# Polynesians May Have Invented Binary Math

How old is the binary number system? Perhaps far older than the invention of computers or even the invention of binary math in the West. The residents of a tiny Polynesian island may have been doing calculations in binary—a number system with only two digits—centuries before it was described by Gottfried Leibniz, the co-inventor of calculus, in 1703.

If you’re reading this article, you are almost certainly a user of the decimal system. That system is also known as base-10 because of its repeating pattern of 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 is followed by 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, and so forth. But the decimal system is not the only counting system available. The Babylonians used base-60. The Mayas used base-20. Some Australian aboriginal groups may have used base-5. And of course, today most counting and calculation is done by computers not in decimal but binary, the base-2 system of zeros and ones.

Each system has subtle advantages depending on what sort of counting and calculations are needed. The decimal system is handy considering that people have 10 fingers. But when it comes to division, other systems are better. Because 10 has only two prime factors (2 and 5), dividing by thirds results in an annoyingly infinite approximation (0.3333 … ) whereas the base-12 counting system produces a nice finite solution. (Indeed, some mathematicians have advocated for a worldwide switch to base-12.) Binary, meanwhile, has a leg up on decimal when it comes to calculation, as Leibniz discovered 300 years ago. For example, although numbers in binary become much longer, multiplying them is easier because the only basic facts one must remember are 1 x 1 = 1 and 0 x 0= 1 x 0 = 0 x 1 = 0.

But Leibniz may have been scooped centuries earlier by the people of Mangareva, a tiny island in French Polynesia about 5000 kilometers south of Hawaii. While studying their language and culture, Andrea Bender and Sieghard Beller, anthropologists at the University of Bergen in Norway, were astonished to find a mathematical system that seems to mix base-10 and base-2. “I was so thrilled that I couldn't sleep that night,” Bender says. It could be not only the first new indigenous arithmetic system discovered in decades, but also the first known example of binary arithmetic developed outside Eurasia.

Like all Polynesians, the people who first settled on Mangareva more than 1000 years ago had a decimal counting system. But, according to Bender and Beller, the islanders added a binary twist over the ensuing centuries. Just like English has a few special words like a dozen for 12 and a score for 20, the Mangarevan language has special words for large groups. But their special counting words are all decimal numbers multiplied by powers of two, which are 1, 2, 4, 8 … . Specifically, takau equals 10; paua equals 20; tataua, 40; and varu, 80. Those big numbers are useful for keeping track of collections of valuable items, such as coconuts, that come in large numbers. Bender and Beller realized that the Mangarevan counting system makes it possible to use binary arithmetic for calculations of large numbers, they report today in the Proceedings of the National Academy of Sciences in a paper that even nonexperts will enjoy reading.

But here’s the catch. Even if the native mathematical system of Mangareva employed binary arithmetic, the current residents of the island no longer use that system. Two centuries of contact with the West has resulted in a complete switch to decimal calculation. Even the Mangarevan language itself is now threatened with extinction. Bender and Beller are relying on their analysis of the language and an account of the traditional counting words written by ethnographers in 1938. They acknowledge that it is impossible to prove exactly when Mangareva developed the system, but the entrenchment of the number terms in the language suggests a far-reaching origin. Unfortunately, the anthropologists may have made their discovery just one generation too late to see Mangarevan math in action.

“The hypothesis advanced by the authors is indeed plausible,” says Rafael Núñez, an anthropologist at the University of California, San Diego, “but the absence of original Mangarevan written records constitutes a real challenge.” However, Núñez notes that ironically, “it is the absence of written practices in this culture that makes the hypothesis plausible.” Keeping track of all those calculations in their heads would have been so much easier with the binary math built into the Mangarevan language, he says.

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### John Bohannon

John is a Science contributing correspondent.

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