Thinking in Numbers: How Maths Illuminates Our Lives Read online

From Mr Baxter I learnt at least one profitable lesson; I learnt how not to teach. This lesson would serve me well on numerous occasions. Two years after I left school, when I was unpeeling the newspaper one morning, I came across an agency’s advert, recruiting for tutors. I had taught English in Lithuania during my gap year and discovered that I liked teaching. So I applied. The interview placed me opposite a lady up in years named Grace, in the office that she kept in her living room. Needlepoint cushions filled the small of my back as I sat before her desk. The wallpaper, if I remember correctly, had a pattern of small birds and honeybees.

  The meeting was brief.

  ‘Do you enjoy helping others to learn something new?’

  ‘Are you mindful of a student’s personal learning style?’

  ‘Could you work according to a set curriculum?’

  Her questions contained their own answers, like the dialogue taught in a foreign language class: ‘Yes, I could work according to a set curriculum’.

  After ten minutes of this she said, ‘Excellent, well you would certainly fit in with us here. For English we already have tutors and there is not much demand for foreign languages. What about primary-school level maths?’

  What about it? I was a taker.

  Grace’s bookings certainly kept me on my toes. My tutoring patch extended to the neighbouring town five miles away, and the bus ride and walk to the farthest homes took as long as the lessons. Nervous, I learned on the job, but the families helped me. I found the children, between the ages of seven and eleven, generally polite and industrious; their parents’ nods and smiles tamed my nerves. After a while I ceased to worry and even began to look forward to my weekly visits.

  Should I admit I had a favourite student? He was a brown-haired, freckled boy, eight years old but small for his age, and the first time I came to the house he fairly shivered with shyness. We started out with the textbooks that the agency loaned me, but they were old and smelly and the leprous covers soon came apart in our hands. A brightly coloured book replaced them, one of the boy’s Christmas gifts, but its jargon was poison to his mind. So we abandoned the books and found some better way to pass the hour together. We talked a lot.

  It turned out that he had a fondness for collecting football stickers and could recite the names of the players depicted on them by heart. With pride he showed me the accompanying album.

  ‘Can you tell me how many stickers you have in there?’ I asked him. He admitted to having never totalled them up. The album contained many pages.

  ‘If we count each sticker one by one it will take quite a long time to reach the last page,’ I said. ‘What if we were to count the stickers two by two instead?’ The boy agreed that would be quicker. Twice as quick, I pointed out, ‘And what if we were to count the stickers in threes? Would we get to the end of the album even faster?’ He nodded. Yes, we would: three times as fast.

  The boy interrupted. ‘If we counted the stickers five at a time, we would finish five times faster.’ He smiled at my smile. Then we opened the album and counted the stickers on the first page, I placing my larger palm over every five. There were three palms of stickers: fifteen. The second page had slightly fewer stickers (two palms and three fingers, thirteen) – so we carried the difference over to the next. By the seventh page we had reached twenty palms: one hundred stickers. We continued turning the pages, putting my palm on to each, and counting along. In all, the number of stickers rose to over eighty palms (four hundred).

  After making light work of the album, we considered the case of a giant counting its pages. The giant’s palm, we agreed, would easily count the stickers by the score.

  What if the same giant wanted to count up to a million? The boy thought for a moment. ‘Perhaps he counts in hundreds: one hundred, two hundred, three hundred . . .’ Did the boy know how many hundreds it would take to reach a million? He shook his head. Ten thousand, I told him. His eyebrows leapt. Finally, he said, ‘He would count in ten thousands then, wouldn’t he?’ I confirmed that he would: it would be like us counting from one to hundred. ‘And if he were really big, he might count in hundred thousands,’ I continued. The giant would reach a million as quickly as we counted from one to ten.

  Once, during a lesson solving additions, the boy hit upon a small but clever insight. He was copying down his homework for us to go through together. The sum was 12 + 9, but it came out as 19 + 2. The answer, he realised, did not change. Twelve plus nine, and nineteen plus two, both equal twenty-one. The fortuitous error pleased him; it made him pause and think. I paused as well, not wishing to talk for fear of treading on his thoughts. Later I asked him for the answer to a much larger sum, something like 83 + 8. He closed his eyes and said, ‘Eighty-nine, ninety, ninety-one.’ I knew then that he had understood.

