Edited image via Wikimedia Commons
Your computer operates on a bunch of 1s and 0s, a system better known as binary notation. But despite powering the digital age, binary is old, having received its first formal description in the seventeenth century by Gottfried Wilhelm Leibniz. However, it turns out that even Leibniz was a little late to the game.
A report in the Proceedings of the National Academy of Sciences claims that hundreds of years before his time, the people of the Polynesian island Mangareva had already been successfully utilizing a binary system to manage their cultural transactions.
The Mangarevan binary (base-2) system was not pure, but part of a more complex system, involving a decimal (base-10) component that is more familiar to us in our daily lives. This combination system was distinct from the decimal numerical systems of the rest of Polynesia, indicating there must have been specific reasons for the development of a mixed system. Those reasons can be found in the cultural milieu of Mangareva.
On the island, write researchers Andrea Bender and Sieghard Beller, “the chiefs secured a tight and dominant tenure over the typically scarce crop lands and requested tributes from their peasants, but also redistributed a considerable proportion of it during public feasts.”
Mangareva is a tiny island that's part of an isolated group in the south Pacific.
The tributes consisted of highly valuable items, like fish, turtles, and octopus. Accordingly, they had a system that allowed for general counting for regular old stuff, via “a highly regular, decimal system with large power numerals,” and counting of those specific, highly valued items via “three binary steps superposed onto a decimal structure.”
“The specific systems… also served the purpose of emphasizing the special status of these objects,” write Bender and Beller.
Though it may seem more complicated to us to deal with both base-2 and base-10—indeed, Bender and Beller note the difficulty inherent in learning such a system—this was actually expedient for the Mangarevans.
Table A shows addition of numbers 1-10 (10 is represented by K), and Table B shows multiples of 10 in the Mangarevan system. To help with Table B, K is worth 10, P is work 20, PK is worth 30, and so on.
“In binary arithmetic, you get by with simple transformations rather than arduous calculations,” Bender explained to me in an email. “The decimal parts of the system helps to keep number representations compact, which is not the case in a purely binary system.”
Together, these two elements make “mental arithmetic” more straightforward, which was important in a society lacking numerical notation, where all calculations needed to be done mentally. This system fit the needs of its people, allowing them to do arithmetic without stretching the boundaries of their cognitive capabilities.
For Bender and Beller, this represented a significant moment not just for the history of mathematics, but also for anthropological understandings of human complexity. As Bender told me, “even in the absence of notational systems or technological advances, people can come up with ingenious solutions for mathematical problems.”
In other words, without notation and without technology, the Mangarevans managed to scoop Leibniz several hundred years ahead of his time.
Though this combination system died out as the influence of the French gained ground in Polynesia beginning in the nineteenth century, these findings are important because, in the words of Bender, they “highlight the relevance of our cultural heritage as humans in all its diversity and… the development of cultural complexity.” Not to mention that expand our understandings of the history of binary, proving that it had its uses well before the time of birth of machines.