In Lincolnshire during medieval times, a pimp plus a dik got you a bumfit. There was nothing dishonourable about this. The words were simply the numbers five, ten and fifteen in a jargon used by shepherds when counting their sheep. The full sequence ran:

1. Yan
   >

   • Tan
   • Tethera
   • Pethera
   • Pimp
   • Sethera
   • Lethera
   • Hovera
   • Covera
   • Dik
   • Yan-a-dik
   • Tan-a-dik
   • Tethera-dik
   • Pethera-dik
   • Bumfit
   • Yan-a-bumfit
   • Tan-a-bumfit
   • Tethera-bumfit
   • Pethera-bumfit
   • Piggot
    

   This is a different way from how we count now, and not just because all the words are unfamiliar. Lincolnshire shepherds organized their numbers in groups of twenty, starting counting with yan and ending with piggot. If a shepherd had more than twenty sheep – and provided he hadn’t sent himself to sleep – he would make note of having completed one cycle by putting a pebble in his pocket, or making a mark on the ground, or scraping a line in his crook. He would then start from the beginning again: ‘Yan, tan, tethera…’ If he had 80 sheep, he would have four pebbles in his pocket, or have marked four lines, at the end. The system is very efficient for the shepherd; he has four small items to represent 80 big ones.

   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 intricate – technique allows you to count up to one less than ten billion – 9,999,999,999. Each finger has nine imaginary points – three on each crease line, as marked on the diagram opposite. These points on the right little finger represent the digits 1 to 9. The points on the right fourth finger take us from 10 to 90. The right middle finger goes from 100 to 900, and so on, with each new finger representing the next power of ten. It is therefore possible to count every single person on Earth with only your fingers, which is one way to have the whole world in your hands.

   Some cultures count using more of their bodies than just fingers and toes. At the end of the nineteenth century an expedition of British anthropologists reached the islands of the Torres Strait, the stretch of water that separates Australia from Papua New Guinea. There they discovered a community that started with ‘right hand little finger’ for 1, ‘right hand ring finger’ for 2 and this continued through the fingers to ‘right wrist’ for 6, ‘right elbow’ for 7 and on through the shoulders, sternum, left arm and hand, feet and legs, ending at ‘right foot little toe’ for 33. Subsequent expeditions and research uncovered many communities in the region with similar ‘body-tally’ systems.

   

    

   In this Chinese system, each finger has nine points, representing the digits 1 to 9 for each order of magnitude, so the right hand can express any number up to 10⁵– 1 when the other hand touches the relevant points. Swapping hands, the numbers continue to 10¹⁰– 1. A ‘zero’ point is not needed on any finger, since when there are no values relating to that finger it is simply left alone by the other hand.

    

   Perhaps the most curious is the Yupno, the only Papuan people for whom each individual owns a short melody that belongs to them like a name, or signature tune. They also have a counting system that enumerates the nostrils, eyes, nipples, belly button and climaxes in 31, for ‘left testicle’, 32, ‘right testicle’ and 33, ‘penis’. While one can ponder the significance of 33 in the three great monotheistic religions (the age when Christ died, the length of King David’s reign and the number of individual beads on a Muslim prayer string), what is particularly intriguing about the Yupno’s phallic number is that they are actually very coy about it. They refer to the number 33 euphemistically in phrases such as ‘the man thing’. Researchers were unable to discover whether women use the same terms, since they are not supposed to know the number system and refused to answer questions. The upper limit in Yupno is 34, which they call ‘one dead man’.

   Base-ten systems have been used in the West for thousands of years. Despite their harmoniousness with our bodies, however, many have questioned whether they are the most sensible base for counting. In fact, some have argued that their physical provenance makes them an actively bad choice. King Charles XII of Sweden dismissed base ten as the product of ‘rustic and simple people’ fumbling around with their fingers. In modern Scandinavia, he believed, a base was needed ‘of more convenience and greater use’. So, in 1716, he ordered the scientist Emanuel Swedenborg to devise a new counting system with a base of 64. He arrived at this formidable number due to the fact that it was derived from a cube, 4 × 4 × 4. Charles, who fought – and lost – the Great Northern War, believed that military calculations, such as measuring the volume of a box of gunpowder, would be made easier with a cube number as a base. Yet his brainwave, wrote Voltaire, ‘could prove only that he loved the extraordinary and the difficult’. Base 64 requires 64 unique names (and symbols) for numbers – an absurdly inconvenient system. Swedenborg therefore simplified the system to base eight and came up with a new notation in which 0, 1, 2, 3, 4, 5, 6, 7 were renamed o, l, s, n, m, t, f, u. In this system, therefore l + l = s, and m × m = so. (The words for the new numbers, however, were rather wonderful. The powers of 8, which would have been written lo, loo, looo, loooo and looooo, were to be pronounced, or yodelled, lu, lo, li, le and la.) In 1718, however, shortly before Swedenborg was due to present the system, a bullet shot the king – and his octonary dream – stone dead.

   

   One dead Yupno.

