Chapter 26 The fourth big professor-5

While at Princeton, I was sitting in the lounge one day and overheard some mathematicians talking about the series of e.When you expand e, you get 1 + x + (x2/2!) + (x3/3!) ten...  Each term in the formula comes from multiplying the previous term by x and dividing by the next number.For example, to get the next term of (x4/4!), you multiply it by x and divide by 5.It's very simple. From a very young age, I loved studying series.I used this series equation to calculate the value of e, and saw for myself how quickly each new term that appeared became very small. I was muttering to myself how easy it is to use this equation to calculate e to any power (or "power").

"Huh, yeah?" they said, "Well, what's e to the power of 3.3?" said a kid—I think it was Tudge. I said, "That's easy. The answer is 27.11." Tudge knew I was unlikely to be able to figure this out mentally: "Hey! How do you calculate? " Another guy said, "You all know Feynman, he's just bluffing, that must be wrong." They ran to find the e-value table, and I took advantage of this gap to calculate a few more decimal places: "27.1126," I said. They found the result in the table: "He got it right! How did you figure it out?"

"I calculated the series one by one and added them up." "No one can do it that fast. You must happen to know the answer. What is e to the power of three?" "Hey," I said, "this is hard work! Only one problem a day!" "Ha! Prove him a liar!" They were delighted. "Well," I said, "the answer is 20.085." They quickly looked up the meter, and I added a few more decimal places at the same time.They all got nervous because I got another question right! So, all the elites in the mathematics world in front of me couldn't figure out how I calculated a certain power of e!Someone said: "He can't really substitute the numbers and add them up one by one-it's too difficult. There must be some tricks in it. You can't just calculate something like e to the 1.4th power." value."

I said, "That's really hard, but well, for your sake, the answer is 4.05." When they look up the e-value table, I give them a few more decimal places and say: "This is the last question for today!" Then he walked out. The truth of the matter is this: I happen to know the values ​​of three numbers - the logarithm of 10 to base e, Loge10 (used to convert numbers from base 10 to base e), which equals 2.3026; Also from radiation research (half-life of radioactive substances, etc.), I know that the logarithm of 2 to the base e (Loge2) is equal to 0.69315. Therefore, I also know that e raised to the 0.7 power is almost equal to 2.Of course, I also know the value of the first power of e, which is 2.71828.

The first number they wanted me to test was e to the power 3.3, which equals e to the power 2.3—that is, 10—multiplied by e, which is 27.18.And while they were busy figuring out what method I was using, I was correcting my answer and calculating an extra 0.0026 because my original calculation used a higher value, 2.3026. I understand that this kind of thing can never happen again, because it was just a matter of luck.But at this time he said e to the power of 3, that is, e to the power of 2.3 times e to the power of 0.7, and I know that is equal to 20 and a little more.And while they were busy worrying about how I did the math, I corrected that 0.693.

After doing these two questions, I really feel that there is no way to calculate another question, because the second question is also calculated by luck, but the number they proposed is the 1.4th power of e, that is, e Multiplying the 0.7 power of 0 by itself is just a little bit more than 4! They never understood how I figured it out. When I got to Los Alamos, I found that Bate was a master at these calculations.For example, one time we were plugging numbers into equations and needed to find the square of 48.Just as I reached out to shake the Macante, he said, "That's 2300." I started working on the computer, and he said, "If you have to be precise, the answer is 2304."

The computer is also 2304, "Wow! It's amazing!" I said. "Don't you know how to square numbers near fifty?" he said: "You first square 50, which is 2500, then subtract the difference between the number you want to calculate and 50 (2 in this example) and multiply by one hundred, so you get 2300. If you want to be more precise, take Add the square of the difference, and that's 2304." In a few minutes, we'll take the cube root of 2.5.Back then, before computing the cube root of any number with a computer, we had to find a first approximation from a table.I opened the drawer to get my watch—more time this time—and he said, "About 1.35."

I tried it on the computer, and I can't go wrong! "How did you figure it out?" I asked. "Do you have any secrets for taking the cube root?" "Oh," he said, "the logarithm of 2.5 is . . . one-third of the logarithm is the logarithm of 1.3, which is . Just interpolate it out." So I discovered: first, he can memorize the logarithmic table; second, if I use the interpolation method like him, it will definitely take much longer than reaching for the watch and pressing the computer.I was so impressed. Since then, I have tried to do the same.I memorized the logarithms of several numbers, and I began to pay attention to many things.For example, someone said, "What is the square of 28?" Then note that the square root of 2 is 1.4, and 28 is 20 times that of 1.4, so the square of 28 must be close to twice 400, that is, around 800.

If anyone wants to know what 1.73 is divided by 1, you can immediately tell him that the answer is 0.577, because 1.73 is about the square root of 3, so 1/1.73 is about the square root of 3 divided by 3, and if What about 1/1.75, which happens to be 4/7, and you know the famous recurring decimal of 1/7, so you get 0.571428... It was great fun doing quick mental calculations with Bethe using all sorts of tricks. Usually what I think, he thinks, and I can rarely count faster than him.And if I figured out a problem, he laughed heartily.No matter what the problem is, he can always calculate it, and the error is almost within 1%.For him, it was a breeze—any number was always close to some number he was already familiar with.

