The Gift of Numbers aka The Housekeeper and the Professor

2

That evening, after I'd got home and put my son to bed, I decided to look for "amicable numbers" on my own. I wanted to see whether they were really as rare as the Professor had said, and since it was just a matter of writing out factors and adding them up, I was sure I could do it, even though I'd never graduated from high school.

But I soon realized what I was up against. Following the Professor's suggestion, I tried using my intuition to pick likely pairs, but I had no luck. I stuck to even numbers at first, thinking the factors would be easier to find, and I tried every pair between ten and one hundred. Then I expanded my search to odd numbers, and then to three-digit numbers as well, still to no effect. Far from being amicable, the numbers seemed to turn their backs on each other, and I couldn't find a pair with even the most tenuous connection-let alone this wonderfully intimate one. The Professor was right: my birthday and his watch had overcome great trials and tribulations to meet each other in the vast sea of numbers.

Soon, every inch of the paper was filled with figures. My method was logical, if a little primitive-yet I ended up with nothing to show for all my work.

I did make one small discovery: the sum of the factors of 28 equals 28.

28: 1 + 2 + 4 + 7 + 14 = 28

Though I wasn't sure this amounted to anything. None of the other numbers I'd tried were the sum of their own factors, but that didn't mean there weren't more out there. I knew it was an exaggeration to call it a "discovery," but for me it was just that. This one line of numbers stretched across the page as if pulled taut by some mysterious intention.

As I got into bed, I finally glanced at the clock. It had been much more than eighty minutes since we'd had our talk about amicable numbers. By now he'd have forgotten all about our secret, and he'd have no idea where the number 220 had come from. I found it difficult to fall asleep.

From a housekeeper's perspective, working for the Professor was relatively easy: a small house, no visitors or phone calls, and only light meals for one man who had little interest in food. At other jobs, I always had to do as much as possible in a short amount of time; but now I was delighted to have so much time to do a truly thorough job of cleaning, washing, and cooking. I learned to recognize when the Professor was beginning a new contest, and how to avoid disturbing him. I polished the kitchen table to my heart's content with a special varnish and patched the mattress on his bed. I even invented various ways to camouflage the carrots in his dinner.

The one thing about the job that was always a little tricky was understanding how the Professor's memory worked. According to the old woman, he remembered nothing after 1975; but I had no idea what yesterday meant to him or whether he could think ahead to tomorrow, or how much he suffered.

It was clear that he didn't remember me from one day to the next. The note clipped to his sleeve simply informed him that it was not our first meeting, but it could not bring back the memory of the time we had spent together.

When I went out shopping, I tried to return home within an hour and twenty minutes. As befit a mathematician, the device in his brain that measured those eighty minutes was more precise than any clock. If an hour and eighteen minutes had passed from the time I walked out the door to the time I got back, I would receive a friendly welcome; but after an hour and twenty-two minutes, we were back to "What's your shoe size?"

I was always afraid of making some careless remark that might upset him. I nearly bit my tongue once when I started to mention something the newspaper had said about Prime Minister Miyazawa. (For the Professor, the prime minister was still Takeo Miki.) And I felt awful about suggesting that we get a television to watch the summer Olympics in Barcelona. (His last Olympics were in Munich.) Still, the Professor gave no sign that this bothered him. When the conversation veered off in a direction he couldn't follow, he simply waited patiently until it returned to a topic he could handle. But, for his part, he never asked me anything about myself, how long I'd been working as a housekeeper, where I came from, or whether I had a family. Perhaps he was afraid of bothering me by repeating the same question again and again.

The one topic we could discuss without any worry was mathematics. Not that I was enthusiastic about it at first. In school, I had hated math so much that the mere sight of the textbook made me feel ill. But the things the Professor taught me seemed to find their way effortlessly into my brain-not because I was an employee anxious to please her employer but because he was a such a gifted teacher. There was something profound in his love for math. And it helped that he forgot what he'd taught me before, so I was free to repeat the same question until I understood. Things that most people would get the first time around might take me five, or even ten times, but I could go on asking the Professor to explain until I finally got it.

