A: The number of years humans have been calculating with decimals.
A crack team of scholars in Beijing learned this last year when they solved a 23-centuries-old puzzle. And it only took them five years.
The complex mathematical problem arose in 2008, when an alumnus of Tsinghua University donated a bundle of ink-inscribed bamboo strips he'd bought at a Hong Kong art market.
But the strips were an inscrutable mess. They were out of order, and some were broken. All were reeking, caked in mud and mold.
Clearly they'd come from a looted tomb. But what were they?
Cracking the Code
On the top floor of Tsinghua's Research and Conservation Center for Excavated Texts, a multidisciplinary team of researchers got together—and got to work. Laying out the 2,500 strips in a climate-controlled room, they spent three painstaking months drying and cleaning them.
"We had to be very careful," says Wen Xing, a paleography professor at Dartmouth College in Hanover, New Hampshire, who was involved from the start. "They were saturated with water, so we had to stop the mold from growing and making them completely rotten. And we had to use soft brushes, to keep the ink on the strips. It was a very difficult process."
But it paid off. They could soon see a vertical line of calligraphy, brushed in black ink, on each strip, which were 20 inches (51 centimeters) long and a half inch (1.27 centimeters) wide. After they applied antioxidizing chemicals, they carbon-dated the batch to 310 B.C.
For the next four years Xing and his colleagues read through every strip, sorting them by their content and calligraphy style—and finding more than 60 discrete texts.
"Most were historical works," says Xing, "including chapters from theBook of Documents, [which is] one of the Five Classics [of the Confucian canon]. There were some military texts too, all written in a beautiful style used in the ancient state of Chu."
Yet 21 of the strips stood out. They were painted with numerical characters, not alphabetical ones. When Feng Lisheng, a math historian at Tsinghua, placed them in the proper order, they formed a base-10 multiplication matrix—the oldest decimal-based calculator in the world.
How It Works
It looks a lot like a modern multiplication table. The top row and the far-right column contain the same 19 numbers: 0.5, the integers 1 through 9, and multiples of 10 from 10 to 90.
To multiply 8 times 7, for example, find the 8 on the top row and the 7 on the far-right column. Follow the numbers beneath the 8 until they intersect with the numbers to the left of the 7. The answer is at the intersection: 56.
"You can see [the answer] at a glance," says Dauben. "And that's probably its great virtue. It's impressive the way this thing is put together."
(The Chinese written system didn't use a symbol for zero, because it didn't need one. When Chinese mathematicians recorded the result of a computation, says Dauben, they used a specific character for each power of ten. So for 57 they would write 5 tens and a 7. But for 507 they would write 5 hundreds and a 7.)
The Tsinghua table also lets you multiply partial numbers between 0.5 and 99.5, though to do that you have to first convert the equations into sums. For instance, (29.5 × 31.5) would be (20 + 9 + 0.5) × (30 + 1 + 0.5). That creates nine separate multiplications (20 × 30, 20 × 1, 20 × 0.5, then 9 × 30, and so on), each of which can be read off the table. Adding up the answers gives you the final result.
But to what end? Feng says he suspects the table was used to calculate land area, crop yields, and taxes. "We can even use the matrix to do divisions and square roots," he says. "But we can't be sure that such complicated tasks were performed at the time."
Guo Shuchun of the Chinese Academy of Sciences calls the table "very advanced for the world at that time, an important discovery in the mathematical history of China—and the world."
On the Timeline of Math
"Mathematics," says Dauben, "has been around since someone looked up and realized there was a sun and a moon and objects around them. The record of human counting goes back to prehistoric caves, to Paleolithic times, with lines indicating times between months or how many animals were killed on a given day."
After humans learned to count, they developed arithmetic. In the West, that started with the ancient Babylonians and Egyptians. According to Mazur, Sumerian cuneiform number writing dates to 3400 B.C. And well before 2000 B.C. both civilizations were adding, subtracting, multiplying, and dividing.
Yet they did it differently than we do. The Babylonian number system was sexagesimal, or base 60—the basis for 60 seconds, 60 minutes, and 360 degrees. And their multiplication tables, says Dauben, were used to compute fractions.
"But they weren't a matrix setup like [the Tsinghua table], where you take a number and can run down the column to any other given number and find out what the product is going to be."
The Egyptians did use a base-10 system like ours—perhaps based on having 10 fingers and 10 toes—but they didn't have place values. So they represented orders of magnitude with different symbols in hieroglyphs (a coiled rope for 100, a lotus flower for 1,000) or a cursive system called Hieratic.
The Chinese weren't far behind.
"China has had written numerals since as early as the Shang Dynasty, circa 1200 B.C. or slightly earlier," says Dauben. Compared with the Greeks of their time, "they made comparable achievements. It's sometimes said that they didn't develop the concept of 'proof' that's fundamental to Euclid and Archimedes, but this is wrong. They [may not have used] an abstract axiomatic method—and much of their math was based on practical concerns like business, bureaucracy, astronomy, and calendars—but they understood the importance of being able to prove that their results were correct."
That's also when decimal tables appeared in Europe. Though records show they existed as early as the 12th century A.D., they weren't used widely until the Renaissance, when the printing press aided their spread.
Sign of the Times
The multiplication table deciphered at Tsinghua wasn't the first one found in China. But it was particularly sophisticated and practical.
Earlier examples, says Dauben, "only list the results of multiplication: 9 times 9 is 81, 9 times 8 is 72, et cetera. What makes the Tsinghua table unique is its matrix structure, and the simplicity with which it allows any multiplications—or divisions or even, possibly, the determinations of square roots—simply by reference to the table.
"It's considerably more advanced than later times tables produced in the Qin Dynasty. Those tables date to between 221 and 206 B.C., and they show simple sentences like the kind you recited in class: 'Two times one is two, two times two is four, two times three is six.' You can't really use sentences to calculate elaborate multiplications, never mind divisions, square roots, et cetera, in the same way you can with a matrix."
The Tsinghua table was made during the Warring States period, says Xing, a century before the first Qin Dynasty emperor, Qin Shi Huang, unified China.
One of the emperor's first undertakings was to try to stamp out the ideas of Confucius and other philosophers he deemed a threat to his authority. He executed scholars, rewrote texts, burned books, and banned private libraries.
The bamboo strips at Tsinghua escaped that fate, probably because they were buried underground in a tomb. Their survival offers us a glimpse of life—historical, intellectual, philosophical—during a formative period in China.
"They tell us about thinking in early China," says Xing. "They were using characters to describe numbers and do calculations. It also helps establish the place-value system, a crucial development in the history of math. This is material evidence of that."
Over the next 50 years, says Dauben, "the archaeology that's coming out of China is going to change our understanding of history. And this [times table is] just one good example. Until this thing turned up, nobody had a clue that the Chinese had been so clever as to present an immediate visual understanding of multiplication."