The Gift of Numbers aka The Housekeeper and the Professor

7

I thought of the Professor whenever I saw a prime number-which, as it turned out, was almost everywhere I looked: price tags at the supermarket, house numbers above doors, on bus schedules or the expiration date on a package of ham, Root's score on a test. On the face of it, these numbers faithfully played their official roles, but in secret they were primes and I knew that was what gave them their true meaning.

Of course, I couldn't always tell immediately whether a number was prime. Thanks to the Professor, I knew the prime numbers up to 100 just by their feel; but when I encountered a larger number that I suspected might be prime, I had to divide it to be sure. There were plenty of cases where a number that looked to be composite turned out to be prime, and just as many others where I discovered divisors for a number that I was certain was prime.

Taking my cue from the Professor, I started carrying a pencil and a notepad around in the pocket of my apron. That way, I could do my calculations whenever the mood struck. One day while I was cleaning in the kitchen in the tax consultants' house, I found a serial number engraved on the back of the refrigerator door: 2311. It looked intriguing, so I took out my notepad, moved aside the detergent and the rags, and set to work. I tried 3, then 7, and then 11. All to no avail. They all left a remainder of 1. Next I tried 13, and 17, and 19, but none of them was a divisor. There was no way to break up 2,311; but, more than that, its indivisibility was positively devious. Every time I thought I had spotted a divisor, the number seemed to slip away, leaving me oddly exhausted yet all the more eager for the hunt-which was always the way with primes.

Once I'd proved that 2,311 was prime, I put the notepad back in my pocket and went back to my cleaning, though now with a new affection for this refrigerator, which had a prime serial number. It suddenly seemed so noble, divisible by only one and itself.

I encountered 341 while I was scrubbing the floor in their office. A blue tax document, Form 341, had fallen under the desk.

My mop stopped in midstroke. It had to be prime. The form was covered with dust from sitting under the desk for so long, but 341 called out to me; it had all the qualities that would have made it a favorite of the Professor.

My employers had gone home and so I set about checking the number in the darkened office. I hadn't really developed a system for finding divisors, and I ended up relying mostly on intuition. The Professor had shown me a method invented by someone named Eratosthenes, who had been the librarian at Alexandria in ancient Egypt, but it was complicated and I'd forgotten how to do it. Since the Professor had such great respect for intuition when it came to numbers, I suspected he would have been tolerant of my method.

In the end, 341 was not a prime: 341 ÷ 11 = 31. A wonderful equation, nonetheless.

Of course, it felt good when a number turned out to be prime. But I wasn't disappointed if it did not. Even when my suspicions proved unfounded, there were still things to be learned. The fact that multiplying two primes such as 11 and 31 yielded a pseudo-prime such as 341, led me in an unexpected direction: I now found myself wondering whether there might be a systematic way to find these pseudo-primes, which so closely resembled true prime numbers.

But despite my curiosity, I set the form on the desk and rinsed my mop in the murky bucket. Nothing would have changed if I'd found a prime number, nor if I'd proven that one wasn't prime. I was still facing a mountain of work. The refrigerator went on keeping things cold, regardless of its serial number, and the person who had filled out Form 341 was still struggling with his tax problems. The numbers didn't make things better; perhaps they even made them worse. Perhaps the ice cream was melting in that refrigerator, I certainly wasn't making any progress mopping the floor, and I suspected my employers would be unhappy with my work. But for all that, there was no denying that 2,311 was prime, and 341 was not.

I remembered something the Professor had said: "The mathematical order is beautiful precisely because it has no effect on the real world. Life isn't going to be easier, nor is anyone going to make a fortune, just because they know something about prime numbers. Of course, lots of mathematical discoveries have practical applications, no matter how esoteric they may seem. Research on ellipses made it possible to determine the orbits of the planets, and Einstein used non-Euclidean geometry to describe the form of the universe. Even prime numbers were used during the war to create codes-to cite a regrettable example. But those things aren't the goal of mathematics. The only goal is to discover the truth." The Professor always said the word truth in the same tone as the word mathematics.

