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.
Change the denominator and the repeating digit and the decimal ends or loops.
Intuition — What Happens When You Divide?
If you only jot down the first few digits after the decimal point, two writings of the same fraction can look like different numbers. Once the fraction is in lowest terms, the question that separates a decimal that ends from one that repeats is whether the denominator’s prime factors are only powers of 2 and 5. Both kinds can be written back as a fraction, so they stay with the rationals; an expansion that goes on forever without a repeating block does not. Asking what primes make up the denominator, before you trust a handful of copied digits, is the habit that keeps those writings from drifting apart.
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Visualize the decimal expansion of 1/n on the number line
A gold point marks 1/n on a number line between 0 and 1, and under the line the expansion of 1 ÷ n is written out to many places. Changing the denominator n from 2 through 12 shifts that point and the digit string together; if a block repeats, the period and its length are written, and if not, a note says the decimal does not repeat. The other picture stacks a one-digit repeating decimal x = 0.aaa…, ten times that value, the subtraction that leaves 9x, and x = a/9; when a reduction is possible, the reduced fraction and a sector of a circle sit beside it. Changing the repeating digit from 1 through 8 updates the numbers in those lines and the share of the circle at the same time.
3
👀 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
After reducing, the denominator must have no primes other than 2 and 5
Convert Repeating Decimals to Fractions
A long tail after the decimal point is not enough reason to call a number rational. An expansion that never repeats cannot be written back as a fraction, and something that looks as if it repeats can still terminate if the denominator, split into primes, leaves only 2 and 5. Writing a few leading digits without an overline on the repeating block makes the same number look like another decimal. Checking the denominator’s primes before you trust the dots is the order that keeps terminating and repeating decimals from being mixed up.
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How to convert 0.nnn... into a fraction
1
One-digit repeat → fraction
0.a̅ = a9
A 1-digit period gives denominator 9
Two-digit repeat → fraction
0.a̅b̅ = ab99
A 2-digit period gives denominator 99
Mixed Repeating Decimals
Mixed Formula
0.ab̅ = 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.
1
Factor the denominator of the reduced fraction into primes.
40 = 2³ × 5
2
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.
1
Let x = 0.444… and form ten times it.
x = 0.444…, 10x = 4.444…
2
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
1
Since 30 = 2 × 3 × 5, the prime factor 3 must cancel for termination.
30 = 2 × 3 × 5
2
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: a dot and an overline mean the same; formulas here use the overline
③Repeating → fraction: use the 10^n subtraction trick