Lucky Numbers

One day at Princeton I was sitting in the lounge and overheard some mathematicians talking about the series for e^x, which is 1 + x + x^2/2! + x^3/3! Each term you get by multiplying the preceding term by x and dividing by the next number. For example, to get the next term after x^4/4! you multiply that term by x and divide by 5. It’s very simple.

When I was a kid I was excited by series, and had played with this thing. I had computed e using that series, and had seen how quickly the new terms became very small.

I mumbled something about how it was easy to calculate e to any power using that series (you just substitute the power for x).

“Oh yeah?” they said. “Well, then what’s e to the 3.3?” said some joker—I think it was Tukey.

I say, “That’s easy. It’s 27.11.”

Tukey knows it isn’t so easy to compute all that in your head. “Hey! How’d you do that?”

Another guy says, “You know Feynman, he’s just faking it. It’s not really right.”

They go to get a table, and while they’re doing that, I put on a few more figures.: “27.1126,” I say.

They find it in the table. “It’s right! But how’d you do it!”

“I just summed the series.”

“Nobody can sum the series that fast. You must just happen to know that one. How about e to the 3?”

“Look,” I say. “It’s hard work! Only one a day!”

“Hah! It’s a fake!” they say, happily.

“All right,” I say, “It’s 20.085.”

They look in the book as I put a few more figures on. They’re all excited now, because I got another one right.

Here are these great mathematicians of the day, puzzled at how I can compute e to any power! One of them says, “He just can’t be substituting and summing—it’s too hard. There’s some trick. You couldn’t do just any old number like e to the 1.4.”

I say, “It’s hard work, but for you, OK. It’s 4.05.”

As they’re looking it up, I put on a few more digits and say, “And that’s the last one for the day!” and walk out.

What happened was this: I happened to know three numbers—the logarithm of 10 to the base e (needed to convert numbers from base 10 to base e), which is 2.3026 (so I knew that e to the 2.3 is very close to 10), and because of radioactivity (mean-life and half-life), I knew the log of 2 to the base e, which is.69315 (so I also knew that e to the.7 is nearly equal to 2). I also knew e (to the 1), which is 2. 71828.

The first number they gave me was e to the 3.3, which is e to the 2.3—ten—times e, or 27.18. While they were sweating about how I was doing it, I was correcting for the extra.0026—2.3026 is a little high.

I knew I couldn’t do another one; that was sheer luck. But then the guy said e to the 3: that’s e to the 2.3 times e to the.7, or ten times two. So I knew it was 20. something, and while they were worrying how I did it, I adjusted for the.693.

Now I was sure I couldn’t do another one, because the last one was again by sheer luck. But the guy said e to the 1.4, which is e to the.7 times itself. So all I had to do is fix up 4 a little bit!

They never did figure out how I did it.

When I was at Los Alamos I found out that Hans Bethe was absolutely topnotch at calculating. For example, one time we were putting some numbers into a formula, and got to 48 squared. I reach for the Marchant calculator, and he says, “That’s 2300.” I begin to push the buttons, and he says, “If you want it exactly, it’s 2304.”

The machine says 2304. “Gee! That’s pretty remarkable!” I say.

“Don’t you know how to square numbers near 50?” he says. “You square 50—that’s 2500—and subtract 100 times the difference of your number from 50 (in this case it’s 2), so you have 2300. If you want the correction, square the difference and add it on. That makes 2304.”

A few minutes later we need to take the cube root of 2Ѕ. Now to take cube roots on the Marchant you had to use a table for the first approximation. I open the drawer to get the table—it takes a little longer this time—and he says, “It’s about 1.35.”

I try it out on the Marchant and it’s right. “How did you do that one?” I ask. “Do you have a secret for taking cube roots of numbers?”

“Oh,” he says, “the log of 2Ѕ is so-and-so. Now one third of that log is between the logs of 1.3, which is this, and 1.4, which is that, so I interpolated.”

So I found out something: first, he knows the log tables; second, the amount of arithmetic he did to make the interpolation alone would have taken me longer to do than reach for the table and punch the buttons on the calculator. I was very impressed.

After that, I tried to do those things. I memorized a few logs, and began to notice things. For instance, if somebody says, “What is 28 squared?” you notice that the square root of 2 is 1.4, and 28 is 20 times 1.4, so the square of 28 must be around 400 times 2, or 800.

If somebody comes along and wants to divide 1 by 1.73, you can tell them immediately that it’s .577, because you notice that 1.73 is nearly the square root of 3, so 1/1.73 must be one-third of the square root of 3. And if it’s ‘/1.75, that’s equal to the inverse of 7/4, and you’ve memorized the repeating decimals for sevenths: .571428.