  Of the other students, I recall the Singh family whom I taught on Wednesday evenings for two hours back-to-back. I remember I could never get on with the father, a businessman who put on executive airs, although the mother treated me with nothing but kindness. They had three children, two boys and a girl; the children were always waiting for me around the table in the living room, still dressed up in their school uniform. The eldest boy was eleven: a bit of a show-off, he had the confidence of an eldest son. His sister regularly deferred to him. In the middle, the second son made a lot of laughter. He seemed to laugh for all the family.

  In the beginning, the trio took the wan bespectacled man before them only half seriously as a teacher. Between them, they had ten years on me. I looked too young and probably sounded it as well, without the smooth patter that comes with experience. All the same, I held my ground. I helped them with their times tables, in which they were far from fluent. They showed surprise when I failed to berate their errors and hesitations. On the contrary, if they were close to the right answer I told them so.

  ‘What is eight times seven?’

  ‘Fifty . . .’ The eldest boy wavered.

  ‘Yes,’ I said, encouraging.

  ‘Fifty-four,’ he ventured.

  ‘Nearly,’ I said. ‘Fifty-six.’

  This hesitation, a habit of many of my students, intrigued me. It suggested not ignorance, but rather indecision. To say that a student has no idea of a solution, I realised, is untrue. Truth is, the learner does have ideas, too many, in fact – almost all of them bad. Without the knowledge necessary to eliminate this mental haze, the learner finds himself confronted with an embarrassment of wrong answers to helplessly pick from.

  What had the boy been thinking, I enquired, when he selected fifty-four as his answer? He admitted having previously considered fifty-three, fifty-six, fifty-seven, and fifty-five (in that order). He had felt sure that fifty-one or fifty-two would be too small an answer, both fifty-eight and fifty-nine, too high. Then I asked him why he had finally preferred fifty-four to fifty-three. He replied that he had thought of the eight in the question, and the fact that fifty is half of one hundred, and that half of eight is four.

  We moved on to discussing the difference between odd and even numbers. Eight was an even number; seven was odd. What happens when we multiply odd by even? The children’s faces showed more hesitation. I suggested several examples: two times seven (fourteen), three times six (eighteen), four times five (twenty), every answer an even number. Could they see why? Yes, said the second son at last: multiplying by an even number was the same thing as creating pairs. Two times seven created a pair of sevens; four times five made two pairs of fives; three times six produced three pairs of threes. What, then, about eight times seven? It meant four pairs of sevens, the boy said.

  Pairs evened every odd number: one sock became two socks, three became six, five became ten, seven became fourteen, and nine became eighteen. The last digit of a pair would always be an even number.

  The mental fog of eight times seven was evaporating. Fifty-three promptly vanished as a contending answer; so too did fifty-seven and fifty-five. That left fifty-four and fifty-six. How then to tell the two apart? The number fifty-four, I
pointed out, was six away from sixty: fifty-four – like sixty – was divisible by six. Like sixty, fifty-four will therefore be the answer to a question containing a six (or a number divisible by six), but neither a seven nor an eight.

  By this process of elimination, which amounts to careful reasoning, only fifty-six remained. Fifty-six is two sevens away from seventy, three eights away from eighty. Eight times seven equals fifty-six.

  My only adult student was a housewife with copper-coloured skin, and a long name with vowels and consonants that I had never before seen in such a permutation. The housewife, Grace informed me on the phone, aspired to professional accountancy. In my mind, this fact made for an unpromising start. The admission of self-interest rubbed up against my naïve vision of mathematics as something playful and inventive. There seemed to me something almost vulgar in the housewife’s sudden interest in numbers, as if she wanted to befriend them only as some people set out to befriend well-connected people.

  Quickly I amended this faulty judgment. The reservations about my new student were unfair. They were the reservations of a children’s tutor – I knew nothing of teaching adults, of anticipating their needs and expectations.

  We were talking together one day about negative numbers, as we sat in her white-tiled kitchen. Like mathematicians of the sixteenth century, who referred to them as ‘absurd’ and ‘fictitious’, she found these numbers difficult to imagine. What could it mean to subtract something from nothing? I tried to explain, but found myself at a frustrating loss for words. Somehow my student understood.

