The Gift of Numbers aka The Housekeeper and the Professor

1

We called him the Professor. And he called my son Root, because, he said, the flat top of his head reminded him of the square root sign.

"There's a fine brain in there," the Professor said, mussing my son's hair. Root, who wore a cap to avoid being teased by his friends, gave a wary shrug. "With this one little sign we can come to know an infinite range of numbers, even those we can't see." He traced the symbol in the thick layer of dust on his desk.

Of all the countless things my son and I learned from the Professor, the meaning of the square root was among the most important. No doubt he would have been bothered by my use of the word countless -too sloppy, for he believed that the very origins of the universe could be explained in the exact language of numbers-but I don't know how else to put it. He taught us about enormous prime numbers with more than a hundred thousand places, and the largest number of all, which was used in mathematical proofs and was in the Guinness Book of Records , and about the idea of something beyond infinity. As interesting as all this was, it could never match the experience of simply spending time with the Professor. I remember when he taught us about the spell cast by placing numbers under this square root sign. It was a rainy evening in early April. My son's schoolbag lay abandoned on the rug. The light in the Professor's study was dim. Outside the window, the blossoms on the apricot tree were heavy with rain.

The Professor never really seemed to care whether we figured out the right answer to a problem. He preferred our wild, desperate guesses to silence, and he was even more delighted when those guesses led to new problems that took us beyond the original one. He had a special feeling for what he called the "correct miscalculation," for he believed that mistakes were often as revealing as the right answers. This gave us confidence even when our best efforts came to nothing.

"Then what happens if you take the square root of negative one?" he asked.

"So you'd need to get -1 by multiplying a number by itself?" Root asked. He had just learned fractions at school, and it had taken a half-hour lecture from the Professor to convince him that numbers less than zero even existed, so this was quite a leap. We tried picturing the square root of negative one in our heads: The square root of 100 is 10; the square root of 16 is 4; the square root of 1 is 1. So the square root of -1 is…

He didn't press us. On the contrary, he fondly studied our expressions as we mulled over the problem.

"There is no such number," I said at last, sounding rather tentative.

"Yes, there is," he said, pointing at his chest. "It's in here. It's the most discreet sort of number, so it never comes out where it can be seen. But it's here." We fell silent for a moment, trying to picture the square root of minus one in some distant, unknown place. The only sound was the rain falling outside the window. My son ran his hand over his head, as if to confirm the shape of the square root symbol.

But the Professor didn't always insist on being the teacher. He had enormous respect for matters about which he had no knowledge, and he was as humble in such cases as the square root of negative one itself. Whenever he needed my help, he would interrupt me in the most polite way. Even the simplest request-that I help him set the timer on the toaster, for example-always began with "I'm terribly sorry to bother you, but…" Once I'd set the dial, he would sit peering in as the toast browned. He was as fascinated by the toast as he was by the mathematical proofs we did together, as if the truth of the toaster were no different from that of the Pythagorean theorem.

It was March of 1992 when the Akebono Housekeeping Agency first sent me to work for the Professor. At the time, I was the youngest woman registered with the agency, which served a small city on the Inland Sea, although I already had more than ten years of experience. I managed to get along with all sorts of employers, and even when I cleaned for the most difficult clients, the ones no other housekeeper would touch, I never complained. I prided myself on being a true professional.

In the Professor's case, it only took a glance at his client card to know that he might be trouble. A blue star was stamped on the back of the card each time a housekeeper had to be replaced, and there were already nine stars on the Professor's card, a record during my years with the agency.

When I went for my interview, I was greeted by a slender, elegant old woman with dyed brown hair swept up in a bun. She wore a knit dress and walked with a cane.

"You will be taking care of my brother-in-law," she said. I tried to imagine why she would be responsible for her husband's brother. "None of the others have lasted long," she continued. "Which has been a terrible inconvenience for me and for my brother-in-law. We have to start again every time a new housekeeper comes… The job isn't complicated. You would come Monday through Friday at 11:00 A.M., fix him lunch, clean the house, do the shopping, make dinner, and leave at 7:00 P.M. That's the extent of it."

