Thinking in Numbers: How Maths Illuminates Our Lives Read online

  ’Twas only striking from the Calendar

  Unborn tomorrow and dead Yesterday

  One day, in his eighth decade, Khayyám tired and lay down to rest. His heavy head swaddled in sheets of turban, he gazed up toward the sky. The light around him slowly faded, and then went out. The year was around 500 of the hegira – the point Al-Tabari had long ago calculated as being the end of the world.

  Counting by Elevens

  ‘Physicians say that thumbs are the master fingers of the hand,’ writes Michel de Montaigne, the Renaissance nobleman who invented the personal essay, in his short piece On Thumbs. So vital did Rome’s ancient rulers consider them, he explains, that war veterans who were missing thumbs received automatic exemption from all future military service.

  Throughout his writings, the Frenchman marvelled at the extent to which we depend upon our hands. A gesture of thumbs-up or down, an index finger pressed to the mouth, the palms held flat open toward the skies: delivered at the right instant all can say more than any word. In another essay Montaigne describes the case of a ‘native of Nantes, born without arms’ whose feet did double duty so well that the man ‘cuts anything, charges and discharges a pistol, threads a needle, sews, writes, takes off his hat, combs his head, plays at cards and dice.’ Elsewhere he observed that hands sometimes seem almost to possess a life of their own, as when his idle fingers would tap during a period of daydreaming, free of conscious effort or instruction.

  Montaigne neglects to mention counting as being among the many useful tasks undertaken by our hands (biographers have pointed out that arithmetic was not his strong suit). Of course, there is much to be said for the idea that our decimal number system originated from the practice of counting with the fingers. In our Latin-derived word ‘digit’ the meanings ‘number’ and ‘finger’ coincide. The Homeric Greek term for counting, pempathai, translates literally: ‘to count by fives’.

  People the world over, numbers at their fingertips, count to ten and by tens (twenty, thirty . . . fifty . . . one hundred . . .). But the manner of their getting to ten varies from hand to hand and from culture to culture. Like many Europeans, I count ‘one’ beginning on the thumb of my left hand and continue past five with the digits of my right till I arrive at the other thumb. Americans, on the other hand, will often start with their left index finger and count five on their left thumb – repeating the procedure on their right hand for the numbers six through ten. Where people read from right to left, as in the countries of the Middle East, counting generally begins on the little finger of the right hand. In Asia, the counter employs a single hand, folding the fingers thumb first to reach five, before unfolding them from the little finger (six) returning to the thumb (ten).

  What, I wonder, would it be like for a person who had not too few fingers – like Montaigne’s Roman veterans – but too many? Would such a person learn to count like you or me? How would it be to count by elevens?

  According to tradition, Henry VIII’s second wife, Anne Boleyn, carried a sixth finger on one of her hands – a medical condition known as polydactyly. Girls of high birth in Tudor society had the attention of tutors and learned to read, write and do sums. Counting on her eleven fingers, however, would have posed Anne certain difficulties.

  To start with, she would have found herself a number word short. Between the ninth finger and the last (which would still be the tenth – 10 meaning ‘1 set and 0 remainders’) her surplus digit would have needed naming. Given that she spent part of her childhood in France, she might have come up with the label dix. So Anne would have counted: one, two, three, four, five, six, seven, eight, nine, dix, ten. Though the numbers’ music sounds a little strange to our ears, in her mind the sequence would have come to feel like second nature. A year before the age of twenty, she would have celebrated (if only mentally) her dixteenth birthday. Counting higher, she would have taken the nineties down a peg, to make room for the dixties, and by way of dixty-dix arrive finally at one hundred.

  Counting in this way produces some funny-looking sums. Subtracting seven from ten (where, given the extra dix, ‘ten’ would to our way of thinking be eleven), for example, would have left Anne not with three, but four. Half of thirteen equals seven. Six squared (6 × 6) is thirty-three (three ‘tens’ of eleven, plus three units): a pretty result.