"Try making a straight line right here," he'd said to me one evening at the dinner table. Using a chopstick for a ruler, I traced a line on the back of an advertising leaflet-our usual source of scrap paper. "That's right. You know the definition of a straight line. But think about it for a minute: the line you drew has a beginning and an end. So it's actually a line 'segment'-the shortest distance connecting two points. A true line has no ends; it extends infinitely in either direction. But of course, a sheet of paper has limits, as do your time and energy, so we use this segment provisionally to represent the real thing. Now furthermore, no matter how carefully you sharpen your pencil, the lead will always have a thickness, so the line you draw with it will have a certain width, it will have a surface area, and that means it will have two dimensions. A real line has only one dimension, and that means it is impossible to draw it on a piece of real paper."

I studied the point of the pencil.

"So you might wonder where you would ever find a real line-and the answer would be, only in here." Again, he pointed at his chest, just as he had when he had taught us about imaginary numbers. "Eternal truths are ultimately invisible, and you won't find them in material things or natural phenomena, or even in human emotions. Mathematics, however, can illuminate them, can give them expression-in fact, nothing can prevent it from doing so."

As I mopped the office floor, my mind churning with worries about Root, I realized how much I needed this eternal truth that the Professor had described. I needed the sense that this invisible world was somehow propping up the visible one, that this one, true line extended infinitely, without width or area, confidently piercing through the shadows. Somehow, this line would help me find peace.

I had just got back from shopping and was about to start dinner for the tax consultants when a call came from the secretary at the Akebono Housekeeping Agency.

"Get right over to that mathematician's house. It seems your son has done something to upset them. I don't know what happened, but get over there now. That's an order from the Director."

She hung up before I'd had time to find out more.

I remembered immediately the curse of the foul ball. At first, I'd mistaken it for good luck when the ball missed Root, but it seemed to have come back to haunt us, to fall right on his head. The Professor had been right: "You should never leave a child alone."

Maybe he had choked on the donut I'd given him as a snack. Or he'd gotten a shock trying to plug in the radio. Frightening images ran through my head. I didn't know what to tell my employer as I ran off to the Professor's, her glare following me out the door.

It had been less than a month since we'd left the cottage. The broken doorbell, the dilapidated furniture, and the overgrown garden were the same, but the minute I stepped inside I had a bad feeling.

It was clear that Root had not been hurt, which came as a relief. He hadn't suffocated or been electrocuted but was sitting next to the Professor at the table, his school backpack at his feet.

The bad feeling was coming from the Professor's sister-in-law, who was sitting across from them. Next to her was a middle-aged woman I had never seen before-my replacement, I assumed. There was something indescribably unpleasant about seeing these intruders in a space occupied, in my memory, by just the three of us, the Professor, Root, and me.

As my feeling of relief faded, I began to realize how odd it was for Root to be here. The widow sat at the table, in the same sort of elegant dress she had worn during my interview. She held her cane firmly in her left hand. Root seemed completely cowed and refused to even look up at me. The Professor had assumed his "thinking" pose, staring intently off into the distance, acknowledging no one.

"I'm sorry to call you away from work," said the widow. "Please, have a seat." She pointed to a chair. I was so out of breath after running from the station that I forgot to give a proper answer. "Please, sit down," she said again. "And you, get our guest some tea, please." The other woman-I had no idea whether or not she was an Akebono employee-got up and went to the stove. The widow's tone was polite, but I could see that she was upset by the way her tongue darted over her lips, and the way her fingers drummed on the table. Unable to think of something to say, I did as I was told and sat down. We were silent for a moment.

"You people…," she began at last, tapping a fingernail on the table again. "What is it you want?" I took a breath before answering.

"Has my son done something wrong?"

Root was staring down at his lap, where he held his Tigers cap, nervously crumpling it in his hands.

