The Gift of Numbers (5 page)

"A problem has a rhythm of its own, just like a piece of music,"
the Professor said. "Once you get the rhythm, you get the sense of
the problem as a whole, and you can see where the traps might be
waiting."

And so Root read in a loud, clear voice: "I bought two handkerchiefs
and two pairs of socks for ¥380. Two handkerchiefs and
five pairs of socks cost ¥710. How much did each handkerchief
and each pair of socks cost?"

"So, where do we start?" asked the Professor.

"Well, it seems pretty hard."

"You're right. This is the trickiest one in your homework today,
but you read it well. The problem consists of three sentences. The
handkerchiefs and socks appear three times each, and you had the
rhythm just right: so many handkerchiefs ... so many socks ... so
many yen; handkerchiefs ... socks ... yen. You made a boring
problem sound just like a poem."

The Professor was unstinting with his praise for Root. He never
seemed to lose patience when time passed and they were making
little progress; and like a miner sifting a speck of gold from the
muddy river bottom, he always found some small virtue to compliment,
even when Root was stuck.

"Well then, suppose we draw a picture of this little shopping
trip. First, there are two handkerchiefs; then two pairs of socks—"

"Those aren't socks!" Root interrupted. "They look more like
overweight caterpillars. Let me draw them."

"I see what you mean. That does look more like a caterpillar."
"He bought the same number of handkerchiefs the second time
but more socks. Five pairs is a lot to draw.... Mine are starting to
look like caterpillars, too."

"No, they're fine. And you're right, only the number of socks
increases, along with the price. Why don't we check to see how
much the price went up?"

"So, you'd subtract ¥380 from ¥710...."

"Always show your work, and do it neatly."

"I usually just scribble on the back of scrap paper."

"But every formula and every number has meaning, and you
should treat them accordingly, don't you think?"

I was sitting on the bed, doing some mending. Whenever they
started Root's homework, I tried to find something to do in the
study in order to be near them. I would iron the Professor's shirts,
or work on a stain in the rug, or snip string beans for supper. If I
was working in the kitchen and heard their laughter drift in from
the other room, I felt terribly excluded—and I suppose I wanted
to be there when anyone was showing kindness to my son.

The sound of the rain seemed louder in the study, as if the sky
were actually lower there. The room was completely private,
thanks to the lush greenery that grew up around the house, and
there was no need to close the curtains even after dark. Their reflections
appeared dimly in windowpanes, and on rainy days the
musty smell in the study was stronger than usual.

"That's right! Then it's just a matter of simple division and
you've got it."

"So, you get the price of the socks first: ¥110."

"Okay, but you've got to be careful now. The handkerchiefs
seem innocent, but they may turn out to be tricky."

"Right. But it's easier to do the sums when the numbers are
small."

The desk was a bit too high, and Root was forced to sit up very
straight as he leaned over his problem, a well-chewed pencil
clutched tightly in his hand. The Professor sat back, legs crossed
and looking relaxed, and his hand drifted to his unshaven chin from
time to time as he watched Root work. He was no longer a frail old
man, nor a scholar lost in his thoughts, but the rightful protector of
a child. Their profiles seemed to come together, superimposed on
one another, forming a single line. The gentle patter of the rain was
punctuated by the scratching of pencil on paper.

"Can I write out the equations separately like this? Our teacher
gets mad if we don't combine them all into one big formula."

"If you're doing them carefully and correctly, he has no reason
to get mad."

"Okay, let's see.... 110 times 2 is 220. Subtract that from
380.... That's 160 ... 160 divided by 2 ... is 80. That's it. One
handkerchief costs ¥80."

"That's right! Well done!"

As the Professor rubbed Root's head, Root glanced up into his
face, not wanting to miss the look of approval and pleasure.

"I'd like to give you a problem myself," said the Professor.
"Would you mind?"

"What?"

"No long faces now. Since we're studying together, I feel like
playing the teacher and giving you homework."

"That's not fair," said Root.

"It's just one little problem. All right? Here it is: What is the
sum of all the numbers from 1 to 10?"

"Okay, I'll let you give me homework if you'll do something for
me. I want you to get the radio fixed."

"The radio?"