  ‘You mean, like a mortgage?’

  I did not know what a mortgage was. Now it was her turn to try to explain. As she spoke, I realised that she knew a lot more about negative numbers than I did. Her words possessed real value: they had the bullion of hard experience to back them up.

  Another time we went over ‘improper’ fractions: top-heavy fractions, like four-thirds (4/3) or seven-quarters (7/4), that help us to think about units in different ways. If we think of the number one as being equivalent to three thirds, for example, then four-thirds is another way of describing one and one-third. Seven-quarters, my student and I agreed, resembled two apples that had been quartered (considering ‘one’ as equal to four quarters) and one of these eight apple-quarters consumed.

  Our hour was soon up, but we kept on talking. We were discussing fractions and what happens when you halve a half of a half of a half, and so on. It amazed us both to think that this halving, theoretically speaking, could continue indefinitely. There was pleasure in confiding our mutual amazement, almost in the manner of gossip. And like gossip, it was something that we both knew and did not know.

  Then she came to a beautiful conclusion about fractions that I shall never forget.

  She said, ‘There is no thing that half of it is nothing.’

  Shakespeare’s Zero

  Few things, to judge by his works, so fascinated William Shakespeare as the presence of absence: the lacuna where there ought to be abundance, of will, or judgment, or understanding. It looms large in the lives of many of his characters, so powerful in part because it is universal. Not even kings are exempt.

  LEAR: What can you say to draw

  A third more opulent than your sisters? Speak.

  CORDELIA: Nothing, my lord.

  LEAR: Nothing?

  CORDELIA: Nothing.

  LEAR: Nothing will come of nothing. Speak again.

  The scene is one of the tensest, most suspenseful moments in theatre, a concentration of tremendous force within a single word. It is the ultimate negation, tossed between the old king and his beloved youngest daughter, compounded and multiplied through repetition. Nothing. Zero.

  Of course, Shakespeare’s contemporaries were familiar with the idea of nothingness, but not with nothingness as a number, something that they could count and manipulate. In his arithmetic lessons William became one of the first generation of English schoolboys to learn about the figure zero. It is interesting to wonder about the consequences of this early encounter. How might the new and paradoxical number have driven his thoughts along particular paths?

  Arithmetic spelled trouble for many schoolmasters of the period. Their knowledge of it was often suspect. For this reason, lessons were probably kept short, often detained till almost the final hour of the afternoon. Squeezed in after long bouts of Latin composition, a list of proverbs or the chanting of prayers, the sums and exercises were mostly drawn from a single textbook: The Ground of Artes by Robert Recorde. Published in 1543 (and again, in an expanded edition, in 1550), Recorde’s book, which included the first material on algebra in the English language, taught ‘the work and practice of Arithmetike, both in whole numbres and Fractions after a more easyer and exarter sorte than any like hath hitherto been sette furth.’

  Shakespeare learnt to count and reckon using Recorde’s methods. He learnt that ‘there are but tenne figures that are used in Arithmetick; and of those tenne, one doth signifie nothing, which is made like an O, and is privately called a Cypher.’ These Arabic numbers – and the decimal place system – would soon eclipse the Roman numerals (the Tudors called them ‘German numbers’), which were often found too cumbersome for calculating.

  German numbers were, of course, letters: I (or j), one; V, five; X, ten; L, fifty; C, one hundred; D, five hundred; and M, one thousand. Six hundred appeared as vj.C and three thousand as CCC.M. Possibly this is why Recorde compares the zero to an O. Years later, Shakespeare would deploy the cipher to blistering effect. ‘Thou art an O without a figure . . . Thou art nothing,’ the Fool tells Lear, after the dialogue with Cordelia that destroys the king’s peace of mind.

  In Shakespeare’s lessons, letters were out, figures (digits) in. Perhaps they were displayed conspicuously on charts, hung up on walls like the letters of the alphabet. Atop their hard benches, ten to a bench, the boys trimmed and dipped their quills in ink, and their feathered fingers copied out the numbers in small, tidy lines. Zeroes spotted every page. But why record something that has no value? Something that is nothing?

