Zu Chongzhi and Pi

Six digits after the decimal point. No machine of any kind. Not even the decimal point we write today. Only a handful of bamboo counting rods laid out on a counting board. Yet this man pinned down pi to six correct digits, without a single one wrong. And for about 900 years after that, no one on earth went past him.

It sounds hard to believe. It should not have been possible.

His name was Zu Chongzhi. He was born in 429 in the south of China, and he served the imperial court as an astronomer, a mathematician and a builder of all kinds of machines. He died around the year 500. His whole life turned on two questions that seem to have nothing to do with each other. How long is a circle? And where should the leap month go?

The first question is so simple that a child could ask it. Take anything round: the rim of a conical hat, the mouth of a water jar, the wheel of an oxcart. Measure the distance around it, then the distance across. The distance around is always the same number of times longer than the distance across. That number never changes, whether the round thing is as small as a coin or as large as the moon. That number is pi. The only trouble is that it never stops. Its digits run on and on without end, and they never repeat in any pattern.

The old way of chasing it sounds strange at first. To measure a curve, they set the curve aside completely. Inside the circle, they drew a shape with many straight sides, each corner touching the rim exactly. With six sides, the shape is still crude and looks nothing like a ring. With twelve sides, it is a little rounder. Each time every side is split in two, the number of sides doubles, and the shape hugs the rim a little more closely. The mathematician Liu Hui, who lived before Zu Chongzhi, opened this path. Later generations believe that Zu Chongzhi followed that path and went much farther.

To reach his number, the shape had to be pushed to tens of thousands of sides.

Think of a fence made of tens of thousands of tiny wooden slats. Each slat is perfectly straight and so short that it looks like a dot, and together they wrap snugly around the curve. To get the number, every slat must be added up, one by one. And finding the length of each slat means taking a square root, something a calculator now does in a second. If he took that path, he had to do it by hand. Hundreds of times. Each time to as many as nine digits. In his day, all such enormous work was done with bamboo rods pushed back and forth across a counting board.

What he left behind was not a single number. It was a clamp. He said that pi is greater than three point one four one five nine two six, and smaller than three point one four one five nine two seven. The two jaws of the clamp sit so close together that almost no gap is left between them. The remarkable part is that he did more than guess right. He showed the truth could only lie inside that narrow gap, and nowhere else.

He also left a much lighter gift. A fraction. 355 divided by 113. People back then called it the close ratio. Its digits hold a neat trick. One, one, three, three, five, five. Write three pairs of odd numbers in a row, cut the line in half, and put the second half on top of the first. A child can learn it in one evening. Yet this almost comically easy fraction is correct to six digits after the decimal point. For more than 900 years, it was the neatest and most accurate way to write pi that humankind had. He left a rougher fraction for everyday use as well, 22 divided by 7, called the approximate ratio.

Here comes the strangest part of the whole story. No one knows how he did it.

He did write it down, in a book called Zhui Shu, which he compiled with his son. More than 100 years after his death, the imperial court put the book on the examination syllabus of the greatest school in the capital. Then it quietly took the book off again, because the students could not keep up. By the early 12th century the book was lost for good, and no one has seen it since. So later generations had his result, but not the road he took to reach it. It is like finding a bridge across a deep gorge, with no footprints left nearby to show how anyone built it.

That number stood unbeaten for a very long time. Only around the start of the 15th century did anyone go past him. In 1424, an astronomer named al-Kashi went much farther, to 16 digits. More than 900 years lay between Zu Chongzhi and al-Kashi. Nine hundred years is about 30 generations, grandparent telling child, child telling grandchild, and still no one had touched that record.

Yet the part of Zu Chongzhi that lives on in Vietnamese homes is not pi. It is the calendar.

In 462, when he was only 33, he presented a new calendar called the Daming calendar. What troubled him most in it was the leap month. A year counted by the moon is about 11 days shorter than a year counted by the sun. Left alone, within a few decades the first day of the year drifts out of spring and lands in the middle of summer. The cure is to slip in an extra leap month now and then. Those before him put in seven leap months every 19 years. He worked it out again and said that 144 leap months in 391 years would fit much more closely. The emperor approved it but died before it came into use. The next emperor listened to an official who opposed it and canceled it. Only in 510, about ten years after Zu Chongzhi had died, was the calendar put into use.

The Daming calendar later faded into the past, like every other calendar. But the question he wrestled with has not gone anywhere. The calendar sheet hanging on the wall of a Vietnamese home, the one with two rows of numbers, one solar and one lunar, still answers that question every day. It is a lunisolar calendar, a heritage shared across East Asia that Vietnamese people keep in their own way. Every few years comes a year of thirteen months, and someone in the house asks which month is doubled this year. It is thanks to those leap months that the weather is always turning warm at Tết, the Lunar New Year festival in Vietnam. And the Mid-Autumn Festival always comes once autumn has arrived. Someone may count on their finger joints to see how many days are left until the last night of the old year. That person is using the very balance of moon and sun that people like Zu Chongzhi worked to refine more than 1,500 years ago.

Today a pocket calculator shows pi to more than 10 digits in the blink of an eye. But the first six digits of that long, long string are exactly what one man, sitting before a handful of bamboo rods, worked out 15 centuries ago. Those six digits have never once needed correcting.