There was something hesitant about the way she said the words brother-in-law. Her tone was polite enough, but her left hand nervously fingered her cane. Her eyes avoided mine, but occasionally I caught her casting a wary glance in my direction.

"The details are in the contract I signed with the agency. I'm simply looking for someone who can help him live a normal life, like anyone else."

"Is your brother-in-law here?" I asked. She pointed with the cane to a cottage at the back of the garden behind the house. A red slate roof rose above a neatly pruned hedge of scarlet hawthorn.

"I must ask you not to come and go between the main house and the cottage. Your job is to care for my brother-in-law, and the cottage has a separate entrance on the north side of the property. I would prefer that you resolve any difficulties without consulting me. That's the one rule I ask you to respect." She gave a little tap with her cane.

I was used to absurd demands from my employers-that I wear a different color ribbon in my hair every day; that the water for tea be precisely 165 degrees; that I recite a little prayer every evening when Venus rose in the night sky-so the old woman's request struck me as relatively straightforward.

"Could I meet your brother-in-law now?" I asked.

"That won't be necessary." She refused so flatly that I thought I had offended her. "If you met him today, he wouldn't remember you tomorrow."

"I'm sorry, I don't understand."

"He has difficulties with his memory," she said. "He's not senile; his brain works well, but about seventeen years ago he hit his head in an automobile accident. Since then, he has been unable to remember anything new. His memory stops in 1975. He can remember a theorem he developed thirty years ago, but he has no idea what he ate for dinner last night. In the simplest terms, it's as if he has a single, eighty-minute videotape inside his head, and when he records anything new, he has to record over the existing memories. His memory lasts precisely eighty minutes-no more and no less." Perhaps because she had repeated this explanation so many times in the past, the old woman ran through it without pause, and with almost no sign of emotion.

How exactly does a man live with only eighty minutes of memory? I had cared for ailing clients on more than one occasion in the past, but none of that experience would be useful here. I could just picture a tenth blue star on the Professor's card.

From the main house, the cottage appeared deserted. An old-fashioned garden door was set into the hawthorn hedge, but it was secured by a rusty lock that was covered in bird droppings.

"Well then, I'll expect you to start on Monday," the old woman said, putting an end to the conversation. And that's how I came to work for the Professor.

Compared to the impressive main house, the cottage was modest to the point of being shabby: a small bungalow that seemed to have been built hastily. Trees and shrubs had grown wild around it, and the doorway was deep in shadows. When I tried the doorbell on Monday, it seemed to be broken.

"What's your shoe size?"

This was the Professor's first question, once I had announced myself as the new housekeeper. No bow, no greeting. If there is one ironclad rule in my profession, it's that you always give the employer what he wants; and so I told him.

"Twenty-four centimeters."

"There's a sturdy number," he said. "It's the factorial of four." He folded his arms, closed his eyes, and was silent for a moment.

"What's a 'factorial'?" I asked at last. I felt I should try to find out a bit more, since it seemed to be connected to his interest in my shoe size.

"The product of all the natural numbers from one to four is twenty-four," he said, without opening his eyes. "What's your telephone number?"

He nodded, as if deeply impressed. "That's the total number of primes between one and one hundred million."

It wasn't immediately clear to me why my phone number was so interesting, but his enthusiasm seemed genuine. And he wasn't showing off; he struck me as straightforward and modest. It nearly convinced me that there was something special about my phone number, and that I was somehow special for having it.

Soon after I began working for the Professor, I realized that he talked about numbers whenever he was unsure of what to say or do. Numbers were also his way of reaching out to the world. They were safe, a source of comfort.

Every morning, during the entire time I worked for the Professor, we repeated this numerical q and a at the front door. To the Professor, whose memory lasted only eighty minutes, I was always a new housekeeper he was meeting for the first time, and so every morning he was appropriately shy and reserved. He would ask my shoe size or telephone number, or perhaps my zip code, the registration number on my bicycle, or the number of brushstrokes in the characters of my name; and whatever the number, he invariably found some significance in it. Talk of factorials and primes flowed effortlessly, seeming completely natural, never forced.