    

   But Charles XII had a valid point. Why should we stick with the decimal system just because it was derived from the number of our fingers and toes? If humans were like Disney characters, for example, and had only three fingers and a thumb per hand, it is almost certain that we would live in a base-eight world: giving marks out of eight, compiling top eight charts and letting eight cents make a dime. Mathematics would not change by having an alternative way to group numbers. The bellicose Swede was correct to ask which base best suits our scientific needs – rather than opting for the one that suits our anatomy.

    

    

   In late 1970s Chicago, Michael de Vlieger was watching the cartoons on Saturday morning TV. A short segment came on. The soundtrack was of disconcerting, off-key piano chords, wah-wah guitar and a menacing bass. Under a full moon and starry night sky a strange humanoid appeared. He had a blue and white striped top hat and tails, blond hair and a stick nose, rather in keeping with the glam-rock fashion of the era. If that wasn’t creepy enough, he had five fingers and a thumb on each hand, and six toes on each foot. ‘It was a little freaky, kind of spooky,’ remembered Michael. The cartoon was Little Twelvetoes, an educational broadcast about base 12. ‘I think the majority of the American population had no idea what was going on. But I thought it was so cool.’

   Michael is now 38. I met him in his office, a business suite above some shops in a residential part of St Louis, Missouri. He has thick black hair with a few shoots of white, a round face, dark eyes and sallow skin. His mum is Filipino, while his father is white, and being a mixed-race kid made him the victim of taunts. A clever and sensitive child with an active imagination, he decided to invent his own language so that his classmates couldn’t read his notebooks. Little Twelvetoes inspired him to do the same with numbers – and he adopted base 12 for personal use.

   Base 12 has 12 digits: 0 to 9 and two extra ones to represent ten and eleven. The standard notation for each of these two ‘transdecimal’ digits is and . So, counting to 12 now goes: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, , , 10. (See table opposite.)

   The new single digits are given new names to avoid confusion, so is called dek and is called el. Also, we give 10 the name do, pronounced doh, and short for dozen, to avoid confusion with 10 in base ten. Counting upwards from do in base 12, or ‘dozenal’, we have do one for 11, do two for 12, do three for 13 all the way up to twodo for 20.

   Michael devised a private calendar using base 12. Each date in this calendar was the number of days, counted in base 12, from the day he was born. He still uses it, and he told me later that I visited him on the 809th day of his life.

   Michael adopted base 12 for reasons of personal security, but he is not alone in having fallen for its charms. Many seriohinkers have argued that 12 is a better base for a number system because the number is more versatile than 10. In fact, base 12 is more than a number system – it is a politico-mathematical cause. One of its earliest champions was Joshua Jordaine, who in 1687 self-published Duodecimal Arithmetick. He claimed that ‘nothing was more natural and genuine’ than counting in twelves. In the nineteenth century high-profile duodeciphiles included Isaac Pitman, who had gained considerable fame for inventing a widespread system of shorthand, and Herbert Spencer, the Victorian social theorist. Spencer urged base-system reform on behalf of ‘working people, people of narrow incomes and the minor shopkeepers who minister to their wants’. The American inventor and engineer John W. Nystrom was also a fan. He described base 12 as ‘duodenal’ – perhaps the most unfortunate double entendre in the history of science.

   

    

   Dozenal numbers from 1 to 100.

    

   The reason that 12 might be considered superior to ten is because of its divisibility. Twelve can be divided by 2, 3, 4 and 6, whereas ten can be divided only by 2 and 5. Advocates of base 12 argue that we are much more likely to want to divide by 3 or 4 than divide by 5 in our daily lives. Consider a shopkeeper. If you have 12 apples, then you can divide them up into two bags of 6, three bags of 4, four bags of 3, or six bags of 2. This is much more user-friendly than 10, which can only be cleanly divided into two bags of 5, or five of 2. The word ‘grocer’, in fact, is a relic of a retailer’s preference for 12 – it comes from ‘gross’, meaning a dozen dozen, or 144. The multi-divisibility of 12 also explains the utility of imperial measure: a foot, which is 12 inches, can be cleanly divided by 2, 3 and 4 – which is quite a plus for carpenters and tailors.

   Divisibility is also relevant to multiplication tables. The easiest tables to learn in any base are the ones of numbers that divide that base. This is why, in base ten, the 2 and 5 times tables – which are just the even numbers and the numbers ending in 5 or 0 – are so painless to recite. Likewise, in base 12 the simplest times tables are also those of its divisors: 2, 3, 4 and 6.

   

    

   If you look at the final digits of each column, you see a striking pattern. The two times table is, again, all the even numbers. The three times table is all the numbers ending in 3, 6, 9 and 0. The four times table is the numbers ending in 4, 8 and 0, and the six times table all the numbers ending 6 or 0. In other words, in base 12 we get the 2, 3, 4 and 6 times tables for free. Since many children have difficulty in learning their times tables, if we converted to base 12 we would be carrying out a great humanitarian act. Or so the argument goes.

