Number Systems • Topic 4 of 4

Decimal Expansions of Rational Numbers

What are decimal expansions of rational numbers? Any rational number (p/q in simplest form, q ≠ 0) has a decimal expansion that is either:

1. Terminating — ends after a finite number of digits (e.g., 1/4 = 0.25, 3/5 = 0.6) 2. Non‑terminating but repeating — goes on forever with a repeating block of digits (e.g., 1/3 = 0.333…, 2/7 = 0.285714285714…)

How to tell if a rational number terminates: When the denominator q (in simplest form) has only the prime factors 2 and/or 5.

  • If q = \(2^{m} \times 5^{n}\), the decimal expansion terminates
  • If q has any other prime factor (3, 7, 11, …), the decimal expansion repeats

Example:

  • 7/20 = 7/(\(2^{2} \times 5\)) → terminates (20 = \(2^{2} \times 5\)) ✓
  • 5/6 = 5/(2 × 3) → repeats (3 is a prime other than 2 and 5)

Repeating decimals notation: A bar is placed over the repeating block. 0.333… = \(0.\overline{3}\) 0.142857142857… = \(0.\overline{142857}\)

┌─────────────────────────────────────────────────────────────┐
│     DECIMAL EXPANSIONS OF RATIONAL NUMBERS - FLOW CHART     │
└─────────────────────────────────────────────────────────────┘

          START WITH RATIONAL p/q (in simplest form)
                        │
                        ▼
               Factorize denominator q
                        │
          ┌─────────────┴─────────────┐
          │                           │
          ▼                           ▼
   q = 2ᵐ × 5ⁿ only           q has any other prime
   (m, n are whole              factor (3, 7, 11, …)
    numbers)
          │                           │
          ▼                           ▼
   ┌─────────────┐              ┌─────────────┐
   │ TERMINATING │              │ REPEATING   │
   │ DECIMAL     │              │ DECIMAL     │
   └─────────────┘              └─────────────┘
          │                           │
          ▼                           ▼
    Example: 7/20 = 0.35        Example: 7/12
    20 = 2² × 5                  12 = 2² × 3 (3 is other prime)
                                  → 0.58333… = 0.58\overline{3}


CLASSIFY THESE FRACTIONS:

┌──────────────┬────────────────────┬──────────────────┐
│  Fraction    │  Denominator (q)   │  Type of Decimal │
├──────────────┼────────────────────┼──────────────────┤
│    1/2       │  2 = 2¹ × 5⁰       │  Terminating     │
│    3/4       │  4 = 2² × 5⁰       │  Terminating     │
│    7/8       │  8 = 2³ × 5⁰       │  Terminating     │
│    1/5       │  5 = 2⁰ × 5¹       │  Terminating     │
│    9/10      │ 10 = 2¹ × 5¹       │  Terminating     │
│    1/3       │  3 (prime 3)       │  Repeating       │
│    2/7       │  7 (prime 7)       │  Repeating       │
│    5/6       │  6 = 2 × 3         │  Repeating       │
│    1/11      │ 11 (prime 11)      │  Repeating       │
└──────────────┴────────────────────┴──────────────────┘


CONVERTING REPEATING DECIMAL TO FRACTION:

     Let x = 0.\overline{142857}
     Multiply by 1,000,000 (6 digits repeat):
     1,000,000x = 142857.\overline{142857}
     Subtract: 1,000,000x − x = 142857
     999,999x = 142857
     x = 142857/999999 = 1/7  ✓
Solved Examples
1
Worked Example
Without actual division, determine if 7/80 has a terminating or repeating decimal.
Solution
  1. Step 1: Write fraction in simplest form: 7/80 is already simplified (HCF(7,80) = 1)
  2. Step 2: Factorize denominator: 80 = 8 × 10 = \(2^{3}\) × (2 × 5) = \(2^{4} \times 5^{1}
  3. Step\) 3: Denominator has only primes 2 and 5

Answer: It is a terminating decimal

2
Worked Example
Express 1/8 as a decimal and classify it.
Solution
  1. Step 1: 1/8 = 1 ÷ 8
  2. Step 2: 8 × 0.125 = 1, so 1/8 = 0.125
  3. Step 3: The decimal ends after 3 digits → terminating

Answer: 0.125 (terminating)

3
Worked Example
Convert the repeating decimal \(0.4\overline{6}\) (0.46666…) into a rational number p/q.
Solution
  1. Step 1: Let x = 0.46666…
  2. Step 2: Multiply by 10 to shift the non‑repeating part: 10x = 4.6666…
  3. Step 3: Multiply by 100 to align repeating block: 100x = 46.6666…
  4. Step 4: Subtract: 100x − 10x = (46.6666…) − (4.6666…) → 90x = 42
  5. Step 5: x = 42/90 = 7/15 (after dividing numerator and denominator by 6)

Answer: 7/15

Key Points

  • Rational numbers (p/q, q ≠ 0) have terminating or repeating decimal expansions
  • Terminating if q (simplest form) has only prime factors 2 and 5
  • Repeating if q has any prime factor other than 2 or 5
  • Repeating decimals are written with a bar over the repeating block: \(0.\overline{3}\)
  • Terminating decimals can be written as repeating with infinite zeros: 0.5 = \(0.5\overline{0}\)
  • Every repeating decimal can be converted back to a fraction using algebra
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.A rational $\tfrac{p}{q}$ (in lowest terms) has a terminating decimal iff $q$ is of the form:
Explanation: Only factors $2$ and $5$ allow termination.
Q2.$\tfrac{3}{8}$ is:
Explanation: $8=2^3$, so it terminates.
Q3.$\tfrac{1}{3}$ is:
Explanation: $0.\overline3$.
Q4.$\tfrac{13}{3125}$ is:
Explanation: $3125=5^5$, so it terminates.
Practice more MCQs → 📝 Assignment · 35 marks