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You probably use 22/7 for π in school. It is easy and good to two decimal places. Fifteen centuries ago a Chinese astronomer gave a second fraction, 355/113, that is correct to six. This post explains who Zu Chongzhi was, checks his numbers, shows with continued fractions why 355/113 is unusually good, and lists the slips students make when they approximate π.
Who was Zu Chongzhi?
The MacTutor biography from the University of St Andrews says he was born in 429 in Jiankang, now Nanjing, into a family of court officials and calendar experts. He learned mathematics largely from Liu Hui's commentary on the Nine Chapters on the Mathematical Art and served as an official under Emperor Xiao-wu.
His main job was the calendar. In 462 he proposed a new one built on a 391-year cycle with an extra month in 144 of those years. MacTutor reports that he took the tropical year as 365.24281481 days, about 50 seconds from the true value. A minister named Tai Faxin attacked it as violating the classics. Zu replied, in MacTutor's translation, that his calendar was "not from spirits or from ghosts, but from careful observations and accurate mathematical calculations." The emperor approved it, but died in 464 before it came into use, and his successor cancelled it.
What did he find out about pi?
Zu's book, the Zhui shu, is lost. Our main evidence is a passage in the History of the Sui dynasty, compiled in the seventh century, which MacTutor quotes. Taking a circle of diameter 10,000,000 units, Zu found its circumference to be more than 31,415,926 and less than 31,415,927. So
3.1415926 < π < 3.1415927.
The same passage gives two fractions: an accurate one, 355 to 113, and an approximate one, 22 to 7. MacTutor estimates that to get these bounds he would have needed an inscribed polygon of 24,576 sides, which is a hexagon doubled 12 times, with hundreds of square roots carried to about nine decimal places. A 2019 paper by Mohammad K. Azarian notes that this six-decimal value was the best anywhere until Jamshid al-Kashi's work in 1424.
How good is 355/113? A worked check
Divide on a calculator:
355 ÷ 113 = 3.14159292...
π = 3.14159265...
The difference is about 0.00000027, or 2.7 × 10⁻⁷. Compare 22/7 = 3.142857..., which is off by about 0.00126. So 355/113 is roughly 4,700 times more accurate, while its denominator is only about 16 times larger.
A brute-force search shows that no fraction with a denominator smaller than 16,604 gets closer to π than 355/113. The next better one is 52163/16604.
Why is 355/113 so good? Continued fractions
Write π as a whole number plus a fraction, then flip the fractional part, and repeat:
π = 3 + 1/(7 + 1/(15 + 1/(1 + 1/(292 + ...))))
Stopping at each stage gives the "convergents" 3, 22/7, 333/106 and 355/113. Each is the best approximation for its size. The next term in the expansion is 292, which is very large. A large term means the convergent just before it is unusually accurate, and that convergent is 355/113.
MacTutor also describes one way Zu might have found it, though nobody knows for sure. If a/b ≤ c/d, then (a + c)/(b + d) lies between them. Starting from 3/1 below π and 22/7 above, try (3x + 22y)/(x + 7y). Setting this close to 3.1415926 gives y ≈ 16x, and with x = 1:
(3 + 22 × 16)/(1 + 7 × 16) = (3 + 352)/(1 + 112) = 355/113.
What mistakes do students make when approximating pi?
Treating 22/7 as exact. It is an approximation, and it is slightly too big. If a question says "take π = 22/7", use it; otherwise keep π symbolic as long as you can.
Rounding before the last step. Rounding π to 3.14 at the start of a long area or volume problem can change the third significant figure of your answer.
Comparing fractions by eye. Always convert to decimals or compute the difference before claiming one approximation is better.
Our guide to areas related to circles for Class 10 has practice where these choices matter.
What happened to his work?
MacTutor reports that after editing by Li Chunfeng, the Zhui shu became a text for the imperial examinations in 656 and was among the Ten Classics reprinted in 1084. It was then dropped because students at the Imperial Academy found it too hard, which probably explains why it was lost in the early twelfth century. Zu worked with his son, Zu Geng, also a mathematician. For a later, longer calculation, read about Jamshid al-Kashi and pi to sixteen digits.
What can a student take from Zu Chongzhi?
He backed every claim with measurement and calculation, and said so when challenged. He also found a compact answer, a single fraction, that held most of the precision of a long decimal. When you check a result, ask both questions: is it accurate, and is there a simpler form? If you get stuck setting up a calculation like this, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer.
Try it: compute 333/106 and 355/113 as decimals and find how far each is from 3.14159265.
