Thursday, September 12, 2013

i Can't Stand It Anymore

Travis Kowalski—South Dakota School of Mines and Technology

Complex numbers are essential tools of mathematics, providing beautiful connections between arithmetic and geometry, algebra and trigonometry, number theory and analysis. Unfortunately, few people outside the cloister of trained mathematicians know this. I teach a course on complex analysis, and each time I am dismayed to find that, even after 15 weeks of demonstrating how the use of complex numbers fundamentally unifies most mathematical concepts learned in undergraduate studies, there is still a nontrivial subset of students who say, “That’s nice and all, Dr. K, but they aren’t real. They’re still imaginary numbers.”

By its very definition the lamentable word “imaginary” describes something that does not exist or is utterly useless. Of course, these derogatory implications were just what Descartes had in mind when he coined the term “imaginary number” in 1637. Two centuries later, Gauss advocated the term “complex number,” but Euler’s introduction of the symbol i means that, no matter whatever else we may choose to call it, the adjective “imaginary” will always be associated with the root of –1. Students know what the letter i stands for. To them, it is a number that is imaginary and therefore irrelevant.

It is a self–fulfilling—and sadly self–defeating—prophecy. And so, it is time to retire i.

If the previous plea of “pejorative prejudice” is a bit of a stretch (or at least, needlessly alliterative), allow me to strengthen it with a bone fide mathematical argument for retiring Euler’s chosen notation. In a standard presentation, the complex number \(a+b\sqrt{-1}\) is identified with the point (a,b) in the plane. The way complex multiplication is defined, the effect on the plane of multiplying by \(a+b\sqrt{-1}\) is exactly the same as left multiplying each point (written as a column vector) by the real matrix
a -b
b a
and so multiplying by the complex unit \(0+1\sqrt{-1}\) is the 90° rotation
0 -1
1 0

Whatever one wishes to label this matrix, the letter I is off limits because I always refers to the multiplicative “identity” matrix. This suggests that I should not be used for the complex unit. In keeping with a consistent lettering scheme, I should represent the complex number whose multiplication coincides with that of I, but that’s just the multiplicative identity—that is, I should denote the real number 1.

What would be a better symbol? Why not just dust off the old $\sqrt{-1}$ notation and use that? Unfortunately, this is a choice fraught with peril. Following the traditional algebraic “rules of radicals,” we end up with paradoxes like \[-1=\sqrt{-1}\sqrt{-1}=\sqrt{(-1)(-1)}=\sqrt{1}=1\] A better proposal comes from multivariable calculus. In the notation of ordered pairs, complex multiplication takes the form (a,b)(c,d) = (ac bd, ad+bc). Using this and vector algebra allows us to write any point in the plane as

(a,b) = (a,0)(1,0)+(b,0)(0,1)

where i=(1,0) and j=(0,1) are the standard basis vectors for the plane. Not only does this provide one more piece of evidence to support the claim that the symbol i ought to refer to 1, it also means (a,b)=a+bj. This looks exactly like the standard form of a complex number with the vector j standing in for \(\sqrt{-1}\). Even more compelling, note that

jjj = (0,1)(0,1) = (-1,0) = -1,

so j is indeed a square root of –1.

Consequently, we should denote the complex unit by j or, if we want to emphasize its role as a complex number rather than a plane vector, the italicized letter j. In fact, electrical engineers already use exactly this same letter j, although their prime motivation is that i is already reserved for current.

Our main motivation is that the letter j doesn’t stand for anything in particular, and it most certainly doesn’t stand for “imaginary.” The symbol simply denotes a complex unit, a number that multiplies against itself to yield –1. It is a blank canvas on which to paint the utility of the complex number system, effectively banishing the confusion and distrust of that other letter, which shall no longer be named.

Travis Kowalski teaches mathematics at the South Dakota School of Mines and Technology.

This article was published in the September 2013 issue of Math Horizons.