0.3̇ · Repeating Decimals
Remainders on a treadmill — the decimal that never ends.
Write 1 ÷ 3 as a decimal — will it ever end?
The unbelievable truth that 0.999… equals exactly 1.
Open the immersive version →Why does this question matter?
Try writing 1÷3 as a decimal and trouble erupts — 0.333… never ends. We needed a way to write an infinite number on finite paper.
The key is the division's remainder. Dividing by 3 leaves remainders of only 0, 1, 2 — a treadmill of finite rooms, so some remainder must return, and from that moment the digits repeat identically. Hence the dot notation (0.3̇) over the repeating block.
More surprising is the reverse. Shift the decimal point with ×10 or ×100 and subtract: the infinite tails cancel completely — every repeating decimal returns to a fraction. The territory of the rationals is settled here.
Pause and try it yourself
Make a prediction first. Change one input at a time and compare what happens with what you expected.
Predict first — write 1 ÷ 3 as a decimal. What happens?
Divide one digit at a time
10 ÷ 3 gives digit 3, remainder 1 — append a 0 to the remainder and divide again. Watch the remainders.
0.
1 ÷ 3
Read the idea from another angle
The culprit behind repetition is the remainder treadmill.
Dividing by n leaves at most n−1 possible remainders — finitely many rooms, so by the pigeonhole principle a revisit is forced. Even why the repeating block is shorter than the denominator follows from this one line.
The trick that beats infinity: shift and subtract.
Multiply x = 0.121212… by 100 and the tail copies itself exactly. In 100x − x the infinite tails cancel — your first experience of finishing an infinite task with finite arithmetic, and the prototype for handling series later.
Now read the definitions with their conditions
What the division machine confirmed, in mathematical language.
Repeating notation
A dot over the repeating block — the agreement that writes infinity in finite symbols.
Back to a fraction
Shift the point by one repeating block and subtract — the infinite tails cancel. Every repeating decimal is rational.
When decimals terminate
If the reduced denominator has any prime factor besides 2 and 5, the decimal must repeat — because 10 = 2×5.
Make one more distinction
0.999… is just a tiny bit less than 1.
0.999… = 1, exactly. Let x=0.999…; then 10x=9.999…, subtract: 9x=9, so x=1. Endless 9s don't mean 'never reaching 1' — they are another name for 1.
A decimal that never ends can't be a fraction.
If it repeats, it always becomes a fraction — 0.121212… = 12/99 = 4/33. Only decimals with no repetition at all (irrationals) escape. The border isn't 'infinite'; it's 'no repeat'.
Where else can this idea help?
The calculator's 0.3333333
The calculator shows only the first eight digits; the true answer is 0.3̇. Separating a machine's display limit from a number's nature starts here.
Splitting a bill three ways
$100 among 3 people is $33.33… — it never lands exactly. In practice someone pays $33.34: handling the 'rounding remainder' is a daily ritual of repeating decimals.
The magic of 1/7: 142857
1/7 = 0.142857142857… — double or triple the block and the same digits reappear, merely rotated (285714, 428571). The most famous number trick repeating decimals ever produced.
Loops and compression
An endlessly repeating waveform compresses into finite data — 'repetition can be written finitely', the very principle of the repeating decimal, echoes in signal compression.
Explain it in your own words
Try another question
Q1Write 2/11 as a decimal.Show answer ▾
0.181818… = 0.1̇8̇. The remainders cycle 2 → 9 → 2 → 9, repeating 18. Check backwards: 18/99 = 2/11 ✓
Q2Is 3/8 terminating or repeating?Show answer ▾
Terminating — 0.375. The denominator 8 = 2³ has only the prime 2. No prime besides 2 and 5 in the denominator → terminates; any other prime (like the 3 in 1/6) → repeats.
Q3What is 0.4999…?Show answer ▾
0.5. Let x=0.4999…; then 10x=4.999…, 9x=4.5, x=0.5. The same principle as 0.999…=1 — endless 9s are always another name for 'one step up'.
Try answering yourself before revealing it — getting it wrong is where learning starts.
What question comes next?
Previous concept
Decimals
The repeating decimal is the 'guest who never leaves' arriving in the world of place value.
← Decimals labLeads to next
Irrational Numbers
Every repeating decimal came home to a fraction. So what of endless decimals with no repeat at all? — the holes in the world of fractions.
Go to the Irrational Numbers lab →Related lab
Decimals
Meet the terminating world first, and 'never ends' lands with full shock.
Go to the Decimals lab →Related lab
Divisors & Multiples
The terminate/repeat border is the denominator's prime factors (2 and 5) — divisor sense is the judge.
Go to the Divisors & Multiples lab →