Alex's Adventures in Numberland - Alex Bellos [17]
In the modern world, of course, we group our numbers in tens, so our number system has ten digits – 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The number of the counting group, which is often also the number of symbols used, is called the base of a number system, so our decimal system is base ten, while the shepherds’ base is 20.
Without a sensible base, numbers are unmanageable. Imagine that the shepherd had a base-one system, which would mean he had only one number word: yan for one. Two would be yan yan. Three would be yan yan yan. Eighty sheep would be yan said 80 times. This system is pretty useless for counting anything above about three. Alternatively, imagine that every number was a separate word so that being able to count up to 80 would require memory for 80 unique words. Now count to a thousand this way!
Many isolated communities still use unconventional bases. The Arara in the Amazon, for example, count in pairs, with the numbers from one to eight as follows: anane, adak, adak anane, adak adak, adak adak anane, adak adak adak, adak adak adak anane, adak adak adak adak. Counting in twos is not much of an improvement over counting in ones. Expressing 100 requires repeating adak 50 times in succession – which would make haggling at the market rather time-consuming. Systems in which numbers are grouped in threes and fours are also found in the Amazon.
The trick of a good base system is that the base number needs to be large enough to be able to express numbrs like 100 without running out of breath, but not so large that we need to overexercise our memories. The most common bases throughout history have been five, ten and twenty, and there is an obvious reason why. These numbers are derived from the human body. We have five fingers on one hand, so five is the first obvious place to take a breath when counting upwards from one. The next natural pause comes at two hands, or ten fingers, and after that at hands and feet, or twenty fingers and toes. (Some systems are composite. The Lincolnshire sheep-counting lexicon, for example, contains base five and ten as well as base 20: the first ten numbers are unique, and the next ten are grouped in fives.) The role that fingers have played in counting is reflected in much number vocabulary, not least the double meaning of digit. For example, five in Russian is piat, and the word for outstretched hand is piast. Similarly, Sanskrit for the word five, pantcha, is related to the Persian pentcha, hand.
Finger counting from Luca Pacioli’s Summa de arithmetica, geometria, proportioni et proportionalita (1494).
From the moment man started to count he was using his fingers as an aid, and it is no exaggeration to credit a great deal of scientific progress to the versatility of our fingers. If humans were born with flat stumps at the ends of our arms and legs, it is fair to speculate that we would not have evolved intellectually beyond the Stone Age. Before the widespread availability of paper and pencil allowed numbers to be easily written down, they were often communicated through elaborate finger-counting sign languages. In the eighth century the Northumbrian theologian the Venerable Bede presented a system to count to a million, which was one part arithmetic, one part jazz hands. Units and tens were represented by the left fingers and thumb; hundreds and thousands on the right. Higher orders were expressed by moving the hands up and down the body – with a rather unpriestly image to represent 90,000: ‘grasp your loins with the left hand, the thumb towards the genitals’, Bede wrote. Much more evocative was the sign for a million, a self-satisfied gesture of achievement and closure: the hands clasped together, fingers intertwined.
Until only a few hundred years ago, no manual of arithmetic was complete without diagrams of finger-counting. Now, while mostly a lost art, the practice continues in some parts of the world. Traders in India who want to conceal their dealings from bystanders use a method of touching knuckles behind a cloak or cloth. In China, an ingenious – if rather overly