"That's right. I want to listen to the ball games. You don't have
a TV and the radio's broken. And we're coming down to the pennant
race."

"Oh, I see ... baseball." The Professor let out a long, slow
breath, his hand still resting on Root's head. "What team do you
like?" he asked at last.

"Can't you tell from my hat?" Root said, picking up the cap
he'd left with his backpack and pulling it over his head. "The
Tigers!"

"The Tigers? Is that right? The Tigers," the Professor murmured.
"Enatsu! Yutaka Enatsu, best pitcher of all time."

"Yes! Good thing you don't like the Giants. Okay, we've got to
get the radio fixed," Root insisted. The Professor seemed to be
muttering something to himself, but I closed the lid of the sewing
box and stood up to announce it was time for dinner.

I finally managed to get the Professor out of the house. Since I'd
come to work, he had not so much as set foot in the garden, let
alone gone for a real outing, and I thought some fresh air would
be good for him.

"It's beautiful outside today," I said, coaxing him. "It makes
you want to go out, get some sun." The Professor was ensconced
in his easy chair with a book. "Why don't we take a walk in the
park and then stop in at the barbershop?"

"And why would we do that?" he said, glancing up at me over
his reading glasses.

"No particular reason. The cherry blossoms are just over in the
park and the dogwood is about to bloom. And a haircut might feel
good."

"I feel fine like this."

"A walk would get your circulation going, and that might help
you come up with some good ideas for your formulas."

"There's no connection between the arteries in the legs and the
ones in the head."

"Well, you'd be much handsomer if you took care of your hair."

"Waste of time," he said, but eventually my persistence got the
better of him and he closed his book. The only shoes in the cupboard
by the door were old leather ones covered in a thin layer of
mold. "You'll stay with me?" he asked several times as I was cleaning
them off. "You can't just leave me while I'm having my hair
cut and come home."

"Don't worry. I'll stay with you the whole time." No matter
how much I polished, the shoes were still dull.

I wasn't sure what to do with the notes the Professor had
clipped all over his body. If we left them on, people were bound to
stare, but since he didn't seem to care, I decided to leave them
alone.

The Professor marched along, staring down at his feet, without
a glance at the blue sky overhead or the sights we passed along the
way. The walk did not seem to relax him, he was more tense than
usual.

"Look," I'd say, "the cherry blossoms are in full bloom." But
he only muttered to himself. Out in the open air, he seemed somehow
older.

We decided to go to the barbershop first. The barber recoiled
at the sight of the Professor's strange suit, but he turned out to be
a kind man. He realized quickly that there must be a reason for the
notes, and after that he treated the Professor like any other customer.
"You're lucky to have your daughter with you," he said, assuming
we were related. Neither of us corrected him. I sat on the
sofa with the men waiting in line for their haircuts.

Perhaps the Professor had an unpleasant memory of going to
the barber. Whatever the reason, he was clearly nervous from the
moment the cape was fastened around his neck. His face went
stiff, his fingers dug into the arms of the chair, and deep creases
lined his forehead. The barber brought up several harmless topics
in an attempt to put him at ease, but it was no use.

"What's your shoe size?" the Professor blurted out. "What's
your telephone number?" The room fell silent.

Though he could see me in the mirror, he craned around from
time to time, checking to see that I'd kept my promise to stay with
him. When the Professor moved his head, the barber was forced
to stop cutting, but he would wait patiently and then go back to
work. I smiled and gave a little wave to reassure the Professor that
I was still there.

The white clippings of hair fell in clumps on the cape and then
scattered to the floor. As he cut and combed away, did the barber
suspect that the brain inside this snowy head could list all the
prime numbers up to a hundred million? And did the customers
on the sofa, waiting impatiently for the strange old man to depart,
have any notion of the special bond between my birthday and the
Professor's wristwatch? For some reason, I felt a secret pride in
knowing these things, and I smiled at the Professor just a bit more
brightly in the mirror.

After the barbershop, we sat on a bench in the park and drank
a can of coffee. There was a sandbox nearby, and a fountain and
some tennis courts. When the wind blew, the petals from the
cherry trees floated around us and the dappled sunlight danced on
the Professor's face. The notes on his jacket fluttered restlessly,
and he stared down into the can as if he'd been given some mysterious
potion.