"I'll ask the questions, if you don't mind. The first thing I'd like to know is why your boy needs to come to my brother-in-law's house." The polish on her perfect nails was flaking off as she tapped on the table.

"I didn't mean to-" Root started, still not looking up.

"The child of a housekeeper who has left our employ," the widow interrupted him. Though she had said "child" more than once, she made no attempt to look at Root-or at the Professor-as though neither of them was in the room.

"I don't think it's a question of 'need,' " I said, still unsure what she was getting at. "I think he just wanted to pay the Professor a visit."

"I borrowed The Lou Gehrig Story from the library, and I wanted to read it with him," Root said, looking up at last.

"Why would a ten-year-old child pay a visit to a sixty-year-old man?" She ignored Root's explanation.

"I'm sorry my son came here without my permission, and I am very sorry if he bothered you. I apologize for failing to supervise him properly."

"That's not the point. I want to know why a housekeeper who has been let go would send her son to see my brother-in-law. What is it you want from him?"

"Want? I'm afraid there's some misunderstanding. He's just a little boy who wanted to visit a friend. He found an interesting book and he wanted to read it with the Professor. Isn't that enough of a reason?"

"I'm sure it is. I'm not implying that the boy had any ulterior motive. I'm asking what you wanted in sending him here."

"I don't want anything, except for my son to be happy."

"Then why do you involve my brother-in-law? You took him out at night, you stayed later than was called for. I don't remember asking you to do any of that."

The housekeeper brought over the tea. She set it in front of us without a word or so much as a clink of the cups and went straight back to the bedroom. It was obvious she would not be taking my side on this.

"I realize that I was out of line, but I can assure you I had no ulterior motive. It was all very innocent."

"Is it about money?"

"Money?!" The word was so unexpected that I nearly shouted it back at her. "How can you say such a thing?"

"I can think of no other reason why you'd indulge my brother-in- law like this."

"Don't be ridiculous!"

"You were fired. You have no business being here!"

"Excuse me," the new housekeeper interrupted, standing in the kitchen doorway. She had removed her apron and was holding her purse. "I'll be going now." She left as quietly as she'd come. We watched as she slipped out the door.

The Professor seemed lost in thought; Root's cap was crushed almost beyond recognition. I took a deep breath.

"It's because we're friends," I said. "Is it a crime to visit a friend?"

"And who is friends with whom exactly?"

"My son and I, with the Professor."

"I'm afraid you've been deceiving yourself," the widow said, shaking her head. "My brother-in-law has no property. He squandered everything on his studies, and he has nothing to show for it."

"And what does that have to do with me?"

"He has no friends, you understand? No one has ever come to visit him."

"Then Root and I are his first friends," I said.

At that moment, the Professor stood up.

"Leave the boy alone!"

He took a scrap of paper from his pocket and jotted something down. Setting it on the table, he walked out of the room. His manner had been utterly resolute, as if he'd decided from the beginning that this was the only course of action. There had been no anger or hesitation, he was calmly determined.

We stared at the note. No one moved. On the paper he had written a single line, one simple formula:

e^πi +1 0

No one spoke. The widow's fingernails had ceased their tapping. Her eyes, so full of suspicion and disdain a moment earlier, now looked at me with a calm, understanding gaze, and I could tell then that she knew the beauty of math.

Not long after that, I received a message from the agency asking me to report for work again at the Professor's house. I could not say whether the widow had a change of heart, or had simply never liked the new housekeeper. I also had no way of determining whether the absurd misunderstanding had been settled or not. But the Professor had now earned his eleventh star.

No matter how many times I went over the strange scene in my mind, it remained a mystery. Why did the widow report me to the agency and have me fired? Why had she reacted so strongly to Root's visit? I was sure she had spied on us from the garden that night after the baseball game, and when I imagined her dragging her bad leg behind her and hiding in the bushes, I almost forgot my anger and felt sorry for her.