Later, even after I had learned the meanings of some of these terms, there was still something pleasant about our daily introductions at the door. I found it reassuring to be reminded that my telephone number had some significance (beyond its usual purpose), and the simple sound of the numbers helped me to start the day's work with a positive attitude.

He had once been an expert in number theory at a university. He was sixty-four, but he looked older and somewhat haggard, as though he did not eat properly. He was barely more than five feet tall, and his back was so badly hunched that he seemed even shorter. The wrinkles on his bony neck looked a little grimy, and his wispy, snow-white hair fell in all directions, half-concealing his plump, Buddhalike ears. His voice was feeble and his movements were slow. If you looked closely, though, you could see traces of a face that had once been handsome. There was something in the sharp line of his jaw and his deeply carved features that was still attractive.

Whether he was at home or going out-which he did very rarely-the Professor always wore a suit and tie. His closet held three suits, one for winter, one for summer, and one that could be worn in spring or fall, three neckties, six shirts, and an overcoat. He did not own a sweater or a pair of casual pants. From a housekeeper's point of view, it was the ideal closet.

I suspect that the Professor had no idea there were clothes other than suits. He had no interest in what people wore, and even less in his own appearance. For him it was enough to get up in the morning, open the closet, and put on whichever suit wasn't wrapped in plastic from the cleaners. All three suits were dark and well-worn, much like the Professor himself, and clung to him like a second skin.

But by far the most curious thing about the Professor's appearance was the fact that his suit was covered with innumerable scraps of notepaper, each one attached to him by a tiny binder clip. Every conceivable surface-the collar, cuffs, pockets, hems, belt loops, and buttonholes-was covered with notes, and the binder clips gathered the fabric of his clothing in awkward bunches. The notes were simply scraps of torn paper, some yellowing or crumbling. In order to read them, you had to get close and squint, but it soon became clear that he was compensating for his lack of memory by writing down the things he absolutely had to remember and pinning them where he couldn't lose them-on his body. His odd appearance was as distracting as his questions about my shoe size.

"Come in then," he said. "I have to work, but you just do whatever it is you have to do." And with that he disappeared into his study. As he turned and walked away, the notes made a dry, rustling sound.

From the bits and pieces of information I gleaned from the nine housekeepers who had come before me, it seemed that the old woman in the main house was a widow, and that her husband had been the Professor's older brother. When their parents had died, his brother had taken over the family textile business, had enlarged it considerably, and willingly assumed the cost of educating a brother who was a dozen years younger. In this way, the Professor had been able to pursue his study of mathematics at Cambridge University. But soon after he had received his doctorate and had found a position at a research institute, his brother had died suddenly of acute hepatitis. The widow, who had no children, decided to close down the factory, put up an apartment building on the land, and live off the rents she collected.

In the years that followed, the Professor and his sister-in-law had settled peacefully into their respective lives-until the accident. A truck driver had dozed off and struck the Professor's car head-on. He had suffered irreversible brain damage and had eventually lost his position at the university. He was forty-seven at the time, and since then he'd had no income except the prize money he earned from solving contest problems in the mathematics journals. For seventeen years he had been completely dependent on the widow's charity.

"You have to feel sorry for the old woman," one of the former housekeepers had said. "Having that strange brother-in-law eat through what her husband left her like some parasite." She'd been sent packing after she complained about the Professor's incessant jabbering about numbers.

The inside of the cottage was as cold and uninviting as the outside. There were just two rooms, an eat-in kitchen and a study that doubled as the Professor's bedroom. It was small, and the wretched condition of the place was striking. The furniture was cheap, the wallpaper was discolored, and the floor in the hall creaked miserably. The doorbell wasn't the only thing that didn't work: just about everything in the house was either broken or on its last legs. The little window in the bathroom was cracked, the knob on the kitchen door was falling off, and the radio that sat on top of the dish cupboard made no sound when I tried to turn it on.