"The person who discovered amicable numbers must have been a genius."

"You might say that: it was Pythagoras, in the sixth century B.C."

"Did they have numbers that long ago?"

"Of course! Did you think they were invented in the nineteenth century? There were numbers before human beings- before the world itself was formed."

We talked about numbers while I worked in the kitchen. The Professor would sit at the kitchen table or relax in the easy chair by the window, while I stirred something on the stove or washed the dishes at the sink.

"Is that so? I'd always thought that human beings invented numbers."

"No, not at all. If that were the case, they wouldn't be so difficult to understand and there'd be no need for mathematicians. No one actually witnessed the first numbers come into being-when we first became aware of them, they'd already been around for a long time."

"And that's why so many smart people try so hard to figure out how they work?"

"Yes, and why human beings seem so foolish and frail compared to whoever or whatever created these numbers." The Professor sat back in his chair and opened one of his journals.

"Well, hunger makes you even more foolish and frail, so we need to feed that brain of yours. Dinner will be ready in a minute." Having finished grating some carrots to mix into his hamburger, I carefully slipped the peelings into the garbage pail. "By the way," I added, "I've been trying to find another pair of amicable numbers besides 220 and 284, but I haven't had any luck."

"The next smallest pair is 1,184 and 1,210."

"Four digits? No wonder I didn't find them. I even had my son help me. I found the factors, and then he added them up."

"You have a son?" The Professor sat up in his chair; his magazine slipped to the floor.

"Yes."

"How old is he?"

"Ten."

"Ten? He's just a little boy!" The Professor's expression had quickly darkened, he was becoming agitated. I stopped mixing the hamburger and waited for what I was sure was coming: a lesson on the significance of the number 10.

"And where is your son now?" he said.

"Well, let's see. He's home from school by now, but he's probably given up on his homework and gone to the park to play baseball with his friends."

"'Well, let's see'! How can you be so nonchalant? It'll be dark soon!"

I was wrong, there would be no revelations about the number 10, it seemed. In this case, 10 was the age of a small boy, and nothing more.

"It's all right," I said. "He does this every day."

"Every day! You abandon your son every day so you can come here to make hamburgers?"

"I don't abandon him, and it's my job to come here." I wasn't sure why the Professor was so concerned about my son, but I went back to my recipe, adding some pepper and nutmeg.

"Who takes care of him when you're not home? Does your husband come home early from work? Does his grandmother watch him?"

"No, unfortunately there's no husband or grandmother. It's just the two of us."

"So he's at home all alone? He sits and waits for his mother in a dark house while you're here making dinner for a stranger? Making my dinner!"

No longer able to control himself, the Professor jumped up from his chair and began circling the table. The notes on his body trembled as he ran his hand nervously through his hair. Dandruff sprinkled on his shoulder. I turned off the soup just as it began to boil.

"You really don't need to worry," I said, trying to sound calm. "We've been doing this since he was much younger. Now that he's ten, he can manage for himself. He has the phone number here, and if he needs help, he knows to ask the landlord downstairs-"

"No, no, no!" The Professor cut me off as he paced around the table. "You should never leave a child alone. What if the heater fell over and started a fire? What if he choked on a candy? Who'd be there to help? Oh! I don't want to think about it. Go home right now! You should make dinner for your child. Go home!" He grabbed my arm and tried to pull me toward the door.

"I'll go," I said, "but I just have to make these hamburgers for you."

"Are you going to stand there frying hamburgers while your child could be dying in a fire? Now listen to me: beginning tomorrow you'll bring your son along with you. He can come straight here from school. He can do his homework, and be near his mother. And don't think you can fool me just because I'll forget by tomorrow."

He pulled off the tag that read "the new housekeeper" and fished a pencil from his pocket. Under the portrait, he added the words "and her son, ten years old."