   The campaign for base 12 should not be conflated with the crusade against metric by fans of imperial measure. Those people who prefer feet and inches over metres and centimetres have no issue as to whether one foot should be 12 inches, or 10 inches, as it would be in dozenal. Historically, however, an underlying theme of the campaign for base 12 has been a jingoistic anti-Frenchness. Perhaps the finest example of such a view was a pamphlet from 1913 by engineer Rear-Admiral G. Elbrow, in which he called the French metric system ‘retrograde’. He published a list of the dates, in base 12, of the kings and queens of England. He also noticed that Britain had been invaded shortly after each decimal millennium – by the Romans in 43 CE and the Normans in 1066. ‘What if, at the beginning of the [third millennium],’ he prophesized, ‘these two [countries] may again appear in the same direction, and this time in conjunction?’ Invasion by France and Italy might be averted, he argued, simply by rewriting the year 1913 as 1135, as it would be in dozenal, thus delaying the third millennium by several centuries.

   The most famous dozenalist call-to-arms, though, was an article in The Atlantic Monthly in October 1934 by the writer F. Emerson Andrews, which led to the formation of the Duodecimal Society of America, or DSA. (It later changed its name to the Dozenal Society of America since ‘duodecimal’ was deemed to be overly reminiscent of the system they were aiming to replace.) Andrews claimed that base ten had been adopted with ‘inexcusable shortsightedness’ and wondered whether it ‘would be so tremendous a sacrifice’ to abandon it. The DSA initially insisted prospective members pass four tests in dozenal arithmetic, although this requirement was quickly dropped. The Duodecimal Bulletin, which continues to this day, is an excellent publication and the only place outside medical literature with articles on hexadactyly, the condition of being born with six fingers. (Which is more common than you might think. About one in every 500 people is born with at least an extra finger or toe.) In 1959 a sister organization, the Dozenal Society of Great Britain, was founded, and a year later the First International Duodecimal Conference was held in France. It was also the last. Still, both societies continue to battle for a dozenal future, seeing themselves as downtrodden militants rallying against the ‘tyranny of ten’.

   Michael de Vlieger’s youthful infatuation with base 12 was not a passing phase; he is the current president of the DSA. In fact, he is so committed to the system that he uses it in his job as a designer of digital architectural models.

   While base 12 certainly makes times tables easier to learn, its greatest advantage is how it cleans up fractions. Base ten is frequently messy when you want to divide. For example, a third of 10 is 3.33…, where the threes go on for ever. A quarter of 10 is 2.5, which needs a decimal place. In base 12, however, a third of 10 is 4 and a quarter of 10 is 3. Nice. Expressed as a percentage, a third becomes 40 percent, and a quarter 30 percent. In fact if you look at how 100 is divided by the numbers 1 to 12, base 12 provides more concise numbers (note that the semi-colons in the right column stand for the ‘dozenal’ point).

    

    

   

    

    

   It is this increased precision that makes base 12 better suited to Michael’s needs. Even though his clients supply him with dimensions in decimal, he prefers to translate them into dozenal. ‘It gives me more choices when dividing into simple ratios,’ he said.

   

    

   ‘Avoiding [messy] fractions helps ensure things fit. Sometimes, because of time constraints or late-breaking changes, I will need to quickly apply a lot of change at a location that doesn’t jive with the grid I initially set up. Thus it’s important to have predictable simple ratios. I’ve got more and cleaner choices with dozenal, and it’s faster.’ Michael even believes that using base 12 gives his business an edge, comparing it to cyclists and swimmers who shave their legs.

   The DSA used to want to replace decimal with dozenal, and its fundamentalist wing still does, but Michael’s ambitions are more modest. He wants simply to show people that there is an alternative to the decimal system, and that perhaps it suits their needs better. He knows that the chances of the world abandoning dix for douze are non-existent. The change would be both confusing and expensive. And decimal works well enough for most people – especially in the computer age, where mental arithmetic skills are less required generally. ‘I would say that dozenal is the optimum base for general computation, for everyday use,’ he added, ‘but I am not here to convert anybody.’

   An immediate goal of the DSA is to get the numerals for dek and el into Unicode, the repertoire of text characters used by most computers. In fact, a major debate in dozenal society is which symbols to use. The DSA standard and were designed in the 1940s by William Addison Dwiggins, one of the US’s foremost font designers and the creator of Futura, Caledonia and Electra. Isaac Pitman preferred and . Jean Essig, a French enthusiast, preferred and . Some practical members would prefer * and # since they are already on the 12 buttons of a touch-tone phone. The number words are also an issue. The Manual of the Dozen System (written in 1960, or 1174 in dozenal) recommends the terms dek, el and do (with gro for 100, mo for 1000, and do-mo, gro-mo, bi-mo and tri-mo for the next highest powers of do). Another suggestion is to keep ten, eleven and twelve and continue with twel-one, twel-two, and so on. Such is the sensitivity over terminology that the DSA is careful not to push any system. Great care is needed not to marginalize devotees of any particular symbol or term.

   Michael’s love of avant-garde bases did not stop at 12. He has toyed with eight, which he sometimes uses when doing DIY at home. ‘I use bases as tools,’ he said. And he has gone up to base 60. For this he had to design 50 extra symbols to add to the ten digits we have already. His purpose was not practical. He described working in base 60 as like going up a high mountain. ‘I can’t live up there. It is too big a grouping. In the valley it is decimal, and there I can breathe. But I can visit the mountain to see what the view is like.’ He has written out tables of factors in a base 60, or sexagesimal, system, and stared in wonder at the patterns they revealed. ‘There definitely is a beauty there,’ he told me.