The mention of money was probably nothing more than a smoke screen. Maybe the widow was jealous. In her own way, she had been lavishing affection on the Professor for years, and to her I was an interloper. Forbidding me to communicate with the main house was her way of preventing me from disturbing their relationship.

I started work again on July 7, the day known as Tanabata, the Star Festival. The notes fluttering on the Professor's jacket as he met me at the door reminded me of the strips of colored paper on which children write their wishes for the festival. My portrait and the square root sign next to it were still clipped to his cuff.

"How much did you weigh at birth?" This question was new to me.

"I was 3,217 grams," I said. Having no idea what my own weight had been, I used Root's.

"Two to the 3,217th minus 1 is a Mersenne prime," he mumbled before disappearing into his study.

During the previous month, the Tigers had managed to climb back into the pennant race. After Yufune's no-hitter, the strength of the pitching staff had given a boost to the offense as well. But at the end of June things started to unravel. They had lost six straight, and the Giants had managed to pass them, bumping the Tigers down to third place.

The housekeeper who had pinch-hit for me had been methodical, and while I had been afraid to disturb the Professor's work and had barely touched the books in his study, she had picked them all up and stuffed them into the bookshelves, stacking any that didn't fit in the spaces above the armoire and under the sofa. Apparently she had a single organizing principle: size. In the wake of her efforts, there was no denying that the room looked neater, but the hidden order behind the years of chaos had been completely destroyed.

I suddenly remembered the cookie tin filled with baseball cards and went to look for it, fearing it had been lost. It was not far from where I'd left it, now being used as a bookend. The cards inside were safe and sound.

But whether the Tigers rose or fell in the rankings, whether or not his study was neat, the Professor remained the same. Within two days, the interim housekeeper's efforts had vanished and the study had returned to its familiar state of disarray.

I still had the note the Professor had written the day of my confrontation with his sister-in-law. She hadn't seen me take it; I'd slipped it safely away into my wallet next to a photograph of Root.

I went to the library to find out about the formula. The Professor would certainly have explained it to me if I'd asked, but I felt that I would have a much deeper understanding if I struggled with it alone for a while. This was only a feeling, but I realized that during my short acquaintance with the Professor I had begun to approach numbers in the same intuitive way I'd learned music or reading. And my feelings told me that this short formula was not to be taken lightly.

The last time I'd been to the library was to borrow books on dinosaurs for a project Root had been assigned during his school vacation last summer. The mathematics section, at the very back of the second floor, was silent and empty.

In contrast to the Professor's books, which showed signs of their frequent use-musty jackets, creased pages, crumbs in the binding-the library books were so neat and clean, they were almost off-putting. I could tell that some of them would sit here forever without anyone cracking their spines.

I took the Professor's note from my wallet.

e^πi + 1 0

His handwriting was unmistakable: the rounded forms, the wavering lines. There was nothing crude or hurried about it; you could sense the care he had taken with the signs and the neatly closed circle of the zero. Written in tiny symbols, the formula appeared almost modest, sitting alone in the middle of the page.

As I studied it more closely, the Professor's formula struck me as rather strange. Although I could only compare it to a few similar formulas-the area of a rectangle is equal to its length times its width, or the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides-this one seemed oddly unbalanced. There were only two numbers-1 and 0-and one operation-addition. While the equation itself was clear enough, the first element seemed too elaborate.

I had no idea where to begin researching this apparently simple equation. I picked up the nearest books and began leafing through them at random. All I knew for sure was that they were math books. As I looked at them, their contents seemed beyond the comprehension of human beings. The pages and pages of complex, impenetrable calculations might have contained the secrets of the universe, copied out of God's notebook.

In my imagination, I saw the creator of the universe sitting in some distant corner of the sky, weaving a pattern of delicate lace so fine that even the faintest light would shine through it. The lace stretches out infinitely in every direction, billowing gently in the cosmic breeze. You want desperately to touch it, hold it up to the light, rub it against your cheek. And all we ask is to be able to re-create the pattern, weave it again with numbers, somehow, in our own language; to make even the tiniest fragment our own, to bring it back to earth.

