Decimal to Fraction and Repeating Decimal Converter

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fraction to decimal repeating or terminating

Fraction to decimal: terminating or repeating?

Use the Fraction to decimal mode: type 1/7, 5/12 or a mixed number and see the decimal with its repeating block in parentheses, the period length and the long division until a remainder repeats. This page explains the test that predicts the result without dividing.

The denominator test and the period length

First reduce the fraction. If the denominator contains only the prime factors 2 and 5, the decimal terminates: 3/8 = 0.375 (8 = 2³), 7/20 = 0.35 and 9/25 = 0.36. If any other factor shows up, such as 3, 7 or 11, the decimal repeats. The number of places before the repeating block is the larger exponent of 2 or 5: 1/12 has 12 = 2² × 3, so two places before the block and equals 0.08(3); 1/6 has 6 = 2 × 3, one place, and equals 0.1(6).
The period length depends only on the part of the denominator left after removing the factors 2 and 5, and never exceeds that number minus 1. For 3 and 9 the period has 1 digit (1/3 = 0.(3), 1/9 = 0.(1)); for 11 it has 2 (1/11 = 0.(09)); for 7 and 13 it has 6 (1/7 = 0.(142857), 1/13 = 0.(076923)); for 17 it has 16; for 19 it has 18; for 81 it has 9 (1/81 = 0.(012345679)); and for 97 it has 96 digits. Fractions like 5/12 and 1/12 share the same period; only the non-repeating part changes.
Long division shows why. At each step the remainder is multiplied by 10 and divided by the denominator. Since only d − 1 nonzero remainders exist, some remainder must repeat within d steps, and from then on the digits repeat. In 1 ÷ 6 the first step leaves remainder 4 and so does the second: the remainder 4 returns at once and the 6 repeats forever. If the remainder reaches zero, the decimal terminates. The tool table marks the row where the remainder repeats.

Checked examples

Denominator with only 2s

Input: 3/8
Expected output: 0,375

8 = 2³: ends after 3 places.

One-digit block

Input: 1/3
Expected output: 0,(3)

Remainder 1 returns at once.

One non-repeating place

Input: 1/6
Expected output: 0,1(6)

6 = 2 × 3.

Two non-repeating places

Input: 1/12
Expected output: 0,08(3)

12 = 2² × 3.

Six-digit block

Input: 1/7
Expected output: 0,(142857)

6 is the maximum for denominator 7.

Another six-digit block

Input: 1/13
Expected output: 0,(076923)

The leading zero belongs to the block.

With an integer part

Input: 22/7
Expected output: 3,(142857)

22/7 = 3 + 1/7.

Full tool FAQ

How do you turn a decimal into a fraction?

Count the decimal places, write the digits without the point as the numerator and use 1 followed by that many zeros as the denominator. Then reduce by the gcd. Example: 0.375 has 3 places, becomes 375/1000 and, dividing by 125, 3/8. With an integer part, 2.5 = 25/10 = 5/2 = 2 1/2.

What is the fraction of a repeating decimal?

How do you convert a purely repeating decimal?

What about a repeating decimal with non-repeating digits first?

Is 0.999... really equal to 1?

How can I tell if a fraction terminates or repeats?

How do I type a repeating decimal?

Does it handle negatives, percentages and mixed numbers?

Why do π and √2 have no fraction?

Is there a size limit?

Frequently asked questions

How can I tell that 5/12 repeats without dividing?

Factor the denominator: 12 = 2² × 3. The factor 3 appears, so it repeats. Two places before the block (exponent 2 of the factor 2) and a one-digit block (because of the 3): 5/12 = 0.41(6).

Is the period always as long as the denominator?

No, it is at most the denominator minus 1 and is often much shorter. 1/9 and 1/3 have period 1, 1/11 has 2. The maximum only happens for some primes, such as 7, 17, 19 and 97.

What does the tool do with huge periods?

It computes the period length from the order of 10 modulo the denominator and shows up to 1000 digits, warning when it cuts. For giant denominators where that passes a computation limit, the length is reported as very long.