   While base 60 seems like the product of an extraordinarily fertile imagination, sexagesimal has historical pedigree. It is actually the most ancient base system that we know of.

    

    

   The simplest form of numerical notation is the tally. It has been used in different forms across the world. The Incas kept count by tying knots on ropes, while cave dwellers painted marks on rocks and, since the invention of wooden furniture, bedposts have – figuratively, at least – been marked with notches. The oldest discovered ‘mathematical artefact’ is believed to be a tally stick: a 35,000-year-old baboon fibula found in a Swaziland cave. The ‘Lebombo bone217; has 29 lines scraped on it, which possibly denote a lunar cycle.

   As we saw in the previous chapter, humans can instantly tell the difference between one item and two, between two items and three, but beyond four it gets difficult. This is true of notches as well. For any convenient system of tally-keeping, the tallies need to be grouped. In Britain, tally convention is to mark four vertical lines and then make the fifth a diagonal crossing through them – the so-called ‘five-bar gate’. In South America, the preferred style is for the first four lines to mark a square and the fifth is a diagonal in the square. The Japanese, Chinese and Koreans use a more elaborate method, constructing the character, which means ‘correct’ or ‘proper’. (The next time you have sushi, ask the waiter to show you how he is tallying your dishes.)

   

    

   Tally systems of the world.

    

   Around 8000 BC a practice of using small clay pieces with markings to refer to objects emerged throughout the ancient world. These tokens primarily recorded numbers of things, such as sheep to be bought and sold. Different clay pieces referred to different objects or numbers of objects. From that moment sheep could be counted without actually being there, which made trade and stock-keeping much easier. It was the birth of what we understand now as numbers.

   In the fourth millennium bc in Sumer, an area now in present-day Iraq, this token system evolved into a script in which a pointed reed was pressed into soft clay. Numbers were first represented by circles or fingernail shapes. By around 2700 bc the stylus had a flat edge and the imprints looked rather like bird footprints, with different imprints referring to different numbers. The script, called cuneiform, marked the beginning of the long history of Western writing systems. It is wonderfully ironic to think that literature was a by-product of a numerical notation invented by Mesopotamian accountants.

   

    

   In cuneiform there were symbols only for 1, 10, 60 and 3600, which means the system was a mixture of base 60 and base ten, as the basic set of cuneiform numbers translates into 1, 10, 60 and 60×60. The question why the Sumerians grouped their numbers in sixties has been described as one of the greatest unresolved mysteries in the history of arithmetic. Some have suggested it was the result of the fusion of two previous systems, with bases five and 12, though no conclusive evidence of this has been found.

   The Babylonians, who made great advances in maths and astronomy, embraced the Sumerian sexagesimal base, and later the Egyptians, followed by the Greeks, based their time-counting methods on the Babylonian way – which is why, to this day, there are 60 seconds in a minute and 60 minutes in an hour. We are so used to telling the time in base 60 that we never question it, even though it really is quite unexplained. Revolutionary France, however, wanted to iron out what they saw as an inconsistency in the decimal system. When the National Convention introduced the metric system for weights and measures in 1793, it also tried to decimalize time. A decree was signed establishing that every day would be divided into ten hours, each containing 100 minutes, each of which contained 100 seconds. This worked out neatly, making 100,000 seconds in the day – compared to 86,400 (60×60×24) seconds. The revolutionary second was, therefore, a fraction shorter thght="0%e normal second. Decimal time became mandatory in 1794 and watches were produced with the numbers going up to ten. Yet the new system was completely bewildering to the populace and abandoned after little more than six months. An hour with 100 minutes is also not as convenient as an hour with 60 minutes, since 100 does not have as many divisors as 60. You can divide 100 by 2, 4, 5, 10, 20, 25 and 50, but you can divide 60 by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30. The failure of decimal time was a small victory for dozenal thinking. Not only does 12 divide into 60 but it also divides into 24, the number of hours in a day.

   

    

   Revolutionary watch with decimal and traditional clock face.

    

   A more recent campaign to decimalize time also flopped. In 1998 the Swiss conglomerate Swatch launched Swatch Internet Time, which divided the day into 1000 parts called beats (equivalent to 1min 26.4secs). The manufacturer sold watches that displayed its ‘revolutionary vision of time’ for a year or so before sheepishly removing them from its catalogue.

   The French and Swiss, however, are not the only Western nations to have had barmy counting procedures in the not too distant past. The tally stick, which became outdated the moment the first Sumerian printed his first cuneiform tablet, was used as a form of British currency until 1826. The Bank of England used to issue souped-up tally sticks that were worth a monetary value based on the distance of a mark from the base. A document written in 1186 by the Lord Treasurer Richard Fitzneal set out the values as:

    

    

   £1000	thickness of the palm of the hand
   £100	breadth of a thumb
   £20	breadth of a little finger
   £1	width of a swollen barleycorn

    

    

   The procedure the Treasury used was, in fact, a system of ‘double tallies’. A piece of wood was split down the middle, giving two parts – the stock and the foil. A value was marked – tallied – on the stock and was also marked on the foil, which acted like a receipt. If I lent some money to the Bank of England, I would be given a stock with a notch indicating the amount – which explains the origin of the words stockholder and stockbroker – while the bank kept the foil, which had a matching notch.

