Class‑8 CBSE Mathematics – Rational Numbers
Goal – Master the concepts, properties, and operations of rational numbers so that you can solve exam‑style problems (IIT‑JEE, CBSE, Olympiads) with confidence.
1. What is a Rational Number?
| Symbol | Definition | Examples |
|---|---|---|
| ℚ | Set of rational numbers | 0, 1, – 3, ½, 4 ⅞, – 7/9, 0.25, 1.333… |
| ℚ⁺ | Positive rationals | 1/2, 3, 4 ⅝ |
| ℚ⁻ | Negative rationals | – 1/3, – 5/2 |
Formal definition:
A numberris rational if it can be expressed as a quotient of two integerspandq(withq ≠ 0):
[ r=\frac{p}{q},\qquad p\in\mathbb{Z},;q\in\mathbb{Z}\setminus{0} ]
Important: The numerator
pmay be any integer (positive, negative or zero); the denominatorqmust be a non‑zero positive integer.
(Ifqwere negative we can multiply numerator and denominator by –1 to make it positive.)
2. Basic Properties
| Property | Statement | Quick Proof |
|---|---|---|
| Closure under addition | If a,b ∈ ℚ then a+b ∈ ℚ. |
Let a = p/q, b = r/s. a+b = p/q + r/s = (ps+rq)/(qs) → integer over integer. |
| Closure under subtraction | Same as addition. | a-b = a+(-b); -b = –r/s = (–r)/s ∈ ℚ. |
| Closure under multiplication | a,b ∈ ℚ ⇒ a·b ∈ ℚ. |
p/q * r/s = (pr)/(qs). |
| Closure under division | If b ≠ 0, a,b ∈ ℚ ⇒ a/b ∈ ℚ. |
a/b = p/q ÷ r/s = p/q * s/r = (ps)/(qr) (denominator ≠ 0). |
| Density | Between any two rationals there exists another rational. | Let a<b. Take m = (floor((a+b)/2) + 1) → ( \frac{2m}{2m} ) or simply c = (a+b)/2; c ∈ ℚ. |
| Countability | ℚ is countable. | Map each fraction to a unique pair of integers; enumerate pairs in a grid. |
| Ordering | ℚ is totally ordered. | If p/q < r/s iff ps < rq (cross‑multiplication). |
3. Operations with Fractions
3.1 Addition / Subtraction
[ \frac{p}{q};\pm;\frac{r}{s} = \frac{ps \pm rq}{qs} ]
Derivation:
[
\frac{p}{q} = \frac{p s}{q s},\qquad
\frac{r}{s} = \frac{r q}{s q}
]
Adding/subtracting gives the common denominator qs.
Example
[
\frac{2}{5} + \frac{3}{7}
= \frac{2\cdot7 + 3\cdot5}{5\cdot7}
= \frac{14+15}{35}
= \frac{29}{35}
]
3.2 Multiplication
[ \frac{p}{q}\times\frac{r}{s} = \frac{pr}{qs} ]
Example
[
\frac{-4}{9}\times\frac{3}{5}
= \frac{-12}{45}
= \frac{-4}{15}
]
3.3 Division
[ \frac{p}{q}\div\frac{r}{s} = \frac{p}{q}\times\frac{s}{r} = \frac{ps}{qr} ]
Example
[
\frac{5}{6}\div\frac{2}{3}
= \frac{5}{6}\times\frac{3}{2}
= \frac{15}{12}
= \frac{5}{4}
]
4. Converting Between Formats
| Type | Conversion |
|---|---|
| Improper Fraction ↔ Mixed Number |
Improper to mixed:
[ \frac{