  Fractions would have proven particularly tricky. Unlike ten, eleven is a prime number: divisible only by itself and one. No precise midway point exists between one and eleven, or between Anne’s ten (equivalent to our eleven) and her one hundred (11 × 11). What then might a half of something mean to someone with eleven fingers? And what about a fifth or a quarter or two-thirds? Possibly such concepts would remain as airy and intangible as a dixth of something might seem to you or me. (We can nonetheless imagine that rote learning would have sufficed to acquire them.)

  Still, I am curious about the intuitions that an eleven-fingered girl might have brought to such ideas. From our two hands, equally endowed, we understand immediately that halves are clean and precise, leaving no remainder. Half of eight is four, not three or five. A right-angle triangle is exactly one half of a square. Prime numbers, by definition, cannot be halved in this way. Might Anne’s hands – with six fingers on one hand, and ‘only’ five on the other – have given her a more approximate, fuzzier feeling for the concept of a half? To a casual remark like, ‘I’ll be along in half an hour’, would the impatient thought have occurred to her, ‘Which half do you mean – the lesser or the greater?’

  At her secret wedding to the King of England, Boleyn would have well understood which half of the couple she represented. Her triumph was infamously short-lived. Within months, the Archbishop of Canterbury had declared the marriage invalid. Roman Catholics denounced her as a scarlet woman.

  The charge of treason brought against her only three years after the wedding, makes no mention of witchcraft. All the same, her enemies alleged that the Queen’s body showed strange warts and growths – the same body that had produced only stillborn foetuses for a male heir. It was in the role of a conspiring adulteress that she knelt in a black damask dress before the executioner’s block on the morning of 19 May 1536.

  It is certainly conceivable that the story of an eleventh finger was a fabrication of Anne’s enemies. A famous portrait by an unknown artist, which now hangs in Hever Castle (Boleyn’s childhood home) in Kent, reveals a striking young woman clutching a rose. The hands peek shyly from long sleeves; the fingers – ten in all – appear, in contrast to the smooth oval face, somewhat ill formed. A slander, or a secret, Anne’s eleven fingers have become an integral part of her legend.

  I was reminded of it recently when a fascinating newspaper article caught my attention. It was about one Yoandri Hernandez Garrido, a thirty-seven-year-old Cuban man with twelve fingers and twelve toes. Apparently, Fidel Castro’s doctor once paid the extra-dextrous boy a visit and declared his hands and feet the most beautiful that he had ever seen.

  In keeping with the Latin American taste for nicknames based on physical appearance, schoolmates gave Garrido the name ‘Veinticuatro’ (twenty-four). He learned to count in class like his friends but one day, he recalls, his primary school teacher asked him for the answer to the sum 5 + 5. In his confusion he answered, ‘Twelve’.

  Garrido tells the reporter that he makes a fine living with his hands. American tourists regularly tender their dollars to have a photo taken with the twelve-fingered Cuban. A few extra bucks can go a long way in Castro’s republic. Proudly, Garrido holds his hands up to the camera. There is generosity in his smile.

  The article makes no mention of the adult Garrido’s arithmetic. Counting time, to give one example, should be simpler with twelve fingers – one digit for every hour on the clock. To find what time is nine hours before four o’clock in the afternoon, he need simply hold down five fingers (9 – 4) on his right hand to reveal the answer: one hand (six fingers) plus the right hand’s thumb: seven a.m. One digit for every month as well: so half-yea
r periods jump from one hand’s finger to the same finger on the other hand.

  We know that the Romans preferred to perform certain calculations using a base of twelve. In his Ars Poetica the poet Horace offers us a vignette of Roman boys learning their fractions.

  Suppose Albinus’s son says: if one-twelfth is taken from five-twelfths, what is left? You might have answered by now: One-third. Well done. You will be able to manage your money. Now add a twelfth: what happens? One-half.

  The Venerable Bede taught his fellow monks to quantify the various periods of time in the Biblical stories using the Roman fractions. One-twelfth, he noted, went by the name of an uncia (from which comes the word ‘ounce’), while the remaining part of 11/12 was called deunx. Dividing into six, the sixth part (1/6) was called sextans, and dextans for the rest (5/6). Quadrans was the Roman name for a quarter (1/4); three-quarters they called dodrans.