   This practice was abandoned barely two centuries ago. In 1834, the Treasury decided to incinerate the obsolete pieces of wood in a furnace under the Palace of Westminster, the seat of British government. The fire, however, spread out of control. Charles Dickens wrote: ‘The stove, overgorged with these preposterous sticks, set fire to the panelling; the panelling set fire to the House of Commons; the two houses [of government] were reduced to ashes.’ Obscure financial instruments have often impacted on the work of government, but only the tally stick has brought down a parliament. When the palace was rebuilt it had a brand new clock tower, Big Ben, which quickly became the most recognizable landmark in London.

    

    

   An argument often used in favour of the imperial system over metric is that the words sound better. A case in point is the measures for wine:

   2 gills = 1 chopin

   2 chopins = 1 pint

   2 pints = 1 quart

   2 quarts = 1 pottle

   2 pottles = 1 gallon

   2 gallons = 1 peck

   2 pecks = 1 demibushel

   2 demibushels = 1 bushel (or firkin)

   2 firkins = 1 kilderkin

   2 kilderkins = 1 barrel

   2 barrels = 1 hogshead

   2 hogsheads = 1 pipe

   2 pipes = 1 tun

    

   This system is base two, or binary, which is usually expressed using the digits 0 and 1. Numbers in binary are the numbers you would use in base ten when only 0 and 1 appear. In other words, the sequence that begins 0, 1, 10, 11, 100, 101, 110, 111, 1000. So, 10 is two, 100 is four, 1000 is eight and so on, with each extra 0 on the end representing multiplication by two. (Which is just like base ten – adding a 0 on the end of a number is multiplication by ten.) In the wine measures, the smallest unit is a gill. Two gills makes a chopin, 4 gills a pint, 8 gills a quart, 16 gills a pottle, etc. The measures replicate perfectly the binary numerals. If a gill is represented by 1, then a chopin is 10, a pint is 100, a quart is 1000 and this carried on all the way to a tun, which is 10,000,000,000,000.

   Binary can claim as its cheerleader the greatest mathematician ever to have fallen in love with a non-standard base. Gottfried Leibniz was one of the most important thinkers of the late seventeenth century, a scientist, philosopher and statesman. One of his duties was as librarian to the court of the Duke of Brunswick in Hanover. Leibniz was so excited with base two that he once wrote a letter to the Duke urging him to cast a silver medallion inscribed with the words Imago Creationis – ‘in the image of the world’ – as a tribute to the binary system. For Leibniz, binary had practical and spiritual relevance. First, he thought that its capacity for describing every number in terms of doubles facilitated a variety of operations. ‘[It] permits the Assayer to weigh all sorts of masses with few weights and could serve in coinage to give more value with fewer pieces,’ he wrote in 1703. Leibniz did admit that binary had some practical drawbacks. The numbers are much longer when written out: 1000 in decimal, for example, is 1,111,101,000 in binary. But he added: ‘In recompense for its length, [binary] is more fundamental to science and gives new discoveries.’ By looking at the symmetries and patterns in binary notation, he claimed, new mathematical insights are revealed, and number theory is richer and more versatile because of it.

   

    

   Design for eibniz’s binary medallion, in Johann Bernard Wiedeburg’s Dissertatio mathematica de praestantia arithmeticae binaria prae decimali (1718). As well as the words Imago Creationis , the Latin reads ‘From nothing comes one and everything, but the one is necessary’.

    

   Second, Leibniz marvelled at how the binary system chimed with his religious views. He believed that the cosmos was composed of being, or substance, and non-being, or nothingness. The duality was perfectly symbolized by the numbers 1 and 0. In the same way that God creates all beings from the void, all numbers can be written in terms of 1s and 0s. Leibniz’s conviction that binary exemplified a fundamental metaphysical truth was – to his great delight – strengthened when later in life he was shown the I Ching, the ancient Chinese mystical text. The I Ching is a book of divination. It contains 64 different symbols, each of which comes with an accompanying commentary. The reader randomly selects a symbol (traditionally by casting yarrow sticks) and interprets the related text – a little like one might read an astrological chart. Each symbol in the I Ching is a hexagram, which means it is composed of six horizontal lines. The lines are either broken or unbroken, corresponding to a yin or a yang. The 64 hexagrams in the I Ching are the full set of combinations of yins and yangs when taken in groups of six at a time.

   A particularly elegant way of ordering the hexagrams is shown opposite. If each yang is written 0 and each yin is 1, then the sequence matches precisely the binary digits from 0 to 63.

   This way of ordering is known as the Fu Hsi sequence. (Strictly speaking, it is the inverse of Fu Hsi, but they are mathematically equivalent.) When Leibniz was made aware of the binary nature of Fu Hsi, it gave him ‘a high opinion of [the I Ching’s] profundity’. Since he thought that the binary system mirrored Creation, his discovery that it also underlay Taoist wisdom meant that Eastern mysticism could now be accommodated within his own Western beliefs. ‘The substance of the ancient theology of the Chinese is intact and, purged of additional errors, can be harnessed to the great truths of the Christian religion,’ he wrote.

