The fraction for π that nothing smaller has beaten in 1,500 years
355/113 agrees with π to seven decimal places. The next fraction that beats it needs a denominator 147 times larger and improves the error by 0.2%.
Some time before the year 500, in the Southern Dynasties, Zu Chongzhi worked out that π lies between 3.1415926 and 3.1415927. He also wrote down a fraction: 355/113. Fifteen centuries later it is still the best fraction for π that anyone can reach with three digits over three, and it is better than it has any obvious right to be.
How much better
355/113 comes out at 3.1415929204 against π’s 3.1415926536. The error is 2.7 × 10⁻⁷, which is seven decimal places from a six-digit fraction.
The way to feel that is to ask what else was available. Sweep every denominator from 1 to 112, take the nearest numerator for each, and keep the closest result. The winner is 333/106, and it is out by 8.3 × 10⁻⁵ — 312 times further away. Something happens between a denominator of 106 and a denominator of 113 that is worth more than every improvement in the hundred steps before it.
And nothing takes the record back for a long time. 355/113 is the closest fraction to π for every denominator up to 16,603. The first fraction that finally beats it is 52163/16604, which needs a denominator 147 times larger to reduce the error by 0.21%.
The 292
The reason is visible the moment π is written as a continued fraction:
π = [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, …]
Truncate that after each term and the fractions that fall out are 3, then 22/7, then 333/106, then 355/113. These are the convergents, and a theorem of the subject says each one is the best rational approximation available at its own size: no fraction with a smaller denominator comes closer. How Close a Fraction Can Get does not cite that theorem so much as carry it out, sweeping every smaller denominator and reporting what it finds.
The term that matters is the 292. A large term means the leftover at that point was very nearly nothing, and a leftover of nearly nothing is another way of saying the fraction already in your hand was very nearly exact. So the convergent immediately before the 292 gets seven decimals. Compare 22/7, which sits after a term of only 7 and manages three.
The size of the next term tells you how good the fraction you are already holding is. Zu Chongzhi had no continued fractions; the method is Euler’s, twelve hundred years later. What he had was the number, and the number is the same either way.
Which numbers are easy
Some irrationals are far more obliging than others, and the expansion says which. √2 is [1; 2, 2, 2, …] and e is [2; 1, 2, 1, 1, 4, 1, 1, 6, …]: patterns that can be stated and proved. Compare each error against the square of its denominator and √2 settles at 0.354, e wanders, and π lurches about depending on which term it has just passed.
The extreme case is the golden ratio, [1; 1, 1, 1, …]. Every term is the smallest a term can be, so every convergent is the least possible improvement on the one before, and the same measurement settles at 0.447. That number is 1/√5, and Hurwitz proved in 1891 that no irrational can stay above it for ever. φ presses right against the ceiling. Calling it the most irrational number sounds like a flourish; it is a measurement, and 0.447 is where it lives.
What nobody knows
π has no pattern of that kind. Its terms have been computed into the billions and no rule has ever been found in them, which leaves an ordinary-sounding question open: do they stay bounded, or is there a term somewhere out there far larger than 292?
If there is, then sitting just past it is a fraction that approximates π better than anything of its size has any business doing, exactly as 355/113 does. Nobody can say whether such a thing exists. No theorem accounts for the 292. It is what π happens to do.
References (2)
- the best-approximation theorem and the role of a large partial quotient Khinchin, A. Ya. (1964). Continued Fractions (3rd ed.). University of Chicago Press.
- the √5 in the approximation bound, and why the golden ratio saturates it Hurwitz, A. (1891). Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche. Mathematische Annalen 39(2), 279–284.