  What do we get if we add one sextans to a dodrans (1/6 + ¾)? As quick as Horace’s schoolboys, or the Venerable Bede’s monks, Garrido’s fingers (or toes) would tell him: one-sixth equals two digits and three-quarters equals nine digits, so 1/6 + ¾ = 11/12 (a deunx).

  I wonder what Garrido makes of our deficient hands, all of us counting to the tune of ten? Does he pity us as the Romans pitied their thumbless veterans? He calls his condition a blessing; it is clear that he would not wish to be any other way.

  Some believe that we should all learn to count like Garrido. A society, formed in the mid-1900s, advocates replacing the decimal system with a ‘dozenal’ one, since twelve divides more easily than ten. Leaving aside itself and one, the number twelve can be divided into four factors: two, three, four and six, compared with the factors of ten: two and five. In England and America the Society still militates for the return of Roman fractions (among other measures), judging their abandonment a gross error.

  Like the Esperantists and the simplified spellers that came before them, the Society’s members dream of a highly rational world comprising pairs, trios, quarters and sextets, a world innocent of messy fractions. Theirs is the charm of the hopeless cause.

  An English queen with eleven fingers, a Cuban man with twelve toes – such stories still incite our wonder and with it our vague sense that something has gone awry.

  Montaigne, typically generous, remedies this view. He recalls once encountering a family who exhibited ‘a monstrous child’ in return for strangers’ coins. The infant had multiple arms and legs, the remains of what we now call a conjoined twin. In the infinite imagination of its Creator, Montaigne supposes, the child is simply one of a kind, ‘unknown to man’. He concludes that, ‘Whatever falls out contrary to custom we say is contrary to nature, but nothing, whatever it be, is contrary to her. Let, therefore, this universal and natural reason expel the error and astonishment that novelty brings along with it.’

  The Admirable Number Pi

  To believe the poet Wislawa Szymborska, I am one in two thousand. The 1996 Nobel laureate offers this statistic in her poem ‘Some Like Poetry’ to quantify the ‘some’. Actually I think she is a tad too pessimistic – I am hardly as rare a reader as that. But I can see her point. Many people consider poetry to be all clouds and buttercups, without purchase on the real world. They are right and they are wrong. Clouds and buttercups exist in poetry, but they are there only because storms and flowers populate the real world too. Truth is, a good poem can be about anything.

  Including numbers. Mathematics, several of Szymborska’s verses show, lends itself to poetry. Both are economical with meaning; both can create entire worlds within the space of a few short lines. In ‘A Large Number’, she laments feeling at a loss with numbers many zeroes long, while her ‘Contribution to Statistics’ notes that ‘out of every hundred people, those who always know better: fifty-two’ but also, ‘worthy of empathy: ninety-nine’. And then there is ‘The Admirable Number Pi’, my favourite poem. It – the poem, and the number – begins: three point one four one.

  Once, in my teens, I confided my admiration for this number to a classmate. Ruxandra was her name. Like the poet’s, her name came from behind the Iron Curtain. Her parents hailed from Bucharest. I knew nothing of Eastern Europe, but that did not matter: Ruxandra liked me. She liked that I was quite different from the other boys. We spent breaks between lessons in the school library, swapping ideas about the future and homework tips. Happily for me, her strongest subject was maths.

  In an access of curiosity, I asked about her favourite number. Her reply was slow; she seemed not to understand my question. ‘Numbers are numbers,’ she said.

  Was there no difference at all for her between the numbers 333, say, and fourteen? There was not.

  And what about the number pi, I persisted; this almost magical number that we learned about in class. Did she not find it beautiful?

  Beautiful? Her face shrank with incomprehension.

  Ruxandra was the daughter of an engineer.

  The engineer and the mathematician have a completely different understanding of the number pi. In the eyes of an engineer, pi is simply a value of measurement between three and four, albeit fiddlier than either of these whole numbers. For his calculations he will often bypass it completely, preferring a handy approximation like 22/7 or 355/113. Precision never demands of him anything beyond a third or fourth decimal place (3.141 or 3.1416, with rounding). Word of other digits past the third or fourth decimal does not interest him; as far as he is concerned, it is as though they do not exist.