"I was right—you look handsome, and more manly."

"That's quite enough of that," said the Professor. For once he
smelled of shaving cream rather than of paper.

"What kind of mathematics did you study at the university?" I
asked. I had little confidence that I would understand his answer;
maybe I brought up the subject of numbers as a way of thanking
him for coming out with me.

"It's sometimes called the 'Queen of Mathematics,' " he said,
after taking a sip of his coffee. "Noble and beautiful, like a queen,
but cruel as a demon. In other words, I studied the whole numbers
we all know, 1, 2, 3, 4, 5, 6, 7 ... and the relationships
between them."

His choice of the word
queen
surprised me—as if he were
telling a fairy tale. We could hear the sound of a tennis ball bouncing
in the distance. The joggers and bikers and mothers pushing
strollers glanced at the Professor as they passed but then quickly
looked away.

"You look for the relationships between them?"

"Yes, that's right. I uncovered propositions that existed out
there long before we were born. It's like copying truths from
God's notebook, though we aren't always sure where to find this
notebook or when it will be open." As he said the words "out
there," he gestured toward the distant point at which he stared
when he was doing his "thinking."

"For example, when I was studying at Cambridge I worked on
Artin's conjecture about cubic forms with whole-number coefficients.
I used the 'circle method' and employed algebraic geometry,
whole number theory, and the Diophantine equation. I was
looking for a cubic form that didn't conform to the Artin conjecture.
... In the end, I found a proof that worked for a certain type
of form under a specific set of conditions."

The Professor picked up a branch and began to scratch something
in the dirt. There were numbers, and letters, and some
mysterious symbols, all arranged in neat lines. I couldn't understand
a word he had said, but there seemed to be great clarity in
his reasoning, as if he were pushing through to a profound truth.
The nervous old man I'd watched at the barbershop had disappeared,
and his manner now was dignified. The withered stick
gracefully carved the Professor's thoughts into the dry earth, and
before long the lacy pattern of the formula was spread out at our
feet.

"May I tell you about something I discovered?" I could hardly
believe the words had come out of my mouth, but the Professor's
hand fell still. Overcome by the beauty of his delicate patterns,
perhaps I'd wanted to take part; and I was absolutely sure he
would show great respect, even for the humblest discovery.

"The sum of the divisors of 28 is 28."

"Indeed ... ," he said. And there, next to his outline of the Artin
conjecture, he wrote: 28 = 1 + 2 + 4 + 7 + 14. "A perfect number."

"Perfect number?" I murmured, savoring the sound of the
words.

"The smallest perfect number is 6: 6 = 1 + 2 + 3."

"Oh! Then they're not so special after all."

"On the contrary, a number with this kind of perfection is rare
indeed. After 28, the next one is 496: 496 = 1 + 2 + 4 + 8 + 16 + 31
+ 62 + 124 + 248. After that, you have 8,128; and the next one after
that is 33,550,336. Then 8,589,869,056. The farther you go, the
more difficult they are to find"—though he had easily followed
the trail into the billions!

"Naturally, the sums of the divisors of numbers
other
than
perfect numbers are either greater or less than the numbers themselves.
When the sum is greater, it's called an 'abundant number,'
and when it's less, it's a 'deficient number.' Marvelous names, don't
you think? The divisors of 18— + 2 + 3 + 6 + 9—equal 21, so it's an
abundant number. But 14 is deficient: 1 + 2 + 7 + 10."

I tried picturing 18 and 14, but now that I'd heard the Professor's
explanation, they were no longer simply numbers. Eighteen
secretly carried a heavy burden, while 14 fell mute in the face of its
terrible lack.

"There are lots of deficient numbers that are just one larger
than the sum of their divisors, but there are no abundant numbers
that are just one smaller than the sum of theirs. Or rather, no one
has ever found one."

"Why is that?"

"The answer is written in God's notebook," said the Professor.

Everything around us was glowing in the sunlight; even the
dried shells of the insects floating in the fountain seemed to glitter.
The most important of the Professor's notes—the one that read
"My memory lasts only eighty minutes"—had come loose, and I
reached over to adjust the clip.