The first two weeks were exhausting, since I had no idea what I was supposed to be doing. The work wasn't physically demanding, and yet at the end of each day my muscles were stiff and my whole body felt heavy. It was always a struggle at each new assignment until I adapted to the rhythm of the work, but the adjustment was especially difficult with the Professor. In most cases, I figured out what sort of person I was dealing with from the things they told me to do, or not to do. I determined where to focus my efforts, how to avoid getting into trouble-how to read the demands of the job. But the Professor never gave me instruction of any kind, as though he did not mind what I did.

On that first day, it occurred to me that I should simply follow what the old woman had said, and start by fixing the Professor's lunch. I checked the refrigerator and the kitchen cupboards, but I found nothing edible except for a box of damp oatmeal and some macaroni and cheese that was four years past its expiration date.

I knocked at the study door. There was no answer, so I knocked again. Still no answer. I knew I shouldn't, but I opened the door and spoke to the Professor's back as he sat at his desk.

"I'm sorry to disturb you," I said.

He gave no sign of having heard me. Perhaps he's hard of hearing, or wearing earplugs, I thought. "What would you like for lunch?" I continued. "Are there… things you like or dislike? Do you have any food allergies?"

The study smelled of books. Half the windows were covered by bookshelves, and piles of books drifted up the walls. A bed with a worn-out mattress was pressed against one wall. There was a single notebook lying open on the desk, but no computer, and the Professor wasn't holding a pen or pencil. He simply stared at a fixed point off in space.

"If there's nothing particular you want, I'll just make something. But please don't hesitate if there's anything I can get for you."

I happened to glance at some of the notes pinned to his suit: "… the failure of the analytic method…," "… Hilbert's thirteenth…," "… the function of the elliptical curve…" Shuffled in among the fragments of obscure numbers and symbols and words was one scrap that even I could understand. From the stains and bent corners of the paper and the rusted edges of the binder clip, I could tell that this one had been attached to the Professor for a long time: "My memory lasts only eighty minutes," it read.

"I have nothing to say," he said, turning suddenly and speaking in a loud voice. "I'm thinking at the moment. Thinking. And to have my thoughts interrupted is like being strangled. Don't you know that barging in here when I'm with my numbers is as rude as interrupting someone in the bathroom?"

I bowed and apologized repeatedly, but I doubt he heard a word of what I said. He had already returned to his fixed point somewhere off in space.

To be shouted at like that on the first day could be a serious problem, and I worried that I might become the tenth star on his file card before I'd even started. I promised myself that I would never disturb him again while he was "thinking."

But the Professor was always thinking. When he came out of the study and sat at the table, when he was gargling in the bathroom, or even when he did his strange stretching exercises, he continued thinking. He ate whatever was set in front of him, mechanically shoveling the food in his mouth and swallowing almost without chewing. He had a distracted, unsteady way of walking. I managed to find the right moment to ask him about things I needed to know-where he kept the wash bucket or how to use the water heater. And I avoided making any unnecessary noise, even breathing too loudly, as I moved about that unfamiliar house, waiting for him to take even a short break from his thinking.

I made a cream stew for dinner, something with vegetables and protein that he could eat with just a spoon-and that he could eat without removing bones or shells. Perhaps it was because he'd lost his parents at such a young age, but he had less than perfect table manners. He never said a word of thanks before he started eating, and he spilled food with almost every bite. I even caught him cleaning his ears with his dirty napkin at the table. He did not complain about my cooking, and he remained silent as he ate. Each time he plunged the spoon into the stew, he looked as if he might lose it in the bowl.

"Would you like some more? I've made plenty." It was careless of me to speak up suddenly like that, to take such a familiar tone, and all I got by way of an answer was a burp. Without so much as a glance in my direction, he got up and disappeared into his study. There was a small pile of carrots at the bottom of his bowl.

