Rational Numbers — Class 8

Clear, concise notes with properties, solved examples, practice questions and answers — mobile-friendly & print-ready.

1. What is a Rational Number?

A rational number is any number that can be expressed as the quotient p/q of two integers p and q, where q ≠ 0. Examples: 3/4, -7/2, 5 (because 5 = 5/1).

Integers are rational
Every integer n = n/1
Terminating decimals
E.g. 0.75 = 3/4
Repeating decimals
E.g. 0.̅3 = 1/3

2. Standard Forms & Representation

Two common forms to represent rational numbers:

  • Fraction form: p/q (in lowest terms)
  • Decimal form: terminating or repeating decimals

Conversion: Decimal → Fraction

For a repeating decimal like 0.̅3 (i.e. 0.333...): let x = 0.̅3. Then 10x = 3.̅3, subtract to get 9x = 3, so x = 1/3.

3. Important Properties of Rational Numbers

PropertyRule / Example
Closure under additionIf a and b are rational, a+b is rational. (e.g. 1/2 + 1/3 = 5/6)
Closure under subtractiona-b is rational.
Closure under multiplicationa×b is rational.
Closure under divisiona/b is rational if b ≠ 0.
Equivalent fractionsp/q = (p×k)/(q×k) for any non-zero integer k.
Reducing to lowest termsDivide numerator and denominator by their GCD.
Note: Rational numbers are dense on the number line — between any two rationals there exists another rational.

4. Solved Examples

Example 1: Add 2/3 and 3/4.
LCM of 3 and 4 = 12 → 2/3 = 8/12, 3/4 = 9/12. Sum = 17/12.
Example 2: Convert 0.125 to a fraction.
0.125 = 125/1000 = 1/8 after simplifying by 125.
Example 3: Show that 0.̅9 = 1.
Let x = 0.̅9, then 10x = 9.̅9; subtract: 9x = 9, so x = 1.

5. Practice Questions (with answers)

  1. Write -7 as a rational number.
    Answer
    -7 = -7/1
  2. Convert 0.̅27 to a fraction.
    Answer
    Let x = 0.̅27 → 100x = 27.̅27 → 99x = 27 → x = 27/99 = 3/11
  3. Simplify 5/6 − 1/4.
    Answer
    LCM(6,4)=12 → 10/12 − 3/12 = 7/12
  4. Is √2 rational?
    Answer
    No — √2 is irrational (cannot be expressed as p/q).
  5. Multiply 3/5 × 10/9.
    Answer
    Cancel 10 and 5 → (3/1) × (2/9) = 6/9 = 2/3

Try these without looking — then expand each answer to show each step clearly.

6. Tips, Tricks & Exam Notes

  • Always reduce fractions to lowest terms before final answer.
  • For repeating decimals, use the x, 10x, 100x method.
  • Watch signs carefully when adding/subtracting mixed signs.
  • Use factorization to find GCD quickly (prime factors).

7. References & Further Reading

  • NCERT Class 8 Mathematics — Chapter: Rational Numbers
  • Any standard Class 8 Maths workbook for extra practice