  Mathematicians know the number pi differently, more intimately. What is pi to them? It is the length of a circle’s round line (its circumference) divided by the straight length (its diameter) that splits the circle into perfect halves. It is an essential response to the question, ‘What is a circle?’ But this response – when expressed in digits – is infinite: the number has no last digit, and therefore no last-but-one digit, no antepenultimate digit, no third-from-last digit, and so on. One could never write down all its digits, even on a piece of paper as big as the Milky Way. No fraction can properly express pi: every earthly calculation produces only deficient circles, pathetic ellipses, shoddy replicas of the ideal thing. The circle that pi describes is perfect, belonging exclusively to the realm of the imagination.

  Moreover, mathematicians tell us, the digits in this number follow no periodic or predictable pattern: just when we might anticipate a six in the sequence, it continues instead with a two or zero or seven; after a series of consecutive nines, it can as easily remain long-winded with another nine (or two more nines or three) as switch erratically between the other digits. It exceeds our apprehension.

  Circles, perfect circles, thus enumerated, consist of every possible run of digits. Somewhere in pi, perhaps trillions and trillions of digits deep, a hundred successive fives rub shoulders; elsewhere occur a thousand alternating zeroes and ones. Inconceivably far inside the random-looking morass of digits, having computed them for a time far longer than that which separates us from the big bang, the sequence 123456789 . . . repeats 123,456,789 times in a row. If only we could venture far enough along, we would find the number’s opening hundred, thousand, million, billion digits immaculately repeated, as though at any instant the whole vast array were to begin all over again. And yet, it never does. There is only one number pi, unrepeatable, indivisible.

  Long after my schooldays ended, pi’s beauty stayed with me. The digits insinuated themselves into my mind. Those digits seemed to speak of endless possibility, illimitable adventure. At odd moments I would find myself murmuring them, a gentle reminder. Of course, I could not possess it – this number, its beauty, and its immensity. Perhaps, in fact, it possessed me. One day, I began to see what this number, transformed by me, and I by it, could turn into. It was then that I decided to commit a multitude of its digits to heart.

  This was easier than it sounds, since big things are often more unusual, more exciting to the attention, and hence more memorable, than small ones. For example, a short word like pen or song
is quickly read (or heard), and as quickly forgotten, whereas hippopotamus slows our eye (or ear) just enough to leave a deeper impression. Scenes and personalities from long novels return to me with far greater insistence and fidelity, I find, than those that originated in short tales. The same goes for numbers. A common number like thirty-one risks confusion with its common neighbours, like thirty and thirty-two, but not 31,415 whose scope invites curious, careful inspection. Lengthier, more intricate digit sequences yield patterns and rhythms. Not 31, or 314, or 3141, but 3 1 4 1 5 sings.

  I should say that I have always had what others call ‘a good memory’. By this, they mean that I can be dependably relied upon to recall telephone numbers, and dates of birthdays and anniversaries, and the sorts of facts and figures that crowd books and television shows. To have such a memory is a blessing, I know, and has always stood me in good stead. Exams at school held no fear for me; the kinds of knowledge imparted by my teachers seemed especially amenable to my powers of recollection. Ask me for the third person subjunctive of the French verb ‘être’, for instance, or better still the story of how Marie Antoinette lost her head, and I could tell you. Piece of cake.

  Pi’s digits henceforth became the object of my study. Printed out on crisp, letter-sized sheets of paper, a thousand digits to a page, I gazed on them as a painter gazes on a favourite landscape. The painter’s eye receives a near infinite number of light particles to interpret, which he sifts by intuitive meaning and personal taste. His brush begins in one part of the canvas, only to make a sudden dash to the other side. A mountain’s outline slowly emerges with the tiny, patient accumulation of paint. In a similar fashion, I waited for each sequence in the digits to move me – for some attractive feature, or pleasing coincidence of ‘bright’ (like 1 or 5) and ‘dark’ (like 6 or 9) digits, for example, to catch my eye. Sometimes it would happen quickly, at other times I would have to plough thirty or forty digits deep to find some sense before working backwards. From the hundreds, then thousands, of individual digits, precisely rendered and carefully weighed, a numerical landscape gradually emerged.