   

    

   Part of the Fu Hsi sequence of the I Ching and its binary equivalent.

    

   Leibniz’s panegyrics on base two were a rather eccentric preoccupation of the pre-eminent polymath of his day. Yet in ascribing a fundamental importance to the system, he was more prescient than even he could ever have imagined. The digital age runs on binary, as computer technology relies at a most basic level on a language comprised of 0s and 1s. ‘Alas!’ wrote mathematician Tobias Dantzig. ‘What was once hailed as a monument to monotheism ended in the bowels of a robot.’

    

    

   ‘Freedom is the freedom to say two plus two equals four,’ wrote Winston Smith, the protagonist of George Orwell’s Nineteen Eighty-Four. Orwell was making a comment not only about freedom of speech in the Soviet Union, but also about mathematics. Two plus two is always four. No one can tell you it isn’t. Mathematical truths cannot be influenced by culture or ideology.

   On the other hand, our approach to mathematics is very much influenced by culture. The selection of base ten, for example, was not premised on mathematical reasons but on physiological ones, the number of our fingers and toes. Language also shapes mathematical understanding in surprising ways. In the West, for example, we are held back by the words we have chosen to express numbers.

   In almost all Western European languages, number words do not follow a regular pattern. In English we say twenty-one, twenty-two, twenty-three. But we don’t say tenty-one, tenty-two, tenty-three – we say eleven, twelve, thirteen. Eleven and twelve are unique constructions, and even though thirteen is a combination of three and ten, the three part comes before the ten part – unlike twenty-three, when the three part comes after the twenty part. Between ten and twenty, English is a mess.

   In Chinese, Japanese and Korean, however, number words do follow a regular pattern. Eleven is written ten one. Twelve is ten two, and so on with ten three, ten four up to ten nine for nineteen. Twenty is two ten, and twenty-one is two ten one. You pronounce numbers in all cases just as you see them written down. So what? Well, it does make a difference at a young age. Experiments have repeatedly shown that Asian children find it easier to learn to count than Europeans. In one study with Chinese and American four-and five-year-olds, the two nationalities performed similarly when learning to count to 12, but the Chinese were about a year ahead with higher numbers. A regular system also makes arithmetic clearer to understand. A simple sum such as twenty-five plus thirty-two when expressed as two ten five plus three ten two is one step closer to the answer already: five ten seven.

   Not all European languages are irregular. Welsh, for example, is just like Chinese. Eleven in Welsh is un deg un (one ten one); twelve is un deg dau (one ten two), and so on. Ann Dowker and Delyth Lloyd at the University of Oxford tested the maths abilities of Welsh-and English-speaking kids from the same Welsh village. While Asian children may be better than American children because of many cultural factors, such as hours spent practising or attitudes towards maths, cultural factors can be eliminated if the children are all living in the same place. Dowker and Lloyd concluded that while general arithmetic performance was more or less equal between Welsh-and English-speakers, the Welsh-speakers did demonstrate better mathematical skills in specific areas – such as reading, comparing and manipulating two-digit numbers.

   German is even more irregular than English. In German, twenty-one is einundzwanzig, or one-and-twenty, twenty-two is zweiundzwanzig, or two-and-twenty, and this continues with the unit value preceding the tens value all the way up to 99. This means that when a German says a number over 100, the digits are not pronounced in a consecutive order: three hundred and forty-five is dreihundertfünfundvierzig, or three-hundred-five-and-forty, which lists the numbers in the higgledy-piggledy form 3-5-4. Such is the level of concern in Germany that this makes numbers more confusing than they have to be, that a campaign group Zwanzigeins (Twenty-one) has been set up to push for a change to a more regular system.

   And it’s not just the positioning of number words, or their irregular forms between eleven and nineteen, that puts the speakers of the main Western European languages at a disadvantage with some Asian language-speakers. We are also handicapped by how long it takes us to say numbers. In The Number Senseyi, er, san, si, wu, liu, qi, ba, jiu. They can be uttered in less than a quarter of a second, so in a two-second span a Chinese-speaker can rattle through nine of them. English number words, by contrast, take just under a third of a second to say (thanks to the frankly cumbersome ‘seven’, with two syllables, and the extended syllable ‘three’), so our limit in two seconds is seven. The record, however, goes to the Cantonese, whose digits are spoken with even more brevity. They can remember ten of them in a two-second period.

   While Western languages seem to be working against any mathematical ease of understanding, in Japan language is recruited as an ally. Words and phrases are modified in order to make their multiplication tables, called kuku, easier to learn. The tradition of these tables originated in ancient China, spreading to Japan around the eighth century. Ku in Japanese is nine, and the name comes from the fact that the tables used to begin at the end, with 9×9 = 81. Around 400 years ago they were changed so that the kuku now begins ‘one one is one’.

   The words of the kuku are simply:

   One one is one

   One two is two

   One three is three…

    

   This carries on to ‘One nine is nine’, and then the twos begin with:

   Two one is two

   Two two is four

    

   And so on to nine nine is eighty-one.