I left that evening-or rather, I was chased out-without having time to wash my hands, let alone clean the kitchen properly. The Professor appeared even angrier than when I had interrupted his thinking. But his anger seemed to hide a deep fear, and I hurried home wondering what I would do if I found the apartment in flames.

Any reticence or wariness I felt for the Professor vanished the moment I saw him with my son, and from that point on I trusted him completely. As I'd promised the evening before, I gave my son a map to the house and told him to come directly from school. It was against agency rules to bring children to the workplace, but there was no denying the Professor.

When my son appeared at the door the next day with his schoolbag on his back, the Professor broke into a wide grin and opened his arms to embrace him. I didn't even have time to point at the line he'd added to his note-"and her son, ten years old." As a mother, it was a joy to see someone so completely embrace my child, and I felt a slight twinge of jealousy that my welcome from the Professor was always much more reserved.

"I'm so glad you've come!" he said, without any of the questions he asked me every morning. Bewildered by the unexpected greeting, my son stiffened, but managed a polite answer. The Professor took off my son's Hanshin Tigers baseball cap and rubbed his head. Then he gave him the nickname before he'd even learned his real one.

"I'm going to call you Root," he said. "The square root sign is a generous symbol, it gives shelter to all the numbers." And he quickly took off the note on his sleeve and made the addition: "The new housekeeper… and her son, ten years old,."

At first I made us name tags, thinking that if the Professor weren't the only one with notes clipped to him he might feel less anxious. I told my son to change his school name tag for one I made that read "." The experiment proved less successful than I'd hoped. No matter how much time passed, I was always the young woman who made painfully slow progress with numbers, and my son would be the boy who simply appeared, and was embraced.

My son soon grew accustomed to the Professor's enthusiastic greeting and even came to enjoy it. He would take off his cap at the door and present the flat top of his head, as if to show how proud he was to be worthy of the name Root. And for his part, the Professor never missed his cue, he mentioned the square root whenever he met my son.

My contract stipulated that I would make dinner for him at six o'clock and leave at seven after finishing the dishes; but the Professor began objecting to this schedule as soon as my son arrived on the scene.

"I won't stand for it! If you have to finish here and then make another meal once you get home, Root won't get his dinner until eight o'clock. That just won't do. It's inefficient; it's illogical. Children should be in bed by eight o'clock. You can't deprive a child of his sleep-that's when he does his growing."

For a mathematician, his argument wasn't very scientific, but I decided to ask the director of the agency if it would be possible to deduct the cost of our dinner from my salary.

The Professor had never before thanked me for my efforts in the kitchen, but his attitude changed when the three of us sat down to dinner together for the first time. His manners were exemplary. He sat up very straight and ate quietly, without spilling so much as a drop of his soup on the table or his napkin-all of which seemed odd, given how terrible his manners had been when it was just the two of us.

"What's the name of your school?" he asked.

"Is your teacher nice?

"How was lunch today?

"What do you want to be when you grow up?"

As he squeezed lemon on his chicken or picked out the carrots from his soup, the Professor would ask Root one question after another, without hesitating, even when the question concerned the past or the future. He was determined to make our dinner hour as peaceful and pleasant as possible. Though Root's answers to his questions were mostly perfunctory, the Professor listened attentively, and it was thanks to his efforts that we ate together without drifting into any awkward silences.

He was not simply humoring a child. Whenever Root would put his elbows on the table or clatter his dishes or commit any other breach of etiquette (all things the Professor had done himself at his earlier solitary meals), the Professor would gently correct him.

"You have to eat more," he said one evening. "A child's job is to grow."

"I'm the shortest one in my class," said Root.

"Don't let that bother you. You're storing up energy, pretty soon you'll have a growth spurt. One of these days, you're going to feel your bones begin to stretch out and grow."

"Did that happen to you?" Root wanted to know.

"No, unfortunately, in my case, all that energy was wasted on other things."