I had a lot of fun trying to do arithmetic fast, by tricks, with Hans. It was very rare that I’d see something he didn’t see and beat him to the answer, and he’d laugh his hearty laugh when I’d get one. He was nearly always able to get the answer to any problem within a percent. It was easy for him—every number was near something he knew.

One day I was feeling my oats. It was lunch time in the technical area, and I don’t know how I got the idea, but I announced, “I can work out in sixty seconds the answer to any problem that anybody can state in ten seconds, to 10 percent!”

People started giving me problems they thought were difficult, such as integrating a function like 1/(1 + x), which hardly changed over the range they gave me. The hardest one somebody gave me was the binomial coefficient of x^10 in (1 + x)^20; I got that just in time.

They were all giving me problems and I was feeling great, when Paul Olum walked by in the hall. Paul had worked with me for a while at Princeton before coming out to Los Alamos, and he was always cleverer than I was. For instance, one day I was absent-mindedly playing with one of those measuring tapes that snap back into your hand when you push a button. The tape would always slap over and hit my hand, and it hurt a little bit. “Geez!” I exclaimed. “What a dope I am. I keep playing with this thing, and it hurts me every time.”

He said, “You don’t hold it right,” and took the damn thing, pulled out the tape, pushed the button, and it came right back. No hurt.

“Wow! How do you do that?” I exclaimed.

“Figure it out!”

For the next two weeks I’m walking all around Princeton, snapping this tape back until my hand is absolutely raw. Finally I can’t take it any longer. “Paul! I give up! How the hell do you hold it so it doesn’t hurt?”

“Who says it doesn’t hurt? It hurts me too!”

I felt so stupid. He had gotten me to go around and hurt my hand for two weeks!

So Paul is walking past the lunch place and these guys are all excited. “Hey, Paul!” they call out. “Feynman’s terrific! We give him a problem that can be stated in ten seconds, and in a minute he gets the answer to 10 percent. Why don’t you give him one?”

Without hardly stopping, he says, “The tangent of 10 to the 100th.”

I was sunk: you have to divide by pi to 100 decimal places! It was hopeless.

One time I boasted, “I can do by other methods any integral anybody else needs contour integration to do.”

So Paul puts up this tremendous damn integral he had obtained by starting out with a complex function that he knew the answer to, taking out the real part of it and leaving only the complex part. He had unwrapped it so it was only possible by contour integration! He was always deflating me like that. He was a very smart fellow.

The first time I was in Brazil I was eating a noon meal at I don’t know what time—I was always in the restaurants at the wrong time—and I was the only customer in the place. I was eating rice with steak (which I loved), and there were about four waiters standing around.

A Japanese man came into the restaurant. I had seen him before, wandering around; he was trying to sell abacuses. He started to talk to the waiters, and challenged them: He said he could add numbers faster than any of them could do.

The waiters didn’t want to lose face, so they said, “Yeah, yeah. Why don’t you go over and challenge the customer over there?”

The man came over. I protested, “But I don’t speak Portuguese well!”

The waiters laughed. “The numbers are easy,” they said.

They brought me a pencil and paper.

The man asked a waiter to call out some numbers to add. He beat me hollow, because while I was writing the numbers down, he was already adding them as he went along.

I suggested that the waiter write down two identical lists of numbers and hand them to us at the same time. It didn’t make much difference. He still beat me by quite a bit.

However, the man got a little bit excited: he wanted to prove himself some more. “Multiplicao!” he said.

Somebody wrote down a problem. He beat me again, but not by much, because I’m pretty good at products.

The man then made a mistake: he proposed we go on to division. What he didn’t realize was, the harder the problem, the better chance I had.

We both did a long division problem. It was a tie.

This bothered the hell out of the Japanese man, because he was apparently very well trained on the abacus, and here he was almost beaten by this customer in a restaurant.

“Raios cubicos!” he says, with a vengeance. Cube roots! He wants to do cube roots by arithmetic! It’s hard to find a more difficult fundamental problem in arithmetic. It must have been his topnotch exercise in abacus-land.

He writes a number on some paper—any old number—and I still remember it: 1729.03. He starts working on it, mumbling and grumbling: “Mmmmmmagmmmmbrrr ”—he’s working like a demon! He’s poring away, doing this cube root.

Meanwhile I’m just sitting there.

One of the waiters says, “What are you doing?”

I point to my head. “Thinking!” I say. I write down 12 on the paper. After a little while I’ve got 12.002.