   So far, this seems very similar to the plain British style of reciting the times tables. In the kuku, however, whenever there are two ways to pronounce a word, the way that flows better is used. For example, the word for one can be in or ichi, and rather than starting the kuku with either in in or ichi ichi, the more sonorous combination in ichi is used. The word for eight is ha. Eight eights should be ha ha. Yet the line in the kuku for 8×8 is happa since it rolls quicker off the tongue. The result is that the kuku is rather like a piece of poetry, or a nursery rhyme. When I visited an elementary school in Tokyo and watched a class of seven-and eight-year-olds practise their kuku, I was struck by how much it sounded like a rap – the phrases were syncopated and jolly. Certainly it bore no relation to how I remember reciting my times tables at school, which was with the metronomic delivery of a steam train going up a hill. Makiko Kondo, the teacher, said that she teaches her pupils kuku with an uptempo rhythm because it makes it fun to learn. ‘First we get them to recite it, and only some time later do they come to understand the real meaning.’ The of the kuku seems to embed the times tables in Japanese brains. Adults told me that they know, for example, that seven times seven is 49 not because they remember the maths but because the music of ‘seven seven forty-nine’ sounds right.

   While the irregularities of Western number words may be unfortunate for budding arithmeticians, they are of extreme interest to mathematical historians. The French for eighty is quatre-vingts, or four-twenties, indicating that ancestors of the French once used a base-20 system. It has also been suggested that the reason why the words for ‘nine’ and ‘new’ are identical or similar in many Indo-European languages, including French (neuf, neuf ), Spanish (nueve, nuevo), German (neun, neu) and Norwegian (ni, ny) is a legacy of a long-forgotten base-eight system, where the ninth unit would be the first of a new set of eight. (Excluding thumbs, we have eight fingers, which could be how such a base developed. Or possibly from counting the gaps between the fingers.) Number words are also a reminder of how close we are to the numberless tribes of the Amazon and Australia. In English, thrice can mean both three times and many times; in French, trois is three and très is very: shadows, perhaps, of our own ‘one, two, many’ past.

    

    

   Whereas certain aspects of number – such as the base, the style of numeral or the form of the words used – have differed widely between cultures, the early civilizations were surprisingly unified in the mechanics of how they counted and calculated. The general method they used is called ‘place value’ and is the principle by which different positions are used to represent different orders of number. Let’s consider what this means in the context of shepherds in medieval Lincolnshire. As I wrote earlier, they had 20 numbers from yan to piggot. Once a shepherd counted 20 sheep, he put a pebble aside and started counting from yan to 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 much easier to do when helped by the flicking of beads. For example, to calculate 3 plus 1 you start with 3 beads, move 1 bead and the answer is right before your eyes – 4 beads. To calculate, say, 31 plus 45 you start with two columns marking 3 and 1, move 4 bead positions up the left column and 5 up the right column. The columns now read 7 and 6, which is the answer, 76. With a little bit of practice and application, it becomes easy to add numbers of any length so long as there are enough coumns to accommodate them. If on any one column the two numbers add up to more than ten, then you will need to move the beads on the column to the left up one position. For example, 9 plus 2 on one column moves to a 1 on the column to the left and a 1 on the original column, expressing the answer, 11. Subtraction, multiplication and division are a little more complicated, but once mastered can be done extremely quickly.

   Until the availability of cheap calculators in the 1980s abacuses were commonly seen on shop counters from Moscow to Tokyo. In fact, during the transition between the manual and electronic eras, a product combining both calculator and abacus was sold in Japan. Addition is usually faster on the abacus since you get your answer as soon as you input the numbers. With multiplication the electronic calculator gives you a slight speed advantage. (The abacus was also a way for the sceptical abacist to check the calculator’s result, just in case he didn’t believe it.)

   

    

   Sharp’s abacus-calculator.

    

   Abacus use has dropped in Japan since the 1970s when, at its peak, 3.2 million pupils a year sat the national soroban proficiency exam. Yet the abacus still remains a defining aspect of growing up, a mainstream extra-curricular activity like swimming, violin or judo. Abacus training, in fact, is run like a martial art. Levels of ability are measured in dans, and there is a competitive structure of local, provincial and national competitions. One Sunday I went to see a regional event. Almost 300 children, aged between 5 and 12, sat at desks in a conference hall with an array of special soroban accessories, like sleek abacus bags. An announcer stood at the front of the hall and dictated, with the intonation of an impatient muezzin, numbers to be added, subtracted or multiplied. It was a knock-out competition that lasted several hours. A chorus of military brass-band music was pumped through the sound system when the trophies – each with a winged figure holding an abacus aloft – were presented to the victors.

   At Miyamoto’s school he introduced me to one of his best pupils. Naoki Furuyama, aged 19, is a former national soroban champion. He was dressed casually, with a light checked shirt over a black T-shirt, and seemed a relaxed and well-adjusted teenager – certainly not the cliché of a socially awkward übergeek. Furuyama can multiply two six-digit numbers together in about four seconds, which is about as long as it takes to say the problem. I asked him what the point was of being able to calculate so fast, since there is no need for such skills in daily life. He replied that it helped his powers of concentration and self-discipline. Miyamoto was standing with us, and he interrupted. What was the point of running 26 miles, he asked me? There was never any need to run 26 miles, but people did it as a way of pushing human performance to the limit. Likewise, he added, there was a nobility in training one’s arithmetical brain as far as one could.

