The fraction 355/113 (its denominator has three digits) is off from π by only ≈ 2.67 × 10⁻⁷, about 4740 times closer than 22/7 (≈ 1.26 × 10⁻³). No fraction equals π, but none with a denominator below 16,604 improves on it. Here you see which fraction suits the denominator you can use, and why it does so well.
The best fraction for each maximum denominator
“Best” needs a criterion. Here it means the fraction with the smallest error whose denominator is no larger than a limit N. The answer changes in stretches, and the tool to approximate a decimal as a fraction runs this search with whatever limit you choose.
| Maximum denominator N | Best fraction | Error against π |
|---|---|---|
| 7 to 56 | 22/7 ≈ 3.142857 | ≈ 1.26 × 10⁻³, above |
| 99 to 105 | 311/99 ≈ 3.141414 | ≈ 1.79 × 10⁻⁴, below |
| 106 to 112 | 333/106 ≈ 3.141509 | ≈ 8.32 × 10⁻⁵, below |
| 113 to 16,603 | 355/113 ≈ 3.141593 | ≈ 2.67 × 10⁻⁷, above |
| 33,102 to 33,214 | 103993/33102 ≈ 3.14159265 | ≈ 5.78 × 10⁻¹⁰, below |
The table leaves out the limits from 1 to 6 and from 57 to 98. Between 57 and 98, intermediate fractions (semiconvergents) such as 289/92 take over, and between 99 and 105 it is 311/99, which is not a convergent of π. The stretch for 355/113 is very long: across more than 16,000 possible denominators, nothing improves on it in any meaningful way. The next fraction to beat it, 52163/16604, only lowers the error from ≈ 2.668 × 10⁻⁷ to ≈ 2.662 × 10⁻⁷. Between 16,604 and 33,101 other fractions take over, and for N = 33,101 the best is 103638/32989 (error ≈ 1.49 × 10⁻⁹), not 103993/33102, which only enters at N = 33,102. It does not stay on top either: from N = 33,215 on, 104348/33215 beats it, with an error of ≈ 3.32 × 10⁻¹⁰.
How far apart the two best-known ones are
The two fractions can be subtracted exactly, without going through π.
Worked example: the distance between the two
How far apart are the fraction with denominator 7 and the one with denominator 113?
- Find the common denominator. 7 and 113 share no factors, so 7 × 113 = 791 works.
- Convert. The first becomes 2486/791 and the second 2485/791.
- Subtract the numerators. 2486 − 2485 = 1.
The distance is 1/791 ≈ 0.00126.
Check: 3.142857 − 3.141593 = 0.001264, and 1 ÷ 791 ≈ 0.001264.
The difference 1/791 also has the form 1 over the product of the denominators (7 × 113), but these two are not neighboring convergents: 333/106 sits between them. 355/113 and 333/106 are neighbors, and that is why their difference is 1/11978, with 11,978 = 106 × 113. Since the first is above π and the second below, π is trapped between the two, so you know the error of either one is smaller than that difference.
The convergents of π
A convergent is what you get by cutting the continued fraction of π, [3; 7, 15, 1, 292, …], and computing the value of the piece. The ellipsis means the chain never ends, and never repeats. The continued fractions calculator makes that cut for any rational number.
| Coefficients | Convergent | Error against π | Side |
|---|---|---|---|
| [3] | 3 | ≈ 1.42 × 10⁻¹ | below |
| [3; 7] | 22/7 | ≈ 1.26 × 10⁻³ | above |
| [3; 7, 15] | 333/106 | ≈ 8.32 × 10⁻⁵ | below |
| [3; 7, 15, 1] | 355/113 | ≈ 2.67 × 10⁻⁷ | above |
| [3; 7, 15, 1, 292] | 103993/33102 | ≈ 5.78 × 10⁻¹⁰ | below |
The convergents fall alternately on one side of π and the other, and the error shrinks each time. The jump from 333/106 to the next one is the most rewarding, because the error drops from ≈ 8.32 × 10⁻⁵ to ≈ 2.67 × 10⁻⁷ while the denominator barely grows, from 106 to 113.
The 292 and the accuracy of the denominator-113 fraction
The error of a convergent is almost 1 divided by its denominator times the next one. For the one with denominator 113, the next denominator is that of 103993/33102, and 1 ÷ (113 × 33,102) ≈ 2.67 × 10⁻⁷, which matches the real error. A large coefficient sends the next denominator soaring: with the 292 it goes from 113 to 33,102, and that is why the previous convergent sits so close to π. With a small coefficient, like the 1 that comes before the 292, the next denominator grows little and the improvement is small.
In terms of measurements, a circle with diameter 1000 has a circumference of 3141.592654 units. With the fraction of denominator 113 you get 3141.592920, a difference of ≈ 0.000267. With the denominator-7 fraction you get 3142.857143, which overshoots by ≈ 1.26.
3.14 is worse than 22/7
3.14 is the fraction 157/50, and it is written so often that it seems finer than it is. Its error is ≈ 1.59 × 10⁻³, larger than that of the denominator-7 fraction (≈ 1.26 × 10⁻³). It falls below π, while that one falls above. To move between decimals and fractions there are the fraction to decimal conversion and the decimal to fraction one. They turn 3.14 into 157/50 exactly. The guide to exact and repeating decimals explains why the decimal 3.142857 (with a bar over 142857) for the denominator-7 fraction repeats and the one for π does not.
No fraction equals π
Johann Heinrich Lambert proved that π is irrational in the 1760s (1761 is the date usually given for his memoir, and 1768 for its publication). Irrational means no fraction of integers equals π, so every approximation has an error greater than zero, and that is why you write ≈ and not = between one of these fractions and π.
Per English Wikipedia, Archimedes showed that π lies between 223/71 and 22/7. Zu Chongzhi is credited with both of the best-known fractions, and the Spanish-language Wikipedia article on π adds that the denominator-113 value was not matched for more than nine centuries, until the 15th century.
Which one to choose for the job
To estimate in your head, 3 or the denominator-7 fraction. For a hand calculation with six good decimals, the one with denominator 113. For a program that needs a very precise fraction, 103993/33102 (with denominators up to 33,214; if you allow more, 104348/33215 improves on it). If the final result must be exact, leave the symbol π in the expression instead of substituting it. Another irrational number that can also be approximated with fractions is the golden ratio, which you will find in the article on the golden ratio and Fibonacci.
This page draws on the English Wikipedia article “Pi”, sections on the history of computation, ancient mathematics and the irrationality of π (accessed October 17, 2026), and on the Spanish-language article Número π for the Zu Chongzhi detail.