I came across a book about Fermat's Last Theorem. As it was a history of the problem, not a mathematical study, I found it easier to follow. I already knew that the theorem had remained unsolved for centuries, but I had never seen it written down:

"For all natural numbers greater than 3, there exist no integers x, y, and z, such that: xⁿ + yⁿ = zⁿ.

Was this all there was to it? At first glance it seemed that any number of solutions could be found. If n = 2, you get the wonderful Pythagorean theorem; did that mean that by simply adding 1 to n, the order was irrevocably lost? As I flipped through the book, I learned that the proposition had never been published in a formal thesis but was something Fermat had scribbled in the margins of another document; apparently he had not left a proof, having run out of space on the page. Since then, many geniuses have tried their hand at solving this most perfect of mathematical puzzles, all to no avail. It seemed sad that one man's whim had been bedeviling mathematicians for more than three centuries.

I was impressed by the delicate weaving of the numbers. No matter how carefully you unraveled a thread, a single moment of inattention could leave you stranded, with no clue what to do next. In all his years of study, the Professor had managed to glimpse several pieces of the lace. I could only hope that some part of him remembered the exquisite pattern.

The third chapter explained that Fermat's Last Theorem was not simply a puzzle designed to excite the curiosity of math fanatics, it had also profoundly affected the very foundations of number theory. And it was here that I found a mention of the Professor's formula. Just as I was aimlessly flipping through pages, a single line flashed in front of me. I held the note up to the page and carefully compared the two. There was no mistake: the equation was Euler's formula.

So now I knew what it was called, but there remained the much more difficult task of trying to understand what it meant. I stood between the bookshelves and I read the same pages several times. When I was confused or flustered, I did as the Professor had suggested and read the lines out loud. Fortunately, I was still the only person in the mathematics section, so no one could complain.

I knew what was meant by π. It was a mathematical constant- the ratio of a circle's circumference to its diameter. The Professor had also taught me the meaning of i. It stood for the imaginary number that results from taking the square root of -1. The problem was e. I gathered that, like π, it was a nonrepeating irrational number and one of the most important constants in mathematics.

Logarithm was another term that seemed to be important. I learned that the logarithm of a given number is the power by which you need to raise a fixed number, called the base, in order to produce the given number. So, for example, if the fixed number, or base, is 10, the logarithm of 100 is 2: 100 = 10² or log₁₀100.

The decimal system uses measurements whose units are powers of ten. Ten is actually known as the "common logarithm." But logarithms in base e also play an extremely important role, I discovered. These are known as "natural logarithms." At what power of e do you get a given number?-that was what you called an "index." In other words, e is the "base of the natural logarithm." According to Euler's calculations: e = 2.71828182845904523536028… and so on forever. The calculation itself, compared to the difficulty of the explanation, was quite simple:

But the simplicity of the calculation only reinforces the enigma of e.

To begin with, what was "natural" about this "natural logarithm"? Wasn't it utterly unnatural to take such a number as your base-a number that could only be expressed by a sign: this tiny e seemed to extend to infinity, falling off even the largest sheet of paper. I could not begin to understand this never-ending number. It seemed as chaotic and random as a line of marching ants or a baby's alphabet blocks, and yet it obeyed its own inner sort of logic. Perhaps there was no fathoming God's notebooks after all. In the entire universe there were only a handful of especially gifted human beings able to understand a tiny part of this order, and then there were the rest of us, who could barely appreciate their discoveries.

The book was so heavy I needed to rest my arms for a moment before flipping back through the pages. I wondered about Leonhard Euler, who was probably the greatest mathematician of the eighteenth century. All I knew about him was this formula, but reading it made me feel as though I were standing in his presence. Using a profoundly unnatural concept, he had discovered the natural connection between numbers that seemed completely unrelated.

If you added 1 to e elevated to the power of π times i, you got 0: e^πi + 1 = 0.

