1. What are Rational Numbers
- A rational number is any number that can be written in the form p/q, where:
- p and q are integers
- q ≠ 0
- All integers are rational numbers because an integer a = a/1
- Decimal representations:
- Terminating decimals → rational
- Recurring/repeating decimals → rational
2. Properties of Rational Numbers
| Property | Meaning | Example | Holds for Rational Numbers? |
|---|---|---|---|
| Closure | Operation stays in ℚ | (2/3 + 5/7) is rational | ✅ Yes for +, – , ×, ÷ (except ÷0) |
| Commutativity | Changing order doesn’t change result | a + b = b + a | ✅ True for + and × |
| Associativity | Grouping doesn’t matter | (a + b) + c = a + (b + c) | ✅ True for + and × |
| Distributivity | a(b + c) = ab + ac | 2(1/3 + 1/5) | ✅ True |
Additive Identity: 0 (a + 0 = a)
Multiplicative Identity: 1 (a × 1 = a)
Additive Inverse: For any a/b, inverse is –a/b
Multiplicative Inverse: For any a/b ≠ 0, inverse is b/a
3. Representation on the Number Line
- Rational numbers lie between integers
- Between any two rational numbers, there are infinitely many rational numbers
Example: Between 2/3 and 3/4, we can take average: (2/3 + 3/4)/2
4. Decimal Expansion
| Type | Example | Nature |
|---|---|---|
| Terminating | 1/4 = 0.25 | ends after finite digits |
| Recurring / repeating | 1/3 = 0.333… | repeats a block |
| Rational ⇔ decimal is terminating or recurring |
5. Comparison of Rational Numbers
- Bring to same denominator
- Or convert to decimal.
✅ Rational Numbers – Solved Examples
✅ Example 1 – Is 7 a rational number?
7 can be written as 7/1.
Since it is in the form p/q (q ≠ 0), 7 is a rational number.
Answer: Yes, 7 is rational.
✅ Example 2 – Identify which are rational numbers:
√2, 5, -3/7, 0, π
- √2 → cannot be written as p/q → irrational
- 5 → can be written as 5/1 → rational
- -3/7 → already in p/q form → rational
- 0 → can be written as 0/1 → rational
- π → non-terminating and non-repeating → irrational
Rational numbers are: 5, -3/7, 0
✅ Properties of Rational Numbers
✅ Closure Property
Check if rational numbers are closed under addition:
(2/3) + (5/6)
LCM of 3 and 6 = 6
2/3 = 4/6
So,
4/6 + 5/6 = 9/6 = 3/2
3/2 is a rational number.
✅ Closed under addition
✅ Commutativity
Check whether:
1/4 + 3/5 = 3/5 + 1/4
Convert both:
1/4 + 3/5
= (5/20) + (12/20)
= 17/20
3/5 + 1/4
= (12/20) + (5/20)
= 17/20
✅ Both sides are equal → addition is commutative.
✅ Associativity
Check if:
(1/2 + 1/3) + 1/6 = 1/2 + (1/3 + 1/6)
Left side:
1/2 + 1/3 = 3/6 + 2/6 = 5/6
5/6 + 1/6 = 6/6 = 1
Right side:
1/3 + 1/6 = 2/6 + 1/6 = 3/6 = 1/2
1/2 + 1/2 = 1
✅ Both equal → associative
✅ Distributive Property
2/3 × (3/4 + 1/2)
Step 1:
3/4 + 1/2 = 3/4 + 2/4 = 5/4
Step 2:
2/3 × 5/4 = 10/12 = 5/6
Now check separately:
(2/3 × 3/4) + (2/3 × 1/2)
= 6/12 + 2/6
= 1/2 + 1/3
= 3/6 + 2/6 = 5/6
✅ Same result
✅ Identity
Additive Identity
(-7/9) + 0 = -7/9
✅ 0 is additive identity
Multiplicative Identity
(11/5) × 1 = 11/5
✅ 1 is multiplicative identity
✅ Inverses
Additive Inverse
Number: 4/9
Additive inverse: -4/9
Because (4/9) + (-4/9) = 0
Multiplicative Inverse
Number: -3/7
Multiplicative inverse: -7/3
Because (-3/7) × (-7/3) = 1
✅ Rational Numbers on Number Line
Place 3/4
- Divide space between 0 and 1 into 4 equal parts
- Mark 3 parts from 0
✅ 3rd division is 3/4
✅ Decimal Form – Terminating
1/8 = 0.125
Stops → terminating decimal
✅ Decimal Form – Recurring
2/3 = 0.666666…
Digit 6 repeats → recurring
✅ Rational Number Between Two Rational Numbers
Between 1/3 and 1/2
Use midpoint:
(1/3 + 1/2) ÷ 2
= (2/6 + 3/6) ÷ 2
= 5/6 ÷ 2
= 5/12
✅ 5/12 lies between 1/3 and 1/2