Free Alex’s Adventures in Numberland by Alex Bellos Page B

piggot all over again. If he had 400 sheep, he would have 20 pebbles, since 20×20 = 400. Now imagine the shepherd had a thousand sheep. If he counted them all he would have 50 pebbles, since 50×20 = 1000. Yet the problem with the shepherd having 50 pebbles is that he has no way to count them, since he cannot count higher than 20!
    A way to solve this is to draw parallel furrows on the ground, as in the figure overleaf. When the shepherd counts 20 sheep he puts a pebble in the first furrow. When he counts another 20 sheep, he puts another pebble in the first furrow. Slowly the first furrow fills up with pebbles. When the time comes to put a twentieth pebble in the furrow, however, he instead puts a single pebble in the second furrow, and clears the first furrow of all pebbles. In other words, one pebble in the second furrow means 20 pebbles in the first – just as one pebble in the first furrow means 20 sheep. A pebble in the second row stands for 400 sheep. A shepherd who has a thousand sheep and uses this procedure will have two pebbles in the second furrow, and ten in the first. By using a place-value system like this one – by which each furrow confers a different value to the pebble in it – he has used only 12 pebbles to count 1000 sheep rather than the 50 he would have needed without it.

     
    Total sheep = (10 × 20) + (2 × 400)0
     
    Place-value counting systems have been used all over the world. Instead of pebbles in furrows, the Incas used beans or grains of maize in trays. North American Indians threaded pearls or shells on different-coloured string. The Greeks and Romans used counters of bone, ivory or metal on tables that had different columns marked out. In India they used marks on sand.
    The Romans also made a mechanical version, with beads sliding in slots, called an abacus. These portable versions spread through the civilized world, though different countries preferred different versions. The Russian schoty has ten beads per rod (except on the row that has four beads, used by cashiers to denote quarter roubles). The Chinese suan-pan has seven, while the Japanese soroban , like the Roman abacus, has just five.
     
     
    About a million children annually in Japan learn the abacus, attending one of 20,000 after-school abacus clubs. One evening in Tokyo I visited one in a west Tokyo suburb. The club was a short walk from a local train line, on the corner of a residential block. Thirty brightly coloured bicycles were parked outside. A large window displayed trophies, abacuses and a line of wooden slats with the calligraphied names of its star pupils.

     
    The Japanese equivalent of ‘reading, ’riting and ’rithmetic’ is yomi, kaki, soroban , or reading, writing, abacus. The phrase dates from the period in Japanese history between the seventeenth and nineteenth centuries when the country was almost totally isolated from the rest of the world. As a new merchant class emerged, which required skills other than proficiency with a samurai sword, so did a culture of private community-run schools that taught language and arithmetic – with the focus on abacus training.
    Yuji Miyamoto’s abacus club is a modern descendant of these older soroban establishments. When I walked in, Miyamoto, who was wearing a dark blue suit and white shirt, was standing in front of a small classroom of five girls and nine boys. He was reading out numbers in Japanese with the breathless syncopation of a horseracing commentator. As the children added them all up, the clatter of beads sounded like a swarm of cicadas.
    In a soroban , there are exactly ten positions of beads per column, representing the numbers from 0 to 9, as shown overleaf.
    When a number is displayed on the soroban , each digit of the number is represented on a separate column using one of the ten positions.

     
    Numbers on the soroban.
     
    The abacus was invented as a way of counting, but it really came into its own as a method for calculation. Arithmetic became