"What other things?"

"On my friends. I had some very close friends, but as it turned out they weren't the sort you could play baseball or kick-the-can with. In fact, playing with them didn't involve moving at all."

"Were your friends sick?"

"Just the opposite. They were big and strong as a rock. But since they lived in my head, I could only play with them there. So I ended up growing a strong brain instead of a strong body."

"I see," said Root. "Your friends were numbers. My mom says you're a great math teacher."

"You're a bright boy. Very bright. That's correct, numbers were my only friends… But that's why you need to get lots of exercise while you're young. Do you understand? And you have to eat everything on your plate, even the things you don't like. And if you're still hungry, you can have anything on my plate, too."

"Thanks!"

Root had never enjoyed dinner as much as he did when we ate with the Professor. He answered the Professor's questions and let him fill his plate to overflowing, and whenever he could, he looked curiously around the room or stole a glance at the notes on the Professor's suit.

Root was a child who had rarely been embraced. When I first saw him in the hospital nursery, I felt something closer to fear than to joy. His eyelids and earlobes and even his feet were still swollen and damp from the amniotic fluid. His eyes were half-closed, but he didn't seem to be asleep. His tiny arms and legs, protruding awkwardly from the oversized gown, flailed from time to time as if in protest at having been left here by mistake.

I was eighteen, ignorant, and alone. My cheeks were sunken from morning sickness that had continued right up to the moment I lay down on the delivery table. My hair stank with sweat, and my pajamas were still stained where my water had broken.

There were fifteen babies in the nursery and he was the only one awake. It was before dawn and the halls were empty except for the women at the nurse's station. His fists had been clenched tight, but at that moment he opened them, and then awkwardly bent them closed again. The small fingernails were dark and discolored with traces of what I assumed was my blood.

"Excuse me," I called, staggering down to the nurse's station. "I'd like to cut my baby's fingernails. He seems to be moving his hands a lot and I'm afraid he'll scratch himself…" Perhaps I was trying to convince myself that I was a good mother.

From the time of my earliest memories, I had no father. My mother had fallen in love with a man she could never marry, and she had raised me by herself. She worked at a reception hall that people hired for weddings. She had started out helping wherever she was needed-bookkeeping, dressing the wedding parties, flower arranging, table coordination-and ended up managing the whole place.

She was a strong woman who hated nothing more than having people think of her daughter as poor and fatherless. Though we were, in fact, poor, she did her best to make us look and feel rich. She asked the women who worked in the dressmaking department to give us scraps of material from which she made all my clothes. She arranged for the organist at the hall to give me piano lessons at a discount. And she brought home the leftover flowers and made pretty arrangements for the apartment.

I suppose I became a housekeeper because I kept house for my mother from the time I was a small child. When I was barely two and not quite potty trained, I would wash out my own panties if I had an accident; and before I was even in elementary school, I was using the knives in the kitchen and cutting up the ingredients to make fried rice. By the time I was ten, I not only took care of the whole apartment, but I was even paying the electric bill and attending meetings of the neighborhood association in my mother's place.

My mother never said a word against my father and always insisted he was a fine man and terribly handsome. He managed a restaurant somewhere, but the specifics were always kept from me. I was given to understand, however, that he was tall, fluent in English, and a connoisseur of opera.

The image I have of my father is that of a statue in a museum. No matter how close I come to him, I can't get his attention, he continues to stare off into the distance without looking down, and he never reaches out his hand to me.

It wasn't until I entered adolescence that it occurred to me how odd it was that the wonderful man my mother described had abandoned us and had never offered even the least bit of economic support. But by that time I had no interest in learning more about him, and I accepted the role of silent accomplice when it came to my mother's illusions.

It was my pregnancy that utterly destroyed those illusions, along with the others she'd carefully stitched together from fabric scraps, piano lessons, and leftover flowers. It happened not long after I'd started my junior year of high school.

