE rational numbers. Method 2 — Common Denominator Method: Write both numbers with a LARGE common denominator. The numerators between them give rational numbers.
Worked Example
Find 5 rational numbers between 1/3 and 1/2. Write with common denominator: 1/3 = 4/12. 1/2 = 6/12. Between 4/12 and 6/12: we have 5/12. But we need 5 numbers. Increase the denominator: Multiply both by 10 → 1/3 = 40/120. 1/2 = 60/120. Between 40/120 and 60/120: 41/120, 42/120, 43/120, ..., 59/120. Five of them: 42/120 (=7/20), 45/120 (=3/8), 48/120 (=2/5), 51/120 (=17/40), 54/120 (=9/20).
'Between ANY two distinct rational numbers, there are INFINITELY MANY rational numbers. This property is called DENSITY of rational numbers.'
7. Common Mistakes to Avoid
- Subtraction is NOT commutative: 5 − 3 ≠ 3 − 5. Don't rearrange subtraction terms freely.
- Division is NOT commutative: 6 ÷ 3 ≠ 3 ÷ 6. Order matters.
- Zero has NO reciprocal: Never write 1/0. 'There is no number that, multiplied by 0, gives 1.'
- Not reducing fractions to simplest form: 4/8 should be written as 1/2. Identify common factors.
8. AP Exam Focus
| Topic | Marks |
|---|---|
| Properties (fill in blanks) | 2-3 |
| Finding rational numbers between two numbers | 3-4 |
| Additive/multiplicative inverse | 1-2 |
| Application of distributive property | 2-3 |
Worked Examples for AP Exam
Example 1 — Using Distributive Property: Evaluate 2/5 × (−3/7) − 1/6 × 3/2 + 1/14 × 2/5. Take 2/5 common from first and third terms: 2/5 × (−3/7 + 1/14) − 1/6 × 3/2 = 2/5 × (−6/14 + 1/14) − 3/12 = 2/5 × (−5/14) − 1/4 = −10/70 − 1/4 = −1/7 − 1/4 = −(4/28 + 7/28) = −11/28.
Example 2 — Finding Inverse: What is the multiplicative inverse of −1⅓? Express as improper fraction: −1⅓ = −4/3. Multiplicative inverse = −3/4. Check: (−4/3) × (−3/4) = 12/12 = 1 ✓.
Example 3 — Additive Inverse Application: Verify that −(−x) = x for x = 3/7. Additive inverse of x = −3/7. Additive inverse of (−3/7) = −(−3/7) = 3/7. So −(−x) = x. TRUE.
Example 4 — Number Line Representation: Represent −2/5 and 3/5 on the number line. Divide units into 5 equal parts. −2/5 is 2 parts LEFT of 0. 3/5 is 3 parts RIGHT of 0.
AP-Specific Context
'Andhra Pradesh students preparing for competitive exams like NMMS, APRJC, and Polycet will encounter rational number problems requiring QUICK mental calculation. Practice simplifying BEFORE multiplying — cancel common factors to reduce the size of numbers. For example, in 12/35 × 25/18, cancel 6 (from 12 and 18) and 5 (from 25 and 35) FIRST: (2/7) × (5/3) = 10/21. The answer is the same but the calculation is much faster.'
Quick Self-Test
- Is subtraction commutative for rational numbers? (Answer: No.)
- Name the additive identity. (Answer: Zero.)
- Find the multiplicative inverse of −7/11. (Answer: −11/7.)
- State the distributive property. (Answer: a × (b + c) = a×b + a×c.)
- Find 3 rational numbers between 1/4 and 1/3. (Answer: Many possible — e.g., 13/48, 14/48, 15/48.)
- Is zero a rational number? (Answer: Yes — 0 = 0/1.)
