I re-read some of Isaac Asimov's non-fiction lately. I started with
Left Hand of the Electron, which was a fun look at which bits of our
univers aren't quite the same as their mirror images, and related
matters. The next I lighted upon was
of the moon. Each of these volumes (which I read as a teenager) is a
series of essays, grouped so that the book is divided into parts by topic.
They're a little out of date on some details, but have aged toleraly well, for
Asimov on Numbers (a collection of essays previously published
separately), Isaac manages (Chapter 8: The Imaginary That Isn't) to fumble badly
on the task of showing the
reality of the square root of minus 1. A
sociology professor had called mathematicians mystics because they believed in
the square root of minus one and Asimov, as a student, had tried
(unsuccessfully) to refute this. As an essayist, revisiting the episode, I'm
afraid he doesn't do much better.
He does at least manage a tolerably good introduction to simple polynomials.
When he comes to look for zeros of the polynomial x.x +1, however, he boldly
If no positive number will do and no negative one either, it is
absolutely essential to define a completely new kind of number; an imaginary
number, if you like; one with its square equal to −1. If such is the
quality of grounds the sociology professor had been given for believing in the
square root of minus one (which, indeed, I consider quite likely), he had every
reason to regard it with contempt. Isaac goes on to introduce an arbitrary
arithmetic on the points of the compass, which makes East the positive real
direction and North the positive imaginary one; but this is no better. Making
up the rules of a game as you go along is no answer and the sociology professor
had every reason to decline to play.
One can arrive at the complex numbers in various ways, without playing make-believe or defining an imaginary thing and making up rules about how it behaves. One can consider the composition of angle-preserving linear transformations (scalings, rotations and their composites; excluding reflections as angle-reversing rather than angle-preserving) of the two-dimensional plane; or one can consider the arithmetic of polynomials when any two that differ by a multiple of x.x +1 are regarded as equivalent (just as a clock represents times the same way when they differ by multiples of the twelve or twenty-four hours that the clock takes to repeat itself). Anyone who accepts the reality of angle-preserving linear transformations of the plane or polynomial functions can then be shown a concrete square root of minus one – in one case, the quarter turn; in the other, the free variable x of the polynomials.
Sadly, this is not how the subject is usually taught in schools – I cannot blame Isaac for trotting out poorly-motivated make-believe (the approach taken by even the usually excellent 3blue1brown) or making up the rules as he goes along, when these are indeed the ways the subject is commonly introduced by mathematics teachers. Both he and the sociology professor were badly served by teachers too willing to make up a formalism without motivating it.
But then, in Chapter 9,
Forget It, he redeems himself admirably, if a
little stridently, by exhorting the anglophone world to give up
its silly system of units and reform its
We cherish our follies only because we are used to
them remains all too true to this day. He begins the chapter with the
daunting thought that, as humanity learns steadily more about the world, there
is ever more for each generation of students to learn; he ends with the
We must make room for expanding knowledge, or at least make as
much room as possible. Surely it is as important to forget the old and useless
as it is to learn the new and important. I couldn't agree more; and it is
also important to find ways to think about what we do understand that are more
accessible and easier to learn.