Someone on the list once mentioned they thought that Richard Feynman once
wrote about finding cube roots with an abacus. I don't recall seeing that, but
did find the passage below in "Surely You're Joking, Mr. Feynman!" A great
read for many reasons but one for this list is that he talks about doing
mental arithmetic. Much of what he says would not apply to today's audience -
memorizing key logarithms, for example, but it's fascinating reading. It
starts slow, but persist. I was sorry when I got to the end. It's available in
inexpensive paperback.
Phil Mahler
Middlesex CC
Bedford, MA
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Lucky Numbers (Richard Feynman)
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 fore- head: "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.
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