One day I was in a particularly good mood, it happened to be lunch time, and I didn't know how, and I announced on a whim: "If anyone can finish his topic in 10 seconds, I can finish it in 60 seconds." Give the answer within 10% of the time!" Everyone began to throw me problems they thought were difficult, such as calculating the integral of 1/(1+x4).But in fact, within the range of x they gave me, the answer didn't vary much.The most difficult problem they asked was to find the binomial coefficient of x10 in (1+x)20, which I did just as time was running out. They were all asking me questions, and I was elated when Oren happened to pass by in the hallway outside the restaurant.In fact, before coming to Los Alamos, we had worked together in Princeton, and he was always smarter than me.For example, one day I was absent-mindedly playing with a steel measuring tape—the kind that rolls back automatically when you press a button on it; my hand. "Wow!" I exclaimed, "I'm so dumb, this thing hits me every time, and I'm still playing with this thing."

He said, "Your grip is wrong," took the tape measure over, pulled it out, buttoned it, rolled it back, it didn't hurt him. "Wow! How did you do that?" I yelled. "Think for yourself!" For the next two weeks, I pressed the tape everywhere I went, and the back of my hand was bruised and bleeding.Finally I couldn't take it anymore. "Olen! I surrender! What the hell method do you use to hold it without pain?" "Who says it doesn't hurt? I hurt too!" I thought I was so stupid that he tricked me into beating myself with a ruler twice! And now that Oren happened to pass by the restaurant, these people were very excited, "Hey, Oren!" They shouted: "Feynman is really good! He can give a question in 1 minute that we can finish in 10 seconds." The answer, the error is 10%. You also come up with a question!" He almost didn't stop, and said, "The value of the tangent function of 10 to the hundredth power." I'm stumped: I have to divide a number with a hundred digits by π.I can't help it! I boasted once: "What other people have to use the perimeter integration method to calculate the integral, I promise to find out in a different way." So Oren offered me a fantastic, damn point.He started with a complex variable function for which he knew the answer, and removed the real part, leaving only the imaginary part, and it turned out to be a problem that must use the integral method of the perimeter!He always frustrates me a lot and is a very smart guy. When I first arrived in Brazil, I once had lunch in a certain restaurant.I don't know what time it is, but I'm the only customer there - I keep running to the restaurant at odd hours.I ate my favorite steak with rice, and 4 waiters were standing around. A Japanese came in.I have seen him wandering around the neighborhood before, selling abacus for a living.He talks to the waiter and challenges him to add faster than anyone else. The waiters are afraid of losing face, so they say, "Really? Why don't you challenge that gentleman over there?" The Japanese came up to me and I protested: "I can't speak Portuguese well!" The waiters are all laughing: "Numbers in Portuguese are easy!" They got me a pen and paper. The man asked a waiter to come up with some numbers for us to add.He won too much because he was listening and adding while I was writing the numbers down. I suggested that the waiter write down two columns of the same number and give them to us at the same time. It doesn't make much difference, he's still a lot faster than me. He got a little carried away and wanted to further prove his ability. "Multiplicao!" He said he wanted to compare multiplication. Someone wrote a problem and he won again, but not by much because my multiplication was pretty good. Then he made a mistake: he suggested that we continue with division.He didn't realize that the harder the problem, the better my chances of winning. We also did a very long division problem.This time we are tied. This annoyed the Japanese, because he seemed to have been well trained in abacus, and now he almost lost it to a customer in the restaurant. "Raios cubicos!" he said, his voice vindictive.Cube root!He wants to use arithmetic to find the cube root!There is probably no more difficult subject in elementary arithmetic.In his abacus world, the cube root must also be his specialty. He wrote a number on the paper - whatever - I remember it being 1729.03.He immediately started to calculate, murmured words, and kept moving! He has begun to calculate the cube root. And I just sit there. A waiter said, "What are you doing?" I pointed, "I was thinking!" I said, and wrote 12 on the paper.After a while I got 12.002. The Japanese wiped the sweat off his brow, "Twelve!" he said. "Oh no!" I say. "More numbers! More numbers!" I fully understand that when finding the cube root with ordinary arithmetic methods, it is much more difficult to find the following numbers than the previous ones. This is hard work. He put his head back to work hard, saying "Ah Gulummah" incessantly, during which I wrote two more numbers.Finally he looked up and said, "12.0!" The waiters were so excited, they said to the Japanese, "Look, he can just think, but you have to use the abacus! And he does more figures!" He was defeated and walked away dejectedly, while the waiter celebrated wildly. How did this customer win the abacus?The title is 1729.03.I happen to know that there are 1728 cubic inches in a cubic foot, so the answer must be a little over 12.The extra 1.03 is about 1 part in 2000, and I learned in calculus class that for small fractions, the cube root exceeds a third of the number, so I just need Calculate what 1/1728 is, and multiply by 4 (that is, divide by 3 and multiply by 12).This is why I can calculate so many decimal places in one go. A few weeks later, the Japanese ran into the living room of the hotel where I was staying. He recognized me and ran over and said, "Tell me, how did you figure out the cube root so quickly?" I told him that this is an approximation method, which has something to do with the error, "For example, you say 28. Then, the cube root of 27 is 3..." He picked up the abacus: da da da da—"Oh! yes," he said. What I found: he doesn't know how to deal with numbers at all.With an abacus, you don't have to memorize a whole bunch of arithmetic combinations; you just need to know how to push the little beads up and down.You don't need to know that 9 plus 7 equals 16, but you just need to remember that when adding 9, you have to push a ten-digit bead up and dial a one-digit bead down.Maybe we are slow to calculate, but we really understand the mystery of numbers. Moreover, he simply could not understand the rationale involved in the approximation method. He did not understand that there are many situations in which no method can find the full cube root, but can approximate it.So I could never teach him how to find the cube root, or even show him how lucky I was that day that he just happened to pick a number like 1729.03!