I looked at the Professor's note again. A number that cycled on forever and another vague figure that never revealed its true nature now traced a short and elegant trajectory to a single point. Though there was no circle in evidence, π had descended from somewhere to join hands with e. There they rested, slumped against each other, and it only remained for a human being to add 1, and the world suddenly changed. Everything resolved into nothing, zero.

Euler's formula shone like a shooting star in the night sky, or like a line of poetry carved on the wall of a dark cave. I slipped the Professor's note into my wallet, strangely moved by the beauty of those few symbols. As I headed down the library stairs, I turned back to look. The mathematics stacks were as silent and empty as ever-apparently no one suspected the riches hidden there.

The next day, I returned to the library to look into something else that had been bothering me for a long time. When I found the bound volume of the local newspaper for the year 1975, I read through it a page at a time. The article I was looking for was in the September 24 edition.

On September 23, at approximately 4:10 P.M., on National Highway… a truck belonging to a local transport company crossed the center line, causing a head-on collision with a car… Professor of Mathematics… suffered severe head injuries and is in critical condition, while his sister-in-law, who was in the passenger seat, is in serious condition with a broken leg. The driver of the truck suffered only minor injuries and is being interviewed by police, who suspect he fell asleep at the wheel.

I closed the volume, remembering the sound of the widow's cane.

I still have the Professor's note, though the photograph of Root has long since faded. Euler's formula comforts me-it is a memento that I still treasure.

I've often asked myself why the Professor wrote this particular formula at that moment. Simply by writing out this one equation and placing it between us, he put an end to the argument between myself and the widow. And as a result, I returned to work as his housekeeper and the Professor renewed his friendship with Root. Had he been calculating this outcome from the beginning? Or, in his confusion, had he simply written a formula at random? There was no way to tell.

What was certain was the Professor's affection for Root. Fearful that Root would think he had caused the argument, the Professor came to his rescue in the only way he knew how. After all these years, I'm still at a loss for words to describe how purely the Professor loved children-except to say that it was as unchangeable and true as Euler's formula itself.

My son's needs always took precedence with the Professor, who only sought to protect him. Watching over my son was the Professor's greatest joy. And Root appreciated the Professor's attentions. He never ignored or took these kindnesses for granted, and acknowledged that they should be fully recognized and respected. I could only marvel at Root's maturity. If I was setting out their snack and gave the Professor a larger portion than Root, he would invariably scold me. It was a matter of principle that the biggest piece of fish or steak or watermelon should go to the youngest person at the table. Even when he was at a critical point with a math problem, he still seemed to have unlimited time for Root. He was always delighted when Root asked a question, no matter what the subject; and he seemed convinced that children's questions were much more important than those of an adult. He preferred smart questions to smart answers.

The Professor also showed concern for Root's physical wellbeing and watched over him with care. He noticed ingrown hairs or boils long before I did; he didn't stare or touch him in order to discover these things, he simply knew and he would tell me discreetly, so as not to worry Root. I can still recall him whispering in my ear as I was working in the kitchen. "Do you think we ought to do something about that boil?" he might murmur, as if the world were coming to an end. "Children have quick metabolisms. It might suddenly swell up and press on his lymph nodes or even block his windpipe." He was especially anxious when it came to Root's health.

"Fine. I'll pop it with a needle," I'd say-casually enough to get him truly angry.

"But what if it gets infected?!"

"I'll disinfect the needle first over the stove," I would say, teasing him. His concern for Root delighted me, although I didn't show it.

"Absolutely not! You can't kill all the germs like that!" He refused to let up until I had agreed to take Root straight to the doctor.

He treated Root exactly as he treated prime numbers. For him, primes were the base on which all other natural numbers relied; and children were the foundation of everything worthwhile in the adult world.

I still take out that note sometimes and study it. On sleepless nights, or lonely evenings, when tears come to my eyes thinking about friends who are no longer here. I bow my head in gratitude for that one line.