Decimal to Fraction and Repeating Decimal Converter

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pi as a fraction 22/7 355/113

π as a fraction: 22/7, 355/113 and how to find the best approximation

In the tool Approximate mode, type a long decimal and pick the maximum denominator: the answer is the closest possible fraction inside the limit, with the error and the table of convergents. The shortcuts for π, √2, e and φ come pre-filled.

Why 22/7 and 355/113, and how the tool finds them

π is irrational: its decimal (3.14159265358979...) never ends or repeats, so no fraction of integers equals it. What exists are approximations. The continued fraction of π is [3; 7, 15, 1, 292, ...], and cutting it at each term gives the convergents 3, 22/7, 333/106, 355/113 and 103993/33102. Each convergent is the best possible approximation among all fractions whose denominator is equal or smaller. That is why 22/7 (error of about 0.0013) appears in textbooks and 355/113 (error of 2.7 in 10 million, six correct places) is the engineers favorite.
The maximum denominator changes the answer. Up to 10, the best fraction for π is 22/7. Up to 100 the best is 311/99, a semiconvergent (error 0.00018), better than 22/7 even though it is not a convergent. Up to 1000 and up to 10000 it is 355/113, because the next convergent, 103993/33102, has a denominator that is too large. The tool evaluates the last convergent that fits and also the leftover semiconvergent, and keeps the one with the smaller exact error.
The same applies to other numbers. For e = 2.71828..., up to 100 the best fraction is 193/71 and up to 1000 it is 1457/536. For the golden ratio φ = 1.61803..., the convergents are ratios of Fibonacci numbers: 13/8, 144/89 and 1597/987. For 3.14, which is a terminating decimal, the exact answer is 157/50, and with a denominator up to 10 the best approximation is 22/7. The tool shows when the result is exact and when it is only an approximation.

Checked examples

π with denominator up to 10

Input: 3,14159265358979, máx. 10
Expected output: 22/7

Error of about 0.0013.

π with denominator up to 100

Input: 3,14159265358979, máx. 100
Expected output: 311/99

Semiconvergent, error 0.00018.

π with denominator up to 1000

Input: 3,14159265358979, máx. 1000
Expected output: 355/113

Error of 2.7 in 10 million.

e with denominator up to 100

Input: 2,71828182845904, máx. 100
Expected output: 193/71

Convergent of the continued fraction of e.

φ with denominator up to 100

Input: 1,61803398874989, máx. 100
Expected output: 144/89

Ratio of Fibonacci numbers.

A terminating decimal that is already exact

Input: 3,14, máx. 100
Expected output: 157/50

314/100 reduced by 2.

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

Is 22/7 equal to π?

No. 22/7 = 3.142857142857..., a repeating decimal with block 142857, and π = 3.14159265... They agree on 3.14 and diverge at the third decimal place. The difference is about 0.0013.

Why is 355/113 so accurate?

Because the next term of the continued fraction of π is 292, a very large number. Large terms mean the previous convergent is already unusually close to the real value, and 355/113 sits right before that 292.

Can I use Approximate mode with a repeating decimal?

You can. The tool converts the repeating decimal to its exact fraction and approximates that fraction. If the fraction already has a denominator inside the limit, the result is exact and shows as an exact value in the table.

How many digits of π do I need to type?

For denominators up to 10000, 10 places are enough, because the decimal error stays far below the fraction error. The shortcuts already carry 14 places, enough for any limit in the tool.

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

Why 22/7 and 355/113, and how the tool finds them

π is irrational: its decimal (3.14159265358979...) never ends or repeats, so no fraction of integers equals it. What exists are approximations. The continued fraction of π is [3; 7, 15, 1, 292, ...], and cutting it at each term gives the convergents 3, 22/7, 333/106, 355/113 and 103993/33102. Each convergent is the best possible approximation among all fractions whose denominator is equal or smaller. That is why 22/7 (error of about 0.0013) appears in textbooks and 355/113 (error of 2.7 in 10 million, six correct places) is the engineers favorite.

The maximum denominator changes the answer. Up to 10, the best fraction for π is 22/7. Up to 100 the best is 311/99, a semiconvergent (error 0.00018), better than 22/7 even though it is not a convergent. Up to 1000 and up to 10000 it is 355/113, because the next convergent, 103993/33102, has a denominator that is too large. The tool evaluates the last convergent that fits and also the leftover semiconvergent, and keeps the one with the smaller exact error.

The same applies to other numbers. For e = 2.71828..., up to 100 the best fraction is 193/71 and up to 1000 it is 1457/536. For the golden ratio φ = 1.61803..., the convergents are ratios of Fibonacci numbers: 13/8, 144/89 and 1597/987. For 3.14, which is a terminating decimal, the exact answer is 157/50, and with a denominator up to 10 the best approximation is 22/7. The tool shows when the result is exact and when it is only an approximation.

Checked examples

π with denominator up to 10

Input: 3,14159265358979, máx. 10
Expected output: 22/7

Error of about 0.0013.

π with denominator up to 100

Input: 3,14159265358979, máx. 100
Expected output: 311/99

Semiconvergent, error 0.00018.

π with denominator up to 1000

Input: 3,14159265358979, máx. 1000
Expected output: 355/113

Error of 2.7 in 10 million.

e with denominator up to 100

Input: 2,71828182845904, máx. 100
Expected output: 193/71

Convergent of the continued fraction of e.

φ with denominator up to 100

Input: 1,61803398874989, máx. 100
Expected output: 144/89

Ratio of Fibonacci numbers.

A terminating decimal that is already exact

Input: 3,14, máx. 100
Expected output: 157/50

314/100 reduced by 2.

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

Is 22/7 equal to π?

No. 22/7 = 3.142857142857..., a repeating decimal with block 142857, and π = 3.14159265... They agree on 3.14 and diverge at the third decimal place. The difference is about 0.0013.

Why is 355/113 so accurate?

Can I use Approximate mode with a repeating decimal?

How many digits of π do I need to type?

For denominators up to 10000, 10 places are enough, because the decimal error stays far below the fraction error. The shortcuts already carry 14 places, enough for any limit in the tool.