Isaac Asimov and imaginary numbers
Tony Pay
Apropos extensions to the number system, I remember a story Isaac Asimov
once told about the square root of minus one. I read it in one of his
books when I was still at school, and found it very striking. But I've
never seen it repeated anywhere else, and I don't remember the title of
the book, or know whether it's still in print, so I thought I'd post the
story here as I remember it, in case it turns out to be new to some of
you. (The story is a bit unlikely, and Asimov comes out of it perhaps
rather too well, but I quite liked that at the time:-)
Anyway, apparently Asimov was studying science and mathematics; but he
had a friend who was studying philosophy, and they often had lunch
together immediately following a lecture that formed a part of his
friend's course.
One day, Asimov happened to arrive early, decided to drop in on the
philosophy lecture, and slipped into the room, to find himself faced
with two lists on the blackboard. One was headed "Realists" and the
other "Mystics", but Asimov was astounded to see the word
"Mathematicians" under the heading "Mystics". So much so, that he
walked forward, put up his hand and asked for an explanation from the
philosophy professor.
"Ah, so we have a mathematical guest who thinks he's a realist!" said
the professor, with a condescending smile. "Mathematicians, my young
friend, are mystics, because they believe in the existence of unreal
objects; for example they believe in the existence of such a thing as
the square root of minus one."
"I know we *call* the square root of minus one imaginary, but it's just
as real as other numbers," said Asimov indignantly. "And we're
certainly not mystics!"
"Very well," said the professor, smirking a little, "if you say that
the square root of minus one is as real as other numbers, come down here
and show us all. Give me the square root of minus one pieces of chalk!"
And all the philosophy students started to laugh.
For a moment Asimov didn't know what to say, and went a bit red, while
the students' laughter grew louder, and the professor's smirk grew
bigger. But then he suddenly shouted out, "OK! I'll do it!" and there
was an immediate silence.
"I'll do it," Asimov went on, "on one condition. And the condition is
that you give me a half piece of chalk to do it with."
"All right," said the professor, a little puzzled. What was going on?
He broke a fresh stick of chalk in two. "Here's your half piece of
chalk. Now what are you going to do?"
"Ah, but wait a minute," said Asimov. "You haven't kept your side of
the bargain yet. This isn't a half piece of chalk. This is *one* piece
of chalk!"
And the students gradually began to laugh again.
"It's a half piece of chalk!" said the professor, getting a bit
flustered. "A new piece of chalk has a regulation length, and this is
half that regulation length, so this is a half piece of chalk."
"Well, now you're springing a *very* arbitrary definition on me," said
Asimov, beginning to enjoy himself. "The *regulation length of a piece
of chalk* enters philosophical debate about reality. Really??? But
even if I were to accept your definition, how can you be sure that this
piece of chalk is *exactly* half the regulation length? You just broke
it casually in two. You'd have to go to a very great deal of trouble to
divide it in half - in fact, you couldn't possibly do what you said
you'd do *exactly*, so I say you don't know what "one half" really
means, in the real world. But either way, if you think you can use a
number like one half to count pieces of chalk, you're wrong."
The professor was speechless.
"But I'm sure you'll agree that doesn't mean that one half isn't a real
number, just because you can't use it to count pieces of chalk with,"
said Asimov. "I'll tell you what: when you have a better idea of what
you mean by one half, or even what you mean by reality for that matter,
we can discuss the square root of minus one."
And he left, and waited for his friend outside in future.
Tony
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_________ Tony Pay
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