Which notation is right: 0.1(6) or 0.1666...?

Both describe the same number. Textbooks use either the block in parentheses or a bar over the digit; the ellipsis shows only the first digits. In the notation options you choose which one appears.

Next steps

Decimal	Fraction
0,25	1/4
0,(3)	1/3
1,2(5)	113/90
-0,8	-4/5

Decimal

Fraction

0,25

1/4

0,(3)

1/3

1,2(5)

113/90

-0,8

-4/5

The denominator test and the period length

First reduce the fraction. If the denominator contains only the prime factors 2 and 5, the decimal terminates: 3/8 = 0.375 (8 = 2³), 7/20 = 0.35 and 9/25 = 0.36. If any other factor shows up, such as 3, 7 or 11, the decimal repeats. The number of places before the repeating block is the larger exponent of 2 or 5: 1/12 has 12 = 2² × 3, so two places before the block and equals 0.08(3); 1/6 has 6 = 2 × 3, one place, and equals 0.1(6).

The period length depends only on the part of the denominator left after removing the factors 2 and 5, and never exceeds that number minus 1. For 3 and 9 the period has 1 digit (1/3 = 0.(3), 1/9 = 0.(1)); for 11 it has 2 (1/11 = 0.(09)); for 7 and 13 it has 6 (1/7 = 0.(142857), 1/13 = 0.(076923)); for 17 it has 16; for 19 it has 18; for 81 it has 9 (1/81 = 0.(012345679)); and for 97 it has 96 digits. Fractions like 5/12 and 1/12 share the same period; only the non-repeating part changes.

Long division shows why. At each step the remainder is multiplied by 10 and divided by the denominator. Since only d − 1 nonzero remainders exist, some remainder must repeat within d steps, and from then on the digits repeat. In 1 ÷ 6 the first step leaves remainder 4 and so does the second: the remainder 4 returns at once and the 6 repeats forever. If the remainder reaches zero, the decimal terminates. The tool table marks the row where the remainder repeats.

Checked examples

Denominator with only 2s

Input: 3/8
Expected output: 0,375

8 = 2³: ends after 3 places.

One-digit block

Input: 1/3
Expected output: 0,(3)

Remainder 1 returns at once.

One non-repeating place

Input: 1/6
Expected output: 0,1(6)

6 = 2 × 3.

Two non-repeating places

Input: 1/12
Expected output: 0,08(3)

12 = 2² × 3.

Six-digit block

Input: 1/7
Expected output: 0,(142857)

6 is the maximum for denominator 7.

Another six-digit block

Input: 1/13
Expected output: 0,(076923)

The leading zero belongs to the block.

With an integer part

Input: 22/7
Expected output: 3,(142857)

22/7 = 3 + 1/7.

Full tool FAQ

How do you turn a decimal into a fraction?

What is the fraction of a repeating decimal?

How do you convert a purely repeating decimal?

What about a repeating decimal with non-repeating digits first?

Is 0.999... really equal to 1?

How can I tell if a fraction terminates or repeats?

How do I type a repeating decimal?

Does it handle negatives, percentages and mixed numbers?

Why do π and √2 have no fraction?

Is there a size limit?

Frequently asked questions

How can I tell that 5/12 repeats without dividing?

Factor the denominator: 12 = 2² × 3. The factor 3 appears, so it repeats. Two places before the block (exponent 2 of the factor 2) and a one-digit block (because of the 3): 5/12 = 0.41(6).

Is the period always as long as the denominator?

No, it is at most the denominator minus 1 and is often much shorter. 1/9 and 1/3 have period 1, 1/11 has 2. The maximum only happens for some primes, such as 7, 17, 19 and 97.

What does the tool do with huge periods?

Which notation is right: 0.1(6) or 0.1666...?

Both describe the same number. Textbooks use either the block in parentheses or a bar over the digit; the ellipsis shows only the first digits. In the notation options you choose which one appears.