"I'll show you one more thing about perfect numbers," he said,
swinging the branch and drawing his legs under the bench to
make more room on the ground. "You can express them as the
sum of
consecutive
natural numbers."

6 = 1 + 2 + 3

28 = 1 + 2 + 3 + 4 + 5 + 6 + 7

496 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14
+ 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26
+ 27 + 28 + 29 + 30 + 31

The Professor reached out to complete the long equation. The
numbers unfolded in a simple, straight line, polished and clean.
The subtle formula for the Artin conjecture and the plain line of
factors for the number 28 blended seamlessly, surrounding us
where we sat on the bench. The figures became stitches in the
elaborate pattern woven in the dirt. I sat utterly still, afraid I might
accidentally erase part of the design. It seemed as though the secret
of the universe had miraculously appeared right here at our
feet, as though God's notebook had opened under our bench.

"Well then," the Professor said at last. "We should probably be
getting home."

"Yes, we should," I said, nodding. "Root will be there soon."

"Root?"

"My son. He's ten years old. The top of his head is flat, so we
call him Root."

"Is that so? You have a son? We can't dawdle then. You should
be there when he gets home from school." With that, he stood
to go.

Just then, there was a cry from the sandbox. A little girl stood
sobbing, a toy shovel clutched in her hand. Instantly, the Professor
was at her side, bending over to comfort her. He tenderly brushed
the sand from her dress.

Suddenly, the child's mother appeared and pushed the Professor
away, picking the girl up and practically running off with her.
The Professor was left standing in the sandbox. I watched him
from behind, unsure how to help. The cherry blossoms fluttered
down, mingling with the numbers in the dirt.

"I did the problem and I got it right. So now you have to keep
your promise and fix the radio." These were the first words out of
Root's mouth as he came through the door. "Here, look," he said,
holding out his math notebook.

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 55

The Professor studied Root's work as though it were a sophisticated
proof. Unable to recall why he had assigned this problem or
what connection it had to repairing the radio, he was perhaps
looking for an answer in the sum itself.

The Professor carefully avoided asking us questions about
things that had happened more than eighty minutes ago. We
would have happily explained the meaning of the homework and
the radio if he had asked, but he preferred to examine the facts
before him and draw his own conclusions. Because he had been—and
in many ways still was—such a brilliant man, he no doubt understood
the nature of his memory problem. It wasn't pride that
prevented him from asking for help but a deep aversion to causing
more trouble than necessary for those of us who lived in the normal
world. When I realized why he was so reluctant to bring up
the subject of his memory, I decided I would say as little as possible
about it, too.

"You've added up the numbers from 1 to 10," he said at last.

"I got it right, didn't I? I checked it over and over, I'm sure it's
right."

"Indeed it is!"

"Good! Then let's go get the radio fixed."

"Now just a minute," said the Professor, coughing quietly as if
to give himself time to think. "I wonder if you could explain to me
how you got the answer?"

"That's easy! You just add them up."

"That's a straightforward way to do it; perfectly reliable, and no
one can argue with that." Root nodded proudly. "But think for a
minute: what would you do if a teacher, say, a
mean
teacher, asked
you to add the numbers from 1 to 100?"

"I'd add them up, of course."

"Naturally you would. You're a good boy, and a hard worker.
So I'm sure you'd come up with the right answer for 1 to 100, too.
But what if that teacher was really cruel and made you find the
sum for 1 to 1,000? Or 1 to 10,000? You'd be adding, adding, and
adding forever while that teacher laughed at you. What would you
do then?" Root shook his head. "But you can't let that evil teacher
get to you," the Professor continued. "You've got to show him
you're the better man."

"But how do you do that?"

"You need to find a simpler way to get the answer that works
no matter how big the numbers get. If you can find it, then I'll get
the radio fixed."

"That's not fair!" Root objected, kicking his chair leg. "That
wasn't part of the deal."

"Root!" I interrupted. "Is that any way to act?" But the Professor
didn't seem to notice his outburst.

"A problem isn't finished just because you've found the right answer.
There's another way to get to 55; wouldn't you like to find it?"

"Not really ... ," said Root, sulking.

"All right, here's what we'll do. The radio is old, and it may
take them a while to get it working again. So how about a contest
to see whether you can find another way to get the sum before the
radio is fixed?"