The man with the abacus wipes the sweat off his forehead: “Twelve!” he says.

“Oh, no!” I say. “More digits! More digits!” I know that in taking a cube root by arithmetic, each new digit is even more work than the one before. It’s a hard job.

He buries himself again, grunting, “Rrrrgrrrrmmmmmm …”

while I add on two more digits. He finally lifts his head to say, “12.0!”

The waiters are all excited and happy. They tell the man, “Look! He does it only by thinking, and you need an abacus! He’s got more digits!”

He was completely washed out, and left, humiliated. The waiters congratulated each other.

How did the customer beat the abacus? The number was 1729.03. I happened to know that a cubic foot contains 1728 cubic inches, so the answer is a tiny bit more than 12. The excess, 1.03, is only one part in nearly 2000, and I had learned in calculus that for small fractions, the cube root’s excess is one-third of the number’s excess. So all I had to do is find the fraction 1/1728, and multiply by 4 (divide by 3 and multiply by 12). So I was able to pull out a whole lot of digits that way.

A few weeks later the man came into the cocktail lounge of the hotel I was staying at. He recognized me and came over. “Tell me,” he said, “how were you able to do that cube-root problem so fast?”

I started to explain that it was an approximate method, and had to do with the percentage of error. “Suppose you had given me 28. Now, the cube root of 27 is 3..

He picks up his abacus: zzzzzzzzzzzzzzz—

“Oh yes,” he says.

I realized something: he doesn’t know numbers. With the abacus, you don’t have to memorize a lot of arithmetic combinations; all you have to do is learn how to push the little beads up and down. You don’t have to memorize 9 + 7 = 16; you just know that when you add 9 you push a ten’s bead up and pull a one’s bead down. So we’re slower at basic arithmetic, but we know numbers.

Furthermore, the whole idea of an approximate method was beyond him, even though a cube root often cannot be computed exactly by any method. So I never could teach him how I did cube roots or explain how lucky I was that he happened to choose 1729.03.

O Americano, Outra Vez!

One time I picked up a hitchhiker who told me how interesting South America was, and that I ought to go there. I complained that the language is different, but he said just go ahead and learn it—it’s no big problem. So I thought, that’s a good idea: I’ll go to South America.

Cornell had some foreign language classes which followed a method used during the war, in which small groups of about ten students and one native speaker speak only the foreign language-nothing else. Since I was a rather young-looking professor there at Cornell, I decided to take the class as if I were a regular student. And since I didn’t know yet where I was going to end up in South America, I decided to take Spanish, because the great majority of the countries there speak Spanish.

So when it was time to register for the class, we were standing outside, ready to go into the classroom, when this pneumatic blonde came along. You know how once in a while you get this feeling, WOW? She looked terrific. I said to myself, “Maybe she’s going to be in the Spanish class—that’ll be great!” But no, she walked into the Portuguese class. So I figured, What the hell—I might as well learn Portuguese.

I started walking right after her when this Anglo-Saxon attitude that I have said, “No, that’s not a good reason to decide which language to speak.” So I went back and signed up for the Spanish class, to my utter regret.

Some time later I was at a Physics Society meeting in New York, and I found myself sitting next to Jaime Tiomno, from Brazil, and he asked, “What are you going to do next summer?”

“I’m thinking of visiting South America.”

“Oh! Why don’t you come to Brazil? I’ll get a position for you at the Center for Physical Research.”

So now I had to convert all that Spanish into Portuguese! I found a Portuguese graduate student at Cornell, and twice a week he gave me lessons, so I was able to alter what I had learned. On the plane to Brazil I started out sitting next to a guy from Colombia who spoke only Spanish: so I wouldn’t talk to him because I didn’t want to get confused again. But sitting in front of me were two guys who were talking Portuguese. I had never heard real Portuguese; I had only had this teacher who had talked very slowly and clearly. So here are these two guys talking a blue streak, brrrrrrr-a-ta brrrrrrr-a-ta, and I can’t even hear the word for “I,” or the word for “the,” or anything.

Finally, when we made a refueling stop in Trinidad, I went up to the two fellas and said very slowly in Portuguese, or what I thought was Portuguese, “Excuse me … can you understand … what I am saying to you now?”

“Pues nгo, porque nгo? ”—” Sure, why not?” they replied.

So I explained as best I could that I had been learning Portuguese for some months now, but I had never heard it spoken in conversation, and I was listening to them on the airplane, but couldn’t understand a word they were saying.

