square root

The Square Root of Complex Conjugation - A Puzzle

John Baez

October 24, 1997

When I teach complex analysis, I like to motivate the complex numbers as follows: multiplication by the real number x is a linear transformation of the real line which amounts to dilating by a factor of x when x is positive, but dilating and reflecting when x is negative.

Thus, if we seek a square root of -1, we seek a linear transformation of the real line which, when done twice, amounts to a reflection.

But there is no such thing.

To save the day, we need to go to the plane and consider rotations. A rotation by 90 degrees, done twice, amounts to a reflection through the origin. Voila: i is born!

Unfortunately, when i was born, so was its evil twin, -i. One corresponds to rotation 90 degrees clockwise, while the other corresponds to rotation 90 degrees counterclockwise. For some strange reason the usual convention is that the sinister -i corresponds to a clockwise rotation, while i is condemned to rotate things counterclockwise. Perhaps there was a mixup of some kind. Of course, there is an obvious excuse for this error: there is a symmetry of the complex numbers, namely complex conjugation, which interchanges i and -i, so it's a bit tricky to tell which is which.

In fact, this clockwise/counterclockwise business is probably a indication that we've made a horrible mistake all along: what we've been calling "-i" is really i, while "i" is -i! But I'm afraid it's too late to undo the damage: if we switched notations now, the confusion would be immense. (Or would it...?)

Anyway, even sticking with the usual notation, this complex conjugation business is a bit peculiar. After all, we started by considering reflection as a linear transformation of the real line which switched 1 and -1, and seeking a square root of this operation. To solve this insolvable problem we went up to two dimensions, invented i, and patted ourselves on the back for being so clever. But now we have i and -i and a reflection which switches them, namely complex conjugation! So in a sense, we are stuck with the same darn problem we started with! What's the square root of complex conjugation?

Of course, complex conjugation isn't complex-linear, but it is a real-linear transformation of the plane, so we can't blithely say that seeking a square root of complex conjugation is a much stupider idea than seeking a square root of -1 was.

Let's see. What's going on? We started with the real numbers, and then we invented i and considered the complex numbers

a + bi

with a,b real. And now I guess we are saying that we should give complex conjugation equal status and find a square root for it.

Let's call complex conjugation "k", as an abbreviation of the German "Konjugation". I guess we are working ourselves into the uncomfortable situation of studying numbers of the form

a + bi + ck

with a,b,c real.

Hmm. Adding these is a snap, but how do we multiply them? If we complex conjugate twice we get back to where we started, so I guess we have:

k2 = 1.

The big problem is multiplying k and i. What's ik? And what's ki? Well, i corresponds to rotating 90 degrees clockwise, and k corresponds to reflecting across the x axis, so if we do first i and then k we get... well, some operation or other. Let's call it j. Following the usual mathematical practice of writing everything backwards (which may have been responsible for the previous mixup somehow) we express this as follows:

ki = j

If we do i and k in the other order, we get some other operation, and after six days of intensive computation one can easily check that this other operation is just the same as doing j and then doing a reflection through the origin. In other words:

ik = -j

Hmm, this is starting to look like the quaternions! But there's a serious problem, namely that k squared is 1 instead of -1. How about j squared? Well,

j2 = kiki = -kkii = 1.


And we still haven't even begun to tackle the actual problem: to find a square root of complex conjugation! I guess for that, we'll have to make up some new gadget... I'll call it h, since l looks too much like 1 and things are already sufficiently confusing. So we want:

h2 = k

But what does this all mean? What's going on?

I leave it as an open-ended puzzle for everyone....

(P.S. - You don't need to tell me the answer. I already know what I think the answer is.)

And we now see that as the factor negative unity simply reverses a line, while the square root of negative unity (employed as a factor) turns it through a right angle, the one operation may be looked upon as in a certain sense a duplication of the other. In other words, twice turning through a right angle, about the same axis, is equivalent to a reversal; or, negative unity, being taken to imply reversal of direction, may be considered as rotation through two right angles, and its square root (the ordinary imaginary or impossible quantity) may thus be represented as the agent which effects a certain quadrantal rotation. - Peter Tait, in his obituary of Sir William Rowan Hamilton.

© 1997 John Baez