Thinking in Numbers: How Maths Illuminates Our Lives (7 page)

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 to 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.

And therefore, like a cipher [zero]

(Yet standing in rich place), I multiply

With one ‘We thank you’ many thousands more

That go before it

Recorde’s book abounded with exercises. Shakespeare and his classmates’ sheets of paper would have quickly turned black with sums. They measured cloth and purchased loaves and counted sheep and paid clergy. But William’s mind returns ceaselessly to the zero. He thinks of ten and how it differs from his father’s ten. To his father, ten (X) is twice five (v): he counts, whenever possible, in fives and tens. To his son, ten (10) is a one (1) displaced, accompanied by a nought. To his father, ten (X) and one (i) have hardly anything in common: they are two values on opposite ends of a scale. But for the boy, ten and one are intimately linked: there is nothing between them.

Ten and one, one and ten.

With a retinue of zeroes, the boy sees, even the humble one becomes enormously valuable. Imagination can reconcile even one and one million, as Shakespeare affirmed in his prologue to
Henry V
, when the chorus stake their claim to represent the multitudes on the field of Agincourt.

O, pardon! Since a crooked figure [digit] may

Attest in little place a million,