Professor Stewart's Hoard of Mathematical Treasures Read Free

the basic operations of arithmetic can be performed by moving the beads in the right way. A skilled operator can add numbers together as fast as someone using a calculator can type them in, and more complicated things like multiplication are entirely practical.
    The Sumerians used a form of abacus around 2500 BC, and the Babylonians probably did too. There is some evidence of the abacus in ancient Egypt, but no images of one have yet been found, only discs that might have been used as counters. The abacus was widely used in the Persian, Greek and Roman civilisations. For a long time, the most efficient design was the one used by the Chinese from the 14th century onwards, called a suànpán. It has two rows of beads; those in the lower row signify 1 and those in the upper row signify 5. The beads nearest the dividing line determine the number. The suànpán was quite big: about 20 cm (8 inches) high and of varying width depending on the number of columns. It was used lying flat on a table to stop the beads sliding into unwanted positions.

    The number 654,321 on a Chinese abacus.
    The Japanese imported the Chinese abacus from 1600, improved it to make it smaller and easier to use, and called it the soroban. The main differences were that the beads were hexagonal in cross-section, everything was just the right size for fingers to fit, and the abacus was used lying flat. Around 1850 the number of beads in the top row was reduced to one, and around 1930 the number in the bottom row was reduced to four.

    Japanese abacus, cleared.
    The first step in any calculation is to clear the abacus, so that it represents 0 . . . 0. To do this efficiently, tilt the top edge up so that all the beads slide down. Then lie the abacus flat on the table, and run your finger quickly along from left to right, just above the dividing line, pushing all the top beads up.

    Japanese abacus, representing 9,876,543,210.
    Again, numbers in the lower row signify 1 and those in the upper row signify 5. The Japanese designer made the abacus more efficient by removing superfluous beads that provided no new information.
    The operator uses the soroban by resting the tips of the thumb and index finger lightly on the beads, one either side of the central bar, with the hand hovering over the bottom rows of beads. Various ‘moves’ must then be learned, and practised, much like a musician learns to play an instrument. These moves are the basic components of an arithmetical calculation, and the calculation itself is rather like playing a short ‘tune’. You can find lots of detailed abacus techniques at: www.webhome.idirect.com/~totton/abacus/pages.htm#Soroban1
    I’ll mention only the two easiest ones.
    A basic rule is: always work from left to right. This is the opposite of what we teach in school arithmetic, where the calculation proceeds from the units to the tens, the hundreds, and so on - right to left. But we say the digits in the left-right order: ‘three hundred and twenty-one’. It makes good sense to think of them that way, and to calculate that way. The beads act as a memory, too, so that you don’t get confused by where to put the ‘carry digits’.
    To add 572 and 142, for instance, follow the instructions in the pictures. (I’ve referred to the columns as 1, 2, 3, from the right, because that’s the way we think. The fourth column doesn’t play any role, but it would do if we were adding, say, 572 and 842, where 5 + 8 = 13 involves a ‘carry digit’ 1 in place 4.

    Set up 572

    Add 1 in column 3

    Add 4 in column 2 ...

    ... and carry the 1

    Add 2 in column 3
    A basic technique occurs in subtraction. I won’t draw where the beads go, but the principle is this. To subtract 142 from 572, change each digit x in 142 to its complement 10 - x. So 142 becomes 968. Now add 968 to 572, as before. The result is 1,540, but of course 572 - 142 is actually 430. Ah, but I haven’t yet mentioned that at each step you subtract 1 from the column one place