  England’s mediaeval monks, translators and copiers of earlier treatises by the Arab world’s mathematicians, had long ago noted the zero’s almost magical quality. One twelfth-century scribe suggested giving it the name ‘chimera’ after the fabulous monster of Greek mythology. Writing in the thirteenth century, John of Halifax explained the zero as something that ‘signifies nothing’ but instead ‘holds a place and signifies for others’. His manuscript proved popular in the universities. But it would take the invention of Gutenberg’s printing press to bring these ideas to a much wider audience. Including the motley boys of King’s New School in Stratford.

  GLOSTER: What paper were you reading?

  EDMUND: Nothing my lord.

  GLOSTER: No? What needed then that terrible despatch of it into your pocket? The quality of nothing hath not such need to hide itself. Let’s see: Come, if it be nothing I shall not need spectacles.

  The quality of nothing. We can picture the proto-playwright grappling with zero. The boy closes his eyes and tries to see it. But it is not easy to see nothing. Two shoes, yes he can see them, and five fingers and nine books. 2 and 5 and 9: he understands what they mean. How, though, to see zero shoes? Add a number to another number, like a letter to another letter, and you create something new: a new number, a new sound. Only, if you add a number to zero nothing changes. The other number persists. Add five zeroes, ten zeroes, a hundred zeroes if you like. It makes no difference. A multiplication by zero is just as mysterious. Multiply a number, any number – three, or four hundred, or 5,678 – by zero, by nothing, and the answer is zero.

  Was the boy able to keep up in his lessons, or did he lag? The Tudor schoolmaster, violence dressed in long cloak and black shoes, would probably have concentrated his mind. The schoolmaster’s cane could reduce a boy’s buttocks to one big bruise. With its rhyming dialogues (even employing the occasional joke or pun) and clear examples intended to give ‘ease t
o the unlearned’, we can only hope that Recorde’s book spared Shakespeare and his classmates much pain.

  Heere you see vj. (six) lines, whiche stande for vj. (six) places . . . the [lowest line] standeth for the first place, and the next above it for the second, and so upwarde, till you come to the highest, whiche is the first line, and standeth for the first place.

  100000

  10000

  1000

  100

  10

  1

  The first place is the place of units or ones, and every counter set in that line betokeneth but one, and the seconde line is the place of 10, for everye counter there standeth for 10. The thirde line the place of hundreds, the fourth of thousands, and so forth.

  Perhaps talk of counters turned the boy’s thoughts to his father’s glove shop. His father would have accounted for all his transactions using the tokens. They were hard and round and very thin, made of copper or brass. There were counters for one pair of gloves, and for two pairs, and three and four and five. But there was no counter for zero. No counters existed for all the sales that his father did not close.

  I imagine the schoolmaster sometimes threw out questions to the class. How do we write three thousand in digits? From Recorde’s book, Shakespeare learned that zero denotes size. To write thousands requires four places. We write: 3 (three thousands), 0 (zero hundreds), 0 (zero tens), 0 (zero units): 3,000. In Cymbeline we find one of Shakespeare’s many later references to place-value (what Recorde in his book called each digit’s ‘roome’).

  Three thousand confident, in act as many –

  For three performers are the file, when all

  The rest do nothing – with this word ‘Stand, stand’,

  Accommodated by the place . . .

  The concept must have fascinated him, ever since his schooldays. A nothing, the boy sees, depends on kind. An empty hand, for example, is a smaller nothing than an empty class or shop, in the same way that the zero in 10 is ten times smaller than the zero in 101. And the bigger the number, the more places and therefore the more zeroes it can contain: ten has one zero, whereas one hundred thousand has five. He thinks, the bigger an empty room, the more the things that can be contained inside: the greater an absence, the greater the potential presence. Subtract (or ‘rebate’ as the Tudors said) one from one hundred thousand and the entire number transforms: five zeroes, five nothings, all suddenly turn to nines (the largest digit): 99,999. Perhaps, like Polixenes in The Winter’s Tale, he sensed already the tremendous potential of self-effacement, understood imagination as shifting from place to place like a zero inside an immense number.