Rational Numbers & Repeating Decimals

Dividing 1 by 3 gives 0.333... forever, and 1 by 7 repeats six digits. Some fractions terminate and others repeat, decided by whether the denominator has only 2 and 5 as prime factors. Multiply a repeating decimal by a power of ten, subtract, and the loop becomes a clean fraction. Slide the denominator and repeating digit to see why a decimal ends or loops.

Intuition — What Happens When You Divide?

Denominator n3

👀 The Secret of Division

①1 ÷ 3 = 0.333... → the 3 repeats forever

②1 ÷ 7 = 0.142857142857... → 6 digits repeat

③Try different denominators and observe which are "finite" and which "repeat"

Finite vs Repeating

🔑 When is it a Finite Decimal?

①If the prime factors of the denominator are only 2 and 5 → finite

②Any other prime factor → repeating

③Example: 1/8 = 1/2³ (finite), 1/6 = 1/(2×3) (repeating)

Finite-Decimal Test

Denominator = 2^a × 5^b only → finite

After reducing, the denominator must have no primes other than 2 and 5

Convert Repeating Decimals to Fractions

Repeating Digit1

One-digit repeat → fraction

0.ā = a9

A 1-digit period gives denominator 9

Two-digit repeat → fraction

0.ab̄ = ab99

A 2-digit period gives denominator 99

Mixed Repeating Decimals

Mixed Formula

0.ab̄ = ab - a90

Split the non-repeating and repeating parts

📝 How It Works

①If x = 0.1666...

②10x = 1.666..., 100x = 16.666...

③100x - 10x = 15 → 90x = 15 → x = 15/90 = 1/6

Work It Out

Example 1

Determine whether the fraction 3/40 is a terminating or repeating decimal.

Factor the denominator of the reduced fraction into primes.

40 = 2³ × 5

If the only prime factors are 2 and 5, it terminates.

only primes 2, 5 ⇒ terminating (3/40 = 0.075)

▸ terminating (0.075)

A reduced fraction terminates exactly when its denominator has only 2 and 5.

Example 2

Express the repeating decimal 0.444… as a reduced fraction.

Let x = 0.444… and form ten times it.

x = 0.444…, 10x = 4.444…

Subtracting gives 9x = 4.

10x − x = 9x = 4 ⇒ x = 4/9

▸ 4/9

Multiply a repeating decimal by a power of 10 and subtract to remove the period.

Exam Wrap-up

Core Conversion Rule

n-digit period → put n nines in the denominator

Add zeros for non-repeating digits

Grade-8 school exam type

What is the smallest natural number a for which a/30 is a terminating decimal?

①1

②2

③3

④5

⑤6

▸ ③ 3

Since 30 = 2 × 3 × 5, the prime factor 3 must cancel for termination.

30 = 2 × 3 × 5

a must be a multiple of 3, so the smallest is 3.

a = 3 (3/30 = 1/10 = 0.1)

🎯 Exam Points

①Finite test: only primes 2, 5 in the reduced denominator

②Notation: dots above the first and last digits of the period

③Repeating → fraction: use the 10^n subtraction trick

④Every rational is either finite or repeating

⑤Non-repeating infinite decimals = irrational (G9 content)

Laws of Exponents

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