At the end of my first day, I noticed a new note on the cuff of his jacket. "The new housekeeper," it said. The words were written in tiny, delicate characters, and above them was a sketch of a woman's face. It looked like the work of a small child-short hair, round cheeks, and a mole next to the mouth-but I knew instantly that it was a portrait of me. I imagined the Professor hurrying to draw this likeness before the memory had vanished. The note was proof of something, that he had interrupted his thinking for my sake.

Over the next few days, I introduced myself by pointing to the note on his cuff. The Professor would be silent for a moment, comparing my face with the picture on his sleeve, trying to recall what the note had meant. At last he would make a little huffing sound and ask me my shoe size and telephone number. But I realized that something dramatic had changed when, at the end of my first week, he came to me with a bundle of papers covered with formulas and numbers, and asked me to send it off to the Journal of Mathematics.

"I'm terribly sorry to bother you, but…"

His tone was polite, and completely unexpected after the way he had scolded me in his study on my first day. It was the first request he had made of me, and he was no longer "thinking," for the moment.

"It's no trouble at all," I told him. I carefully copied the mysterious foreign address onto the envelope and ran off happily to the post office.

When I returned, the Professor wasn't thinking anymore. He was stretched out in the easy chair by the kitchen window, and as he rested I was finally able to clean the study. I opened the windows and took his quilt and pillow out into the garden to air. And then I ran the vacuum cleaner at full throttle. The room was cluttered and chaotic, but comfortable.

I was not surprised to find balls of hair and moldy Popsicle sticks behind the desk, or a chicken bone resting on top of one of his bookshelves. And yet, the room was filled by a kind of stillness. Not simply an absence of noise, but an accumulation of layers of silence, untouched by fallen hair or mold, silence that the Professor left behind as he wandered through the numbers, silence like a clear lake hidden in the depths of the forest.

But despite its relative comfort, if you had asked me whether it was an interesting room, I would have had to say no. There was not a single object to spark the imagination, no trinkets from the Professor's past, no mysterious photographs or decorations that might have amused a housekeeper.

I attacked the bookshelves with the duster. Group Theory. Algebraic Number Theory. Studies in Number Theory… Chevalley, Hamilton, Turing, Hardy, Baker… So many books and not one I wanted to read. Half of them were in foreign languages, and I couldn't even make out the titles on the spines. A few notebooks were stacked on the desk, along with a scattering of pencil stubs and binder clips. How could he think at such a characterless desk? The residue from an eraser was the only evidence of the work that he had done here.

As I wiped away the dust, arranged the notebooks, and gathered up the clips, it occurred to me that a mathematician ought to have some sort of expensive compass you couldn't find in an ordinary stationery shop, or an elaborate slide rule. The seat of the chair was worn down where the Professor sat.

"When is your birthday?"

That evening after dinner, he did not disappear immediately into his study. Though I was busy cleaning up, he seemed to be looking for a topic of conversation.

"February twentieth."

"Is that so?"

The Professor had picked the carrots out of his potato salad and had left them on the plate. I cleared and wiped the table, noticing that he still seemed to spill a great deal, even when he wasn't thinking. It was spring, but still chilly once the sun set, so the oil heater was burning in the corner.

"Do you send a lot of articles to magazines?" I asked.

"I wouldn't call them 'articles.' They're just puzzles for amateur mathematicians. Sometimes there's even a prize. Wealthy men who love mathematics put up the money." He looked down, checking his suit in various places, and his gaze fell on a note clipped to his left pocket. "Oh, I see. I sent a proof to the Journal of Mathematics today."

It had been much more than eighty minutes since I'd made my trip to the post office.

"Oh, dear!" I said. "If it's a contest, I should have sent it express mail. If it doesn't get there first, I suppose you don't get the prize."

"No, there was no need to send it express. Of course, it's important to arrive at the correct answer before anyone else, but it's just as important that the proof is elegant."

"I had no idea a proof could be beautiful… or ugly."