The boy was someone I'd met at my after-school job, a college student majoring in electrical engineering. He was a quiet and cultured young man, but he lacked the decency to take responsibility for what had happened. The mysterious knowledge of electricity that had attracted me to him in the first place proved useless, and he became just another careless man who vanished from my world.

Once my pregnancy became obvious, there was nothing I could do to appease my mother's anger, even though we now shared the experience of giving birth to a fatherless child. It was a melodramatic sort of anger. Her feelings seemed to block out my own. I left home in the twenty-second week of my pregnancy and I lost all contact with her.

When I brought my baby home from the hospital, it was to public housing that had been set up for single mothers, and the only person who welcomed us was the woman who served as matron for the institution. I folded up the one picture I had saved of the baby's father and stuffed it into the little wooden box they had given me at the hospital to hold the umbilical cord.

As soon as I'd managed to get the baby into a day care center, I went straight to the Akebono Housekeeping Agency and arranged for an interview. It was the only job I could think of that matched my limited skills.

Shortly before Root entered elementary school, my mother and I reconciled: a fancy backpack arrived in the mail for him. This happened at the same time that I had left the single mothers' home and set up house for ourselves. My mother was still managing the wedding hall. But just as our troubles seemed to be over and I'd begun to see how comforting it could be to have a grandmother for my child, my mother suddenly died of a brain hemorrhage-which may be why I was even happier than Root himself when I saw the Professor hug him.

The three of us soon fell into a pleasant routine. There was no change in my schedule or workload, other than making more food for dinner. Fridays were the busiest, as I had to prepare food for the weekend and store it away in the freezer. I would make meat loaf and mashed potatoes, or poached fish and vegetables, and explain repeatedly what went with what and how to defrost the food, although the Professor never quite figured out how to use the microwave. Nevertheless, when I arrived on Monday morning, all the food I'd prepared was gone. The meat loaf and fish had somehow been thawed and eaten, and the dirty dishes had been washed and put away in the cupboard. I was sure that the old woman took care of the Professor when I wasn't there, but as long as I was around, she never made an appearance. I had no idea why she had placed such a firm restriction on communication between her house and the Professor's, but I decided that my next challenge was to figure out how to get to know her.

The Professor's problems, on the other hand, were all mathematical. He never seemed particularly proud of his accomplishments, even when he had spent a long time solving an equation that had won both the prize money and my praise.

"It was just a little puzzle," he would say, "a game"; and his tone sounded more sad than modest. "The person who made the problem already knew the answer. Solving a problem for which you know there's an answer is like climbing a mountain with a guide, along a trail someone else has laid. In mathematics, the truth is somewhere out there in a place no one knows, beyond all the beaten paths. And it's not always at the top of the mountain. It might be in a crack on the smoothest cliff or somewhere deep in the valley."

In the afternoon, when he heard Root's voice at the door, the Professor came out of his study, no matter how absorbed he was in his work. Though he had always hated to have his "thinking" interrupted, he now seemed more than willing to give it up for my son.

Most days, however, Root simply delivered his backpack and went off to the park to play baseball with his friends, and the Professor would retreat dejectedly to his study.

So the Professor seemed particularly cheerful when the weather turned rainy and he was able to help Root with his math homework.

"I think I'm a little smarter when I'm in the Professor's office," Root told me. There were no bookshelves in the little apartment where we lived, so the Professor's study, with its stacks of volumes lining the walls, seemed magical to him. The Professor would sweep aside the notebooks and clips and eraser shavings on his desk to make space for Root, and then he would open the textbook.

How is it possible for a professor of advanced mathematics to teach a child in elementary school? The Professor was especially gifted, he had the most wonderful way of teaching fractions and ratios and volume, and it occurred to me that all parents should be giving this kind of help to their children.

Whether it was a word problem or just a simple calculation, the Professor made Root read it aloud first.

"353 × 840 =…

"6239 ÷ 23 =…

"4.62 + 2.74 =…

"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.