   Some parents send their children to abacus club because it is a way to improve school maths results. But that does not completely explain the abacus’s popularity. Other after-school clubs provide more targeted maths tuition – Kumon, for example, a method of ploughing through worksheets that started in Osaka in the early 1950s, is now followed by more than four million children around the world. Abacus club is fun. I saw that in the faces of the pupils at Miyamoto’s school. They clearly enjoyed their dexterity at flicking the beads with speed and precision. The Japanese hetage of the soroban generates national pride. Yet the real joy of the abacus, I thought, is more primal: it has been used for thousands of years and, in some cases, is still the fastest way to do sums.

    

    

   After a few years of using an abacus, when you are so familiar with the positioning of the beads, it becomes possible to perform calculations simply by visualizing an abacus in your head. This is called anzan, and Miyamoto’s top pupils have all learned it. The feat was amazing to watch – even though there was nothing to see. Miyamoto read out numbers to a totally silent, still classroom and within seconds the students raised their hands with the answers. Naoki Furuyama told me that he visualizes an abacus with eight columns. In other words, his imaginary abacus can display every number from 0 to 99,999,999.

   Miyamoto’s abacus club is one of the best in the country in terms of the dans of its pupils and their achievements in national tournaments. Its speciality, however, is anzan. A few years ago Miyamoto decided to devise a type of arithmetical challenge that could only be answered using anzan. When you read out a sum to a pupil, for example, it can be answered in many different ways: using a calculator, pencil and paper, an abacus or anzan. Miyamoto wanted to show that there were some circumstances when anzan was the only possible method.

   His solution was the computer game Flash Anzan, which he demonstrated for me. He told the class to get ready, pressed play and the pupils stared at a TV screen at the front of the room. The machine beeped three times to indicate it was about to start, and then the following 15 numbers appeared, one at a time. Each number appeared for only 0.2 seconds, so the whole thing was over in three seconds:

   164

   597

   320

   872

   913

   450

   568

   370

   619

   482

   749

   123

   310

   809

   561

    

   The numbers flashed by so quickly I barely had time to register them. Yet as soon as the last number flashed, Naoki Furuyama smiled and said the sum of the numbers was 7907.

   It is impossible to solve a Flash Anzan challenge with a calculator or an abacus since there is no time to remember the digits being flashed at you, let alone type them into a machine or arrange beads. Anzan does not require you to remember the digits. All you do is shift the beads in your brain whenever you see a new number. You start with 0, then on seeing 164 instantly visualize the abacus on 164. On seeing 597 the internal abacus rearranges to the sum, which is 761. After 15 additions you cannot remember any of the flashed numbers nor the intermediate sums, but the imaginary abacus in your head will show the answer: 7907.

   The wow factor of Flash Anzan has made it a national fad, and Nintendo has even released a Flash Anzan game for its DS consoles. Miyamoto showed me some clips from a Flash Anzan TV game show in which teenage anzan stars battled it out in front of screaming fans. Miyamoto says his game has helped recruit many new pupils to abacus clubs all over Japan. ‘People didn’t realize what you could do with soroban skills,’ he said. ‘With all this coverage, now they do.’

   Neural imaging scans show that the parts of the brain activated by the abacus, or anzan, are different from the parts activated by normal arithmetical calculations and language. Traditional ‘pen and paper’ arithmetic depends on neural networks associated with linguistic processing. The soroban relies on networks associated with visuospatial information. Miyamoto simplifies this as ‘soroban uses the right brain, normal maths uses the left brain’. Not enough scientific research has been done to understand what benefits this segregation brings, or how it relates to general intelligence, concentration or other skills. Yet it does explain an astonishing phenomenon: that soroban experts are able to multitask in the most incredible way.

   Miyamoto met his wife, a former national soroban champion, when they frequented the same abacus club as youngsters. Their daughter, Rikako, is a soroban prodigy. Pity her if she wasn’t. At age eight, she completed her top dan – a level that only one in 100,000 people ever achieve in their lifetimes. Rikako, who is now aged nine, was in class. She was wearing a pastel-blue top, and her fringe came down to her glasses. She looked very alert and pursed her lips as a sign of concentration.

   Shiritori is a Japanese word game that starts with a person saying shiritori and each subsequent person must say a word that starts with the last syllable of the previous word. So, a possible second word would be ringo (apple), because it begins with ri. Miyamoto asked Rikako and the girl next to her to play shiritori with each other at the same time as playing a game of Flash Anzan in which 30 three-digit numbers were to be displayed in 20 seconds. The machine sounded its introductory pips and the girls’ dialogue went:

   Ringo

   Gorira (gorilla)

   Rappa (trumpet)

   Panda (panda bear)

   Dachou (ostrich)

   Ushi (cow)

   Shika (deer)

   Karasu (crow)

   Suzume (sparrow)

   Medaka (killifish)

   Kame (turtle)

   Medama yaki (fried egg)

    

   At the end of the 20 seconds, Rikako said: 17,602. She had been able to add up the 30 numbers and play shiritori simultaneously.