art by Arlene Kim Suda

pay attention...

pay attention…

Leonhard Euler

Posted on December 18, 2011 by aks

I love the story below about Euler’s number (e) from Richard Feynman’s book “Surely You’re Joking, Mr Feynman!”

Even more enlightening is this online course by Sal Khan at the Khan Academy that does a good job summing up why there is something poetic about Euler’s number and his formula. (I love the last lines of the lecture…it really is worth listening all the way to the end.)



One day at Princeton I was sitting  in  the lounge and  overheard  some mathematicians talking about the series for ex, which is 1 + x +  x2/2! + x3/3!  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 x4/4! 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.

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