The counting scheme, described today (Dec. 16) in the journal Proceedings of the National Academy of Sciences, uses both decimal and binary numbers, so it isn't a complete binary system from zero to infinity. But the binary portion of the system may have helped ancient people keep track of an elaborate trading network between distant Pacific Islands.
"Those were probably the numbers that were most frequent in their trading and redistribution systems," said study co-author Andrea Bender, a cognitive scientist at the University of Bergen in Norway. "For that specific range, it was helpful to have these binary steps that make mental arithmetic much easier — they didn't have a writing or notational system, so they had to do everything in their mind."
One of the most famous, and avant-garde, mathematicians of the 17th century, Gottfried Wilhelm Leibniz, invented a binary numeral system and showed that it could be used in a primitive calculating machine. Nowadays, binary numbers — a base-2 system where each position is typically written as a 0 or 1 — form the backbone of all modern computing systems.
But new evidence suggests some remote Polynesian islanders may have beat the famous mathematician to the numerical punch line by several centuries.
Bender and her colleague Sieghard Beller were looking through a dictionary from Mangareva, an island with less than 2,000 inhabitants, just 7 square miles (18 square kilometers) large, located about halfway between Easter Island and Tahiti.
"It's only a tiny spot in a vast ocean," Bender told LiveScience.
The researchers noticed Mangarevans had words for numerals 1 through 10. But for numbers 20 through 80, they used a binary system, with separate, one-word terms for 20, 40 and 80. For really large numbers, they used powers of 10 up to at least 10 million.
As an example, to calculate 50 + 70 (which is 120), the Mangarevan system would take the words for 10 (takau)+40 (tataua) and then add it to the word for 10 (takau) + 20 (paua) + 40 (tataua), which would be expressed as 80 (varu) + 40 (tataua).
Solving mental arithmetic
The researchers next looked at the number systems in related Polynesian languages and deduced the Mangarevan system likely evolved to help people solve complex mental arithmetic to support a trading and tribute system that died out in the mid-1400s.
Up until that time, Mangarevans traded across long distances for items such as turtles, octopuses, coconut and breadfruit with people on the Marquesas Islands, Hawaii and the islands around Tahiti. Commoners had to tribute these items to higher-ranking people, all the way to the king, who would then redistribute the bounty at large feasts.
The numbering scheme may be the only known example of an extensive binary numeral system that predates Leibniz. (People in Papua New Guinea also use a binary system, but they don't use words for powers of two, meaning their system doesn't count very high, Bender said.)
"What is fascinating about it is that they show very clearly and very carefully that you can have a very complex number system being used in a culture without needing notation," said Heike Wiese, a cognitive scientist and linguist at the University of Potsdam in Germany, who was not involved in the study.
Ghose, Tia. 2014. “Math Surprise: Remote Islanders Invented Binary Number System”. Live Science. Posted: December 16, 2013. Available online: http://www.livescience.com/41985-binary-math-invented-before-leibniz.html