*Nature*.

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*A tiny island in the Pacific was already using a kind of binary arithmetic in the Middle Ages*

Binary arithmetic, the basis of all digital computation today, is usually said to have been invented at the start of the eighteenth century by the German mathematician Gottfried Leibniz. But a new study shows that a kind of binary was already in use three hundred years earlier among the people of the tiny Pacific island of Mangareva in French Polynesia.

The discovery, made by consulting historical records of the now almost wholly assimilated Mangarevan culture and language and reported in the

*Proceedings of the National Academy of Sciences*[1], suggests that some of the advantages of the binary system adduced by Leibniz might create a cognitive motivation for this system to arise spontaneously even in a society without advanced science and technology.

Pure binary arithmetic works in base 2 rather than the conventional base 10 (the latter quite possibly a consequence of counting on ten fingers). This means that numbers are enumerated as powers of 2: instead of units, tens, hundreds (10**2) and thousands (10**3), the digits of a binary number refer to 1 (2**0), 2 (2**1), 4 (2**2), 8 (2**3) and so on.

Every whole number can be represented in this way using just 1s and 0s, which is why they can be encoded in computers in a system of on-off electrical pulses or switches. The number 13 in binary is 1101 (2**3+2**2+(0x2)+1), for example.

Leibniz pointed out in 1703 that to do simple arithmetic in binary, such as addition and multiplication, you don’t need to remember a whole lot of ‘facts’ about numbers, such as 5+4=9 or 6x7=42. Instead, you need only apply a few simple rules. For addition, say, you just add the 1s and 0s, remembering that 1+1=1 in the next position: 100+101=1001.

The downside to binary is that large numbers require lots of digits. But according to psychologists Andrea Bender and Sieghard Beller of the University of Bergen in Norway, the Mangarevan people found an ingenious answer to that, which they were apparently using even before 1450 AD.

Mangareva is a volcanic island first settled around 500-800 AD, which probably had a population of several thousand before substantial interactions with Europeans began in the eighteenth century. Its highly stratified society survived mostly on seafood and root crops, and needed a number system to quantify large transactions in trade and tributes to chieftains.

Only about 600 Mangarevan speakers now remain on the island, and in any case its indigenous number system has long been superseded by Arabic digits owing to French colonialism. But Bender and Beller have reconstructed it from descriptions written by (mostly European) authors in the nineteenth and early twentieth centuries [2].

They find that the former Mangarevans combined a base 10 with a binary system. They had number words for 1 to10, and then for 10 multiplied by several powers of 2: 10 (takau, denoted K in the new work), 20 (denoted P), 40 (T) and 80 (V). In this notation, for example, 70 is TPK and 57 is TK7.

Bender and Beller show that this system retains the key arithmetical simplifications of true binary, in that you don’t need to memorize lots of number facts but just to enact a few simple rules, such as 2xK=P and 2xP=T.

There are complications with the system too, but the authors argue that “the advantages outweigh the disadvantages.”

Cognitive scientist Rafael Nuñez of the University of California at San Diego points out that some notion of binary systems is actually older than Mangarevan culture. “It can be traced back to at least ancient China, around the 9th century BC”, he says – it can be found in the I Ching, which inspired Leibniz. Nuñez adds that “other ancient groups, such as the Maya, used sophisticated combinations of binary and decimal systems to keep track of time and astronomical phenomena. Thus, the cognitive advantages underlying the Mangarevan counting system may not be unique.”

All the same, say Bender and Beller, a ‘mixed’ system like this isn’t easy or obvious to create. “It’s puzzling that anybody would come up with such a solution, especially on a tiny island with a small population”, Bender and Beller say. “But this very fact also demonstrates just how important culture is for the development of numerical cognition”, they add – for example, how in this case dealing with big numbers can motivate inventive solutions.

Nuñez agrees that the study shows “the primacy of cultural factors underlying the invention of number systems, and the diversity in human numerical cognition.”

*References*

1. Bender, A. & Beller, S.

*Proc. Natl. Acad. Sci. USA*doi:10.1073/pnas.1309160110 (2013).

2. Bender, A.,

*J. Polynesian Soc.*122, in press (2013).