Binary, or base two, is the number system that computer systems use, as opposed to the decimal, or base 10, system used in our day-to-day lives.
Binary is generally associated with high technology and modern mathematics.
However, in a new paper published Dec. 15 in the Proceedings of the National Academy of Sciences, psychologists Andrea Bender and Sieghard Beller discuss how centuries ago, a Polynesian culture on the small island of Mangareva developed a binary system to facilitate counting and calculations.
Mangareva is located in French Polynesia, about 1,000 miles southeast of Tahiti. Humans settled the island in three main waves — two waves of Polynesian settlement between 500 and 800 CE and between 1150 and 1450 CE, and a third wave after European colonization in the 19th century.
Mangarevan society, like many Polynesian societies, was based around a strict hierarchy of chiefs and peasants. The economy was built around trade, tributes, and feasts — peasants would offer the chief tributes of staple food products, particularly turtles, fish, coconuts, octopuses, and breadfruit, and these goods would be redistributed by the chief at large feasts.
This economic organisation made counting very important in Mangareva and other similar Polynesian cultures — keeping track of trade, tribute, and feast goods was essential to the system, and must have been quite difficult in the absence of written notation.
All Polynesian cultures, including the Mangarevans, had a general counting system for day-to-day affairs. This system was a decimal system, based on powers of 10, similar to our own.
However, for those important tribute goods — turtles, fish, coconuts, octopuses, and breadfruit — the Mangarevans developed a special counting system, based partially on binary. This system was recorded by the French missionaries who came to the island in the 19th century, and the paper’s authors note some of the missionaries’ ironic role in both recording, and, by introducing literacy and Arabic numerals, leading to the extinction of this unique system.
The authors relied on the missionaries’ reports, anthropological inference, and an abstract analysis of the system, to get an idea of how it worked and how it was used.
The special counting system is a hybrid of a decimal system and a binary system. Decimal systems like the one we use are based on powers of 10 — we have 10 digits (0, 1, 2, …, 9) and we count higher numbers by using digit multiples of powers of 10 — 234 is two hundreds, plus three tens, plus four ones.
Binary systems like those used by computers are based on powers of two. There are only two digits — 1 and 0, and place-value is based on the powers of two: 1, 2, 4, 8, 16, 32, and so on. Counting in binary, we start with 1, and then two is 10: 2 + 0. Three is 11: 2 + 1. Four is a power of two, and is written 100: 4 + 0 + 0. Five is 101: 4 + 0 + 1.
The Mangarevan system combined the decimal and binary systems in a unique way. Small numbers — one through nine — are represented by their normal digit words. But, for medium size numbers, the system switches over to binary. The Mangarevans had special words for 10, 20, 40, and 80 — the first few powers of two, multiplied by 10. For larger numbers, the system switched back to decimal, taking decimal multiples of 80.
So, a number like 112 would be represented as the Mangarevan language equivalent of “80 20 10 two”: 112 = 80 + 20 + 10 + 2. Another example: 361 would be “four eighties, 40, one”: 361 = 4 x 80 + 40 + 1.
The big advantage of this hybrid system is that much of arithmetic becomes much easier, especially in a culture without writing. Addition in a decimal system has a pretty large number of rules that we have to just memorize to be able to efficiently add. For example, 5 + 6 = 11 is something that gets drilled into most schoolchildren, as it is pretty clearly impractical to start at 5 and count one at a time up 6 more every time we want to add these two numbers.
In the Mangarevan system, addition in the decimal parts works just like this.
But, when adding in the binary part of the Mangarevan system, there are only two basic addition rules. If a power of two number is added to itself, you get the next power of two number up: 20 plus 20 equals 40. If a power of two number is added to a different power of two number, just include both in the sum: 20 plus 10 is just “20 10”.
This makes addition very straightforward in the Mangarevan system, a very useful property for counting up amounts of tribute goods.
The Mangarevan system is impressive, since it shows how different cultures can develop diverse number systems based on their needs. The human mind, and the human capacity for numeracy, is an incredibly creative and flexible thing.
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