“Oh,” they said with a laugh, “Nao e Portugues! E Ladдo! Judeo!” What they were speaking was to Portuguese as Yiddish is to German, so you can imagine a guy who’s been studying German sitting behind two guys talking Yiddish, trying to figure out what’s the matter. It’s obviously German, but it doesn’t work. He must not have learned German very well.

When we got back on the plane, they pointed out another man who did speak Portuguese, so I sat next to him. He had been studying neurosurgery in Maryland, so it was very easy to talk with him—as long as it was about cirugia neural, o cerebreu, and other such “complicated” things. The long words are actually quite easy to translate into Portuguese because the only difference is their endings: “-tion” in English is “-зao” in Portuguese; “-ly” is “-mente,” and so on. But when he looked out the window and said something simple, I was lost: I couldn’t decipher “the sky is blue.”

I got off the plane in Recife (the Brazilian government was going to pay the part from Recife to Rio) and was met by the father-in-law of Cesar Lattes, who was the director of the Center for Physical Research in Rio, his wife, and another man. As the men were off getting my luggage, the lady started talking to me in Portuguese: “You speak Portuguese? How nice! How was it that you learned Portuguese?”

I replied slowly, with great effort. “First, I started to learn Spanish… then I discovered I was going to Brazil.

Now I wanted to say, “So, I learned Portuguese,” but I couldn’t think of the word for “so.” I knew how to make BIG words, though, so I finished the sentence like this: “CONSEQUENTEMENTE, apprendiPortugues!”

When the two men came back with the baggage, she said, “Oh, he speaks Portuguese! And with such wonderful words: CONSEQUENTEMENTE!”

Then an announcement came over the loudspeaker. The flight to Rio was canceled, and there wouldn’t be another one till next Tuesday—and I had to be in Rio on Monday, at the latest.

I got all upset. “Maybe there’s a cargo plane. I’ll travel in a cargo plane,” I said.

“Professor!” they said, “It’s really quite nice here in Recife. We’ll show you around. Why don’t you relax—you’re in Brazil.”

That evening I went for a walk in town, and came upon a small crowd of people standing around a great big rectangular hole in the road—it had been dug for sewer pipes, or something—and there, sitting exactly in the hole, was a car. It was marvelous: it fitted absolutely perfectly, with its roof level with the road. The workmen hadn’t bothered to put up any signs at the end of the day, and the guy had simply driven into it. I noticed a difference: When we’d dig a hole, there’d be all kinds of detour signs and flashing lights to protect us. There, they dig the hole, and when they’re finished for the day, they just leave.

Anyway, Recife was a nice town, and I did wait until next Tuesday to fly to Rio.

When I got to Rio I met Cesar Lattes. The national TV network wanted to make some pictures of our meeting, so they started filming, but without any sound. The cameramen said, “Act as if you’re talking. Say something—anything.”

So Lattes asked me, “Have you found a sleeping dictionary yet?”

That night, Brazilian TV audiences saw the director of the Center for Physical Research welcome the Visiting Professor from the United States, but little did they know that the subject of their conversation was finding a girl to spend the night with!

When I got to the center, we had to decide when I would give my lectures—in the morning, or afternoon.

Lattes said, “The students prefer the afternoon.”

“So let’s have them in the afternoon.”

“But the beach is nice in the afternoon, so why don’t you give the lectures in the morning, so you can enjoy the beach in the afternoon.”

“But you said the students prefer to have them in the afternoon.”

“Don’t worry about that. Do what’s most convenient for you! Enjoy the beach in the afternoon.”

So I learned how to look at life in a way that’s different from the way it is where I come from. First, they weren’t in the same hurry that I was. And second, if it’s better for you, never mind! So I gave the lectures in the morning and enjoyed the beach in the afternoon. And had I learned that lesson earlier, I would have learned Portuguese in the first place, instead of Spanish.

I thought at first that I would give my lectures in English, but I noticed something: When the students were explaining something to me in Portuguese, I couldn’t understand it very well, even though I knew a certain amount of Portuguese. It was not exactly clear to me whether they had said “increase,” or “decrease,” or “not increase,” or “not decrease,” or “decrease slowly.” But when they struggled with English, they’d say “ahp” or “doon,” and I knew which way it was, even though the pronunciation was lousy and the grammar was all screwed up. So I realized that if I was going to talk to them and try to teach them, it would be better for me to talk in Portuguese, poor as it was. It would be easier for them to understand.