"Of course it can," he said. Getting up from the table, he came over to the sink where I was washing the dishes and peered at me as he continued. "The truly correct proof is one that strikes a harmonious balance between strength and flexibility. There are plenty of proofs that are technically correct but are messy and inelegant or counterintuitive. But it's not something you can put into words-explaining why a formula is beautiful is like trying to explain why the stars are beautiful."

I stopped washing and nodded, not wanting to interrupt the Professor's first real attempt at conversation.

"Your birthday is February twentieth. Two twenty. Can I show you something? This was a prize I won for my thesis on transcendent number theory when I was at college." He took off his wristwatch and held it up for me to see. It was a stylish foreign brand, quite out of keeping with the Professor's rumpled appearance.

"It's a wonderful prize," I said.

"But can you see the number engraved here?" The inscription on the back of the case read President's Prize No. 284.

"Does that mean that it was the two hundred and eighty-fourth prize awarded?"

"I suppose so, but the interesting part is the number 284 itself. Take a break from the dishes for a moment and think about these two numbers: 220 and 284. Do they mean anything to you?"

Pulling me by my apron strings, he sat me down at the table and produced a pencil stub from his pocket. On the back of an advertising insert, he wrote the two numbers, separated strangely on the card.

220

284

"Well, what do you make of them?"

I wiped my hands on my apron, feeling awkward, as the Professor looked at me expectantly. I wanted to respond, but had no idea what sort of answer would please a mathematician. To me, they were just numbers.

"Well…," I stammered. "I suppose you could say they're both three-digit numbers. And that they're fairly similar in size-for example, if I were in the meat section at the supermarket, there'd be very little difference between a package of sausage that weighed 220 grams and one that weighed 284 grams. They're so close that I would just buy the one that was fresher. They seem pretty much the same-they're both in the two hundreds, and they're both even-"

"Good!" he almost shouted, shaking the leather strap of his watch. I didn't know what to say. "It's important to use your intuition. You swoop down on the numbers, like a kingfisher catching the glint of sunlight on the fish's fin." He pulled up a chair, as if wanting to be closer to the numbers. The musty paper smell from the study clung to the Professor.

"You know what a factor is, don't you?"

"I think so. I'm sure I learned about them at some point…"

"For 220 is divisible by 1 and by 220 itself, with nothing leftover. So 1 and 220 are factors of 220. Natural numbers always have 1 and the number itself as factors. But what else can you divide it by?"

"By 2, and 10…"

"Exactly! So let's try writing out the factors of 220 and 284, excluding the numbers themselves. Like this."

220: 1 2 4 5 10 11 20 22 44 55 110 142 71 4 2 1: 284

The Professor's figures, rounded and slanting slightly to one side, were surrounded by black smears where the pencil had smudged.

"Did you figure out all the factors in your head?" I asked.

"I don't have to calculate them-they just come to me from the same kind of intuition you used. So then, let's move on to the next step," he said, adding symbols to the lists of factors.

220: 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = =142 + 71 + 4 + 2 + 1: 284

"Add them up," he said. "Take your time. There's no hurry."

He handed me the pencil, and I did the calculation in the space that was left on the advertisement. His tone was kind and full of expectation, and it didn't seem as though he were testing me. On the contrary, he made me feel as though I were on an important mission, that I was the only one who could lead us out of this puzzle and find the correct answer.

I checked my calculations three times to be sure I hadn't made a mistake. At some point, while we'd been talking, the sun had set and night was falling. From time to time I heard water dripping from the dishes I had left in the sink. The Professor stood close by, watching me.

"There," I said. "I'm done."

220: 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284

220 = 142 + 71 + 4 + 2 + 1: 284

"That's right! The sum of the factors of 220 is 284, and the sum of the factors of 284 is 220. They're called 'amicable numbers,' and they're extremely rare. Fermat and Descartes were only able to find one pair each. They're linked to each other by some divine scheme, and how incredible that your birthday and this number on my watch should be just such a pair."

We sat staring at the advertisement for a long time. With my finger I traced the trail of numbers from the ones the Professor had written to the ones I'd added, and they all seemed to flow together, as if we'd been connecting up the constellations in the night sky.