📚 Class‑8 CBSE Mathematics – Rational Numbers
(Full Study Notes – Markdown)
1. Simple Explanation
A rational number is any number that can be written as a fraction
[
\frac{a}{b}
]
where:
| Condition | Meaning |
|---|---|
| (a) is an integer | numerator |
| (b) is a non‑zero integer | denominator |
| The fraction may be in simplest form (no common factor with (b)) |
Examples:
- ( \frac{3}{4},; \frac{-5}{2},; \frac{7}{1} )
- Whole numbers (e.g. (5 = \frac{5}{1})) and integers (e.g. (-3 = \frac{-3}{1})) are also rational.
- Any decimal that terminates or repeats (e.g. (0.75, 0.\overline{3})) is rational.
2. Key Points
| # | Concept | Important Facts |
|---|---|---|
| 1 | Closure | Adding, subtracting, multiplying, or dividing two rational numbers (except division by 0) gives another rational number. |
| 2 | Number Line | Rational numbers are dense: between any two rationals there lies another rational. |
| 3 | Ordering | If (\frac{a}{b} < \frac{c}{d}) (with (b,d>0)) then (ad < bc). |
| 4 | Operations | Use the same rules as for integers; when denominators differ use LCM. |
| 5 | Simplification | Reduce (\frac{a}{b}) by dividing numerator and denominator by their greatest common divisor (GCD). |
| 6 | Negative Fractions | The minus sign can be placed before the numerator, denominator, or the whole fraction. |
| 7 | Decimal‑Fraction Conversion | A terminating decimal (0.\overline{abc}) with (n) digits becomes (\frac{abc}{10^n}). A repeating decimal (0.\overline{abc}) becomes (\frac{abc}{999}). |
| 8 | Common Denominator | LCM of denominators is the smallest common denominator. |
| 9 | Dividing by a Fraction | Multiply by its reciprocal: (\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}). |
| 10 | Special Forms | 0 is rational ((\frac{0}{1})). |
3. Worked Example
Problem:
Compute
[
\frac{7}{12} ;+; \frac{5}{18} ;-; \frac{3}{4}
]
Step‑by‑Step
-
Find a common denominator
Denominators: (12, 18, 4).
LCM of (12, 18, 4) is (36). -
Rewrite each fraction with denominator 36
| Original | Equivalent (denominator 36) |
|---|---|
| (\frac{7}{12}) | (\frac{7 \times 3}{12 \times 3} = \frac{21}{36}) |
| (\frac{5}{18}) | (\frac{5 \times 2}{18 \times 2} = \frac{10}{36}) |
| (\frac{3}{4}) | (\frac{3 \times 9}{4 \times 9} = \frac{27}{36}) |
- Perform the addition/subtraction
[ \frac{21}{36} + \frac{10}{36} - \frac{27}{36} = \frac{21 + 10 - 27}{36} = \frac{4}{36} ]
- Simplify the result
[ \frac{4}{36} = \frac{1}{9}\quad(\text{divide numerator & denominator by }4) ]
Answer
[ \boxed{\frac{1}{9}} ]
4. Quick Revision Summary
- Definition: (\displaystyle \frac{a}{b}) where (a,b \in \mathbb Z,; b\neq 0).
- Whole numbers & integers are rational (denominator = 1).
- Termination / Repetition in decimals → rational.
- Operations:
- Add/Subtract: use LCM, convert fractions.
- Multiply: multiply numerators & denominators.
- Divide: multiply by reciprocal.
- Ordering Rule: (\frac{a}{b} < \frac{c}{d}) iff (ad < bc) (with positive denominators).
- Conversion:
- Terminating decimal (0.\overline{abc}): (\displaystyle \frac{abc}{10^n}).
- Repeating decimal (0.\overline{abc}): (\displaystyle \frac{abc}{999}).
- Simplification: Always reduce fractions using GCD.
- Key Properties: Closure, commutative, associative, distributive, additive & multiplicative identities, inverses.
Tip for Exam:
- Always check the simplest form before writing the final answer.
- For division, remember “multiply by the reciprocal.”
- Use the ordering rule to compare two fractions directly without converting to decimals.
Happy studying! 🚀