During that first time in Brazil, which lasted six weeks, I was invited to give a talk at the Brazilian Academy of Sciences about some work in quantum electrodynamics that I had just done. I thought I would give the talk in Portuguese, and two students at the center said they would help me with it. I began by writing out my talk in absolutely lousy Portuguese. I wrote it myself, because if they had written it, there would be too many words I didn’t know and couldn’t pronounce correctly. So I wrote it, and they fixed up all the grammar, fixed up the words and made it nice, but it was still at the level that I could read easily and know more or less what I was saying. They practiced with me to get the pronunciations absolutely right: the “de” should be in between “deh” and “day”—it had to be just so.

I got to the Brazilian Academy of Sciences meeting, and the first speaker, a chemist, got up and gave his talk—in English. Was he trying to be polite, or what? I couldn’t understand what he was saying because his pronunciation was so bad, but maybe everybody else had the same accent so they could understand him; I don’t know. Then the next guy gets up, and gives his talk in English!

When it was my turn, I got up and said, “I’m sorry; I hadn’t realized that the official language of the Brazilian Academy of Sciences was English, and therefore I did not prepare my talk in English. So please excuse me, but I’m going to have to give it in Portuguese.”

So I read the thing, and everybody was very pleased with it.

The next guy to get up said, “Following the example of my colleague from the United States, I also will give my talk in Portuguese.” So, for all I know, I changed the tradition of what language is used in the Brazilian Academy of Sciences.

Some years later, I met a man from Brazil who quoted to me the exact sentences I had used at the beginning of my talk to the Academy. So apparently it made quite an impression on them.

But the language was always difficult for me, and I kept working on it all the time, reading the newspaper, and so on. I kept on giving my lectures in Portuguese—what I call “Feynman’s Portuguese,” which I knew couldn’t be the same as real Portuguese, because I could understand what I was saying, while I couldn’t understand what the people in the street were saying.

Because I liked it so much that first time in Brazil, I went again a year later, this time for ten months. This time I lectured at the University of Rio, which was supposed to pay me, but they never did, so the center kept giving me the money I was supposed to get from the university.

I finally ended up staying in a hotel right on the beach at Copacabana, called the Miramar. For a while I had a room on the thirteenth floor, where I could look out the window at the ocean and watch the girls on the beach.

It turned out that this hotel was the one that the airline pilots and the stewardesses from Pan American Airlines stayed at when they would “lay over”—a term that always bothered me a little bit. Their rooms were always on the fourth floor, and late at night there would often be a certain amount of sheepish sneaking up and down in the elevator.

One time I went away for a few weeks on a trip, and when I came back the manager told me he had to book my room to somebody else, since it was the last available empty room, and that he had moved my stuff to a new room.

It was a room right over the kitchen, that people usually didn’t stay in very long. The manager must have figured that I was the only guy who could see the advantages of that room sufficiently clearly that I would tolerate the smells and not complain. I didn’t complain: It was on the fourth floor, near the stewardesses. It saved a lot of problems.

The people from the airlines were somewhat bored with their lives, strangely enough, and at night they would often go to bars to drink. I liked them all, and in order to be sociable, I would go with them to the bar to have a few drinks, several nights a week.

One day, about 3:30 in the afternoon, I was walking along the sidewalk opposite the beach at Copacabana past a bar. I suddenly got this treMENdous, strong feeling: “That’s just what I want; that’ll fit just right. I’d just love to have a drink right now!”

I started to walk into the bar, and I suddenly thought to myself, “Wait a minute! It’s the middle of the afternoon. There’s nobody here, There’s no social reason to drink. Why do you have such a terribly strong feeling that you have to have a drink?”—and I got scared.

I never drank ever again, since then. I suppose I really wasn’t in any danger, because I found it very easy to stop. But that strong feeling that I didn’t understand frightened me. You see, I get such fun out of thinking that I don’t want to destroy this most pleasant machine that makes life such a big kick. It’s the same reason that, later on, I was reluctant to try experiments with LSD in spite of my curiosity about hallucinations.

Near the end of that year in Brazil I took one of the air hostesses—a very lovely girl with braids—to the museum. As we went through the Egyptian section, I found myself telling her things like, “The wings on the sarcophagus mean such-and-such, and in these vases they used to put the entrails, and around the corner there oughta be a so-and-so …” and I thought to myself, “You know where you learned all that stuff? From Mary Lou”—and I got lonely for her.

I met Mary Lou at Cornell and later, when I came to Pasadena, I found that she had come to Westwood, nearby. I liked her for a while, but we used to argue a bit; finally we decided it was hopeless, and we separated. But after a year of taking out these air hostesses and not really getting anywhere, I was frustrated. So when I was telling this girl all these things, I thought Mary Lou really was quite wonderful, and we shouldn’t have had all those arguments.