I for Imaginary numbers

Abstract

This paper surveys the history of complex numbers from their initial discovery to their complete acceptance and their use in contemporary physics and mathematics. The initial rejection of these new numbers by figures as important as Leibniz and Cauchy is what strikes us most and makes us reflect on how distrust, by even competent people, sometimes delays the adoption of important and valuable content.

Appendix

From The Confusions of Young Törless by Robert Musil [10, pp. 104–107].

During the mathematics period Törless was suddenly struck by an idea.

For some days past he had been following lessons with special interest, thinking to himself: ‘If this is really supposed to be preparation for life, as they say, it must surely contain some clue to what I am looking for, too.

It was actually of mathematics that he had been thinking, and this even before he had had those thoughts about infinity.

And now, right in the middle of the lesson, it had shot into his head with searing intensity. As soon as the class was dismissed he sat down beside Beineberg, who was the only person he could talk to about such things.

“I say, did you really understand all that stuff?”

“What stuff?”

“All that about imaginary numbers.”

“Yes. It’s not particularly difficult, is it? All you have to do is remember that the square root of minus one is the basic unit you work with.”

“But that’s just it. I mean, there’s no such thing. The square of every number, whether it’s positive or negative, produces a positive quantity. So there can’t be any real number that could be the square root of a minus quantity.”

“Quite so. But why shouldn’t one try to perform the operation of working out the square root of a minus quantity, all the same? Of course it can’t produce any real value, and so that’s why one calls the result an imaginary one. It’s as though one were to say: someone always used to sit here, so let’s put a chair ready for him today too, and even if he has died in the meantime, we shall go on behaving as if he were coming.”

“But how can you when you know with certainty, with mathematical certainty, that it’s impossible?”

“Well, you just go on behaving as if it weren’t so, in spite of everything. It’ll probably produce some sort of result. And after all, where is this so different from irrational numbers—division that is never finished, a fraction of which the value will never, never, never be finally arrived at, no matter how long you may go on calculating away at it? And what can you imagine from being told that parallel lines intersect at infinity? It seems to me if one were to be over-conscientious there wouldn’t be any such thing as mathematics at all.”

“You’re quite right about that. If one pictures it that way, it’s queer enough. But what is actually so odd is that you can really go through quite ordinary operations with imaginary or other impossible quantities, all the same, and come out at the end with a tangible result!”

“Well, yes, the imaginary factors must cancel each other out in the course of the operation just so that does happen.”

“Yes, yes, I know all that just as well as you do. But isn’t there still something very odd indeed about the whole thing? I don’t quite know how to put it. Look, think of it like this: in a calculation like that you begin with ordinary solid numbers, representing measures of length or weight or something else that’s quite tangible—at any rate, they’re real numbers. And at the end you have real numbers. But these two lots of real numbers are connected by something that simply doesn’t exist. Isn’t that like a bridge where the piles are there only at the beginning and at the end, with none in the middle, and yet one crosses it just as surely and safely as if the whole of it were there? That sort of operation makes me feel a bit giddy, as if it led part of the way God knows where. But what I really feel is so uncanny is the force that lies in a problem like that, which keeps such a firm hold on you that in the end you land safely on the other side.”

Beineberg grinned. “You’re starting to talk almost like the chaplain, aren’t you?”