by Yoko Ogawa
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: eπi + 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.
8
It was on the day of the Star Festival that the Tigers lost their seventh game in a row, 1-0 against Taiyo.
I'd had no trouble falling back into the rhythm of the job, despite my month away. And because of the Professor's memory problem, he immediately forgot my misunderstanding with his sister-in-law. For him, no trace of the trouble remained.
I transferred the notes to his summer suit, taking care to fasten them in the same positions, and I rewrote those that were torn or faded.
"In an envelope in the desk, second drawer from the bottom."
"Theory of Functions, 2nd edition, pp. 315-372 and Commentary on Hyperbolic Functions, volume IV, chapter 1, § 17."
"Medicine to take after meals in manila envelope, on the left in the sideboard."
"Spare razor blades next to the mirror above the sink."
"Thank for the cake."
Some of the notes were out of date-it had been a month since Root had brought the Professor a little steamed bun he had baked in his home economics class-but it seemed wrong to throw them out. I treated them all with equal respect.
As I read through them, I realized how hard it was for the Profossor to simply get through the day, and how carefully he hid the enormous efforts he made. I tried to work as quickly as possible and not to linger over the notes. When they were all reattached, his summer suit was ready.
For a few weeks, the Professor had been working on an extremely difficult problem, one that would pay the largest cash prize in the history of the Journal of Mathematics to the reader who solved it. Indifferent to money, the Professor took pleasure in the difficulty of the problem itself. Checks from the journal were left unopened on the hall table, and when I asked him if he wanted me to cash his prize money at the post office, he shrugged. In the end, I asked the agency to forward them to his sister-in-law.
Just by looking at the Professor, I could tell that the new problem was especially hard. The intensity of his thought seemed to be near breaking point. He would vanish into the study as though he were literally retreating into his mind, and I imagined that his body might actually vaporize into pure contemplation and disappear. But then the sound of his pencil scratching across the paper would break the stillness and reassure me-the Professor was still with us and was making some progress with the proof.
I tried to imagine how he could work through a problem like this over such a long period of time-he basically had to start again from the beginning every morning. To compensate for the loss of his thoughts from the day before, he had only an ordinary notebook and the scribbled notes that covered his body like a cocoon. Since the accident, math was his life, so perhaps it was also what led him to sit down at his desk each day and return to the problem in front of him.
I was considering all of this while making dinner when the Professor suddenly appeared. Usually, when he was wrestling with a problem, I hardly saw him. I wasn't sure whether I would be interrupting his thinking if I spoke to him, so I continued seeding the peppers and peeling the onions. He walked over, leaned against the counter, folded his arms, and stood there staring at my hands. I felt awkward with him watching me, so I went to get some eggs out of the refrigerator, and a frying pan.
"Did you need something?" I asked at last, no longer able to stand the silence.
"No, go on," he said. His tone was reassuring. "I like to watch you cook," he added.
I wondered if the problem had proven so difficult his brain had blown a fuse-but I broke the eggs into a bowl and beat them with my chopsticks. I went on stirring after the spices had dissolved and the lumps were gone, only stopping when my hand had grown numb.
"Now what are you going to do?" he asked quietly.
"Well…," I said, "next…, uh, I have to fry the pork." The Professor's sudden appearance had disrupted my usual routine.
"You're not going to cook the eggs now?"
"No, it's best to let them sit, so the spices blend in."
We were alone, Root was off playing in the park. The afternoon sun divided the garden into patches of shadow and dappled light. The air was still, and the curtains hung limply by the open window. The Professor was watching me with the intense stare he normally reserved for math. His pupils were so black they looked transparent, and his eyelashes seemed to quiver with each breath. He was gazing at my hands, which were only a few feet away, but he might have been staring off into distant space. I dusted the pork filets in flour and arranged them in the pan.
"Why do you have to move them around like that?"
"Because the temperature at the center of the pan is higher than at the edges. You have to move them every so often to cook them evenly."
"I see. No one gets the best spot all the time-they have to compromise."
He nodded as if I had just revealed a great secret. The aroma of cooking meat drifted up between us.
I sliced some peppers and onions for the salad and made an olive oil dressing. Then I fried the eggs. I had planned to sneak some grated carrot into the dressing, which now proved impossible with the Professor watching me. He said nothing, but he seemed to hold his breath while I cut the lemon peel in the shape of a flower. He leaned in closer as I mixed the vinegar and oil, and I thought I heard him sigh when I set the piping hot omelet on the counter.
"Excuse me," I said at last,
unable to control my curiosity. "But I'm wondering what you find so interesting."
"I like to watch you cook," he said again. He unfolded his arms and looked out the window for the spot where the evening star would appear. Then he went back to his study without a sound. The setting sun shone on his back as he walked away.
I looked at the food I had just finished preparing and then at my hands. Sautéed pork garnished with lemon, a salad, and a soft, yellow omelet. I studied the dishes, one by one. They were all perfectly ordinary, but they looked delicious-satisfying food at the end of a long day. I looked at my palms again, filled suddenly with an absurd sense of satisfaction, as though I had just solved Fermat's Last Theorem.
The rainy season came to an end, Root's summer vacation began, and still the Professor struggled with his proof. I was eagerly looking forward to the day he would ask me to mail it to the magazine.
The weather had turned hot. The cottage had neither airconditioning nor a cross breeze. Root and I tried not to complain, but we were no match for the Professor's stoicism. At noon, on the hottest day, he would sit at his desk with the doors closed, never removing his jacket-as if he were afraid that all the work he'd done on the proof would crumble if he slipped out of his coat. The notes on his suit had wilted, and he was covered in a painful-looking heat rash, but when I came in with a fan, or suggested a cold shower, or more barley tea, he would chase me out in exasperation.
Once his summer vacation started, Root would come with me to the cottage in the morning. Given my recent run-in with the widow, I thought it best not to increase the amount of time he spent with me at work, but the Professor wouldn't hear of it. He was absolutely convinced that a child on vacation had to be where his mother could watch over him. Root, however, much preferred to be at the park playing baseball with his friends or at the pool, so he was almost never with us.