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Posted: 17122013 21:24 Post subject: Early Polynesians used binary to ease mental arithmetic 



Quote:  Early Polynesians used binary to ease mental arithmetic
http://www.newscientist.com/article/dn24766earlypolynesiansusedbinarytoeasementalarithmetic.html#.UrC_yfRdXtk
19:00 16 December 2013 by Jacob Aron
Polynesian islanders spoke the language of computers centuries before the first programmer was born. It seems that inhabitants of Mangareva island in French Polynesia created their own particular hybrid of decimal and binary number systems to do mental arithmetic.
The binary system can represent any number using a string of 1s and 0s. Though it is used by computers today, it was first described by the 17th century mathematician Gottfried Leibniz.
First you create a series of columns, each one devoted to a different power of 2, starting with 1 (20), followed by 2 (21), 4 (22), 8 (23) and so on. You then put 1s in the columns needed to make up the target number and 0s in the rest. So 2 is represented by 1 0, while 8 is represented by 1 0 0 0, and 13 is represented by 1 1 0 1, and so on.
Andrea Bender and Sieghard Beller at the University of Bergen in Norway studied the Mangarevan language, which dates back to AD 1500, or even earlier. The pair say that, as in the decimal system, there are Mangarevan words for the numbers 1 through 9. Beyond those, the islanders only had words for 10 (takau), 20 (paua), 40 (tataua) and 80 (varu) – the binary powers multiplied by 10. So they used the binary system to count in 10s, but added 1 to 9 in the normal way.
Numerical mastery
For example, instead of the widespread system of adding the tens column in the sum 73+80 by remembering that 7+8=15, under the Mangarevan system you combine "forty twenty ten three" with "eighty" to get "eighty forty twenty ten three", or 153.
In this way, the binary component of their counting system may have simplified calculations, since you only need to remember how to combine four numbers when counting in 10s. "They invented these binary steps, which make calculations easier," Bender suggests.
She says the system may originally have come about because the islanders tallied culturally important items such as coconuts and octopuses in groups of 1, 2, 4 and 8. But Mangarevans also traded items over long distances – including as far as Hawaii – and in bulk, and so would have needed a way to efficiently count much larger numbers.
Bender's team is particularly intrigued by the way that the islanders combined the two number systems. "Mangarevans had found a way to compensate for the downsides of a purely binary system by mixing decimal and binary steps in a wellbalanced manner, thus demonstrating numerical mastery on an advanced level," they write. "Mangarevan deserves a prominent position in theorising on numerical cognition."
Mangarevan is still spoken by a very few people today, but with only 600 speakers left as of 2011, it is classified as "in trouble". And modern speakers no longer use the old counting system.
Journal reference: PNAS, DOI: 10.1073/pnas.1309160110
Additional reporting by Celeste Biever and Victoria Jaggard 

