I’m preaching to the choir here, if to anyone at all, I suppose. Still, it should be said. Descartes coined the term “imaginary number” to deride the invention of a number such that squared becomes -1. The name stuck and has done countless harm to the notion, fooling pupil and school teacher alike into implicitly accepting the formalist position: “that which can be defined, precisely, exists. End of story.” It is no wonder that many have been suspicious of the invention of imaginary constructs that can do the impossible. I have a rhetorical question to justify imaginary numbers to my students. The question is:

Can you think of contexts where a particular class of number is inapplicable?

Think about this for a moment. By “class of number” I mean, for example, negative numbers. Negative numbers, we learn, are great for expressing debt. They apply perfectly well, also, to linear orders where there is no smallest element (temperature isn’t the best choice, since Kelvin has a smallest possible value… year is better, since although we estimate a beginning of the universe, we can’t count days since then, so BC years are appropriately negative). What negative numbers don’t seem to apply to are numbers of family members, times you can iterate a process, dimensions of a space, distances. It is no wonder they too were suspect when first conceived.

How about the class of numbers which are not 1? Possibly it makes no sense to talk of multiple universes (I am of this camp, nevermind the fashionable physics which suggests the idea). For if there is a vantage point from which to observe 2 universes, then causally the two are part of your universe, even if they are entirely independent of each other. We might (should) define universe to be all those things which cause change in what we can perceive (to my mind this is possibly an asymmetric relation: a realm can exist which is broadcasted to us, upon which we have no impact, so that it is part of our universe but we are not part of its). Then to any observer there is, by definition, but one universe.

For number of family members and iterations of a process, it doesn’t generally make sense even to apply positive rational numbers. Particularly if the process is instantaneous. Can you wake up 2 and a third times? Well, not without defining what it means to wake up a non-integer number of times, it certainly is not given. Is it okay that certain numbers have limited scopes to the context to which they apply?

The idea of iterating a process, when put in mathematical terms, gives rise to the old subject of solving functional equations. If I have a function f:R->R from the real line to itself, I can define the square of the function, that is f^2, to be the function composed with itself. Then I can ask if there exists a function g so that g^2=f, i.e., does f have a square root? If f is the function of sending x to -x then normally there exists no function which halves this process (for example, sending x to the average of x and -x certainly doesn’t work). For 1×1 matrices, multiplication and composition are the same thing. So for linear maps from R to R, finding half of the inverting map, [-1], is exactly finding a square root of -1. If you extend the real line to be sitting in a plane, then it becomes clear that you can rotate the plane 90 degrees in either direction to give a process which when iterated twice sends x to -x. This mysterious fake number, i, is nothing more than rotation of a plane by 90 degrees. Do rotations by 90 degrees exist? If anything does, they damn well do. Teachers have been heard saying “i is important because it solves x^2=-1, which you want to be able to do for engineering problems” It is a good teacher that says “i is important because it is a compass point for any object with a quality that takes values in the circle, which certainly includes many physics and engineering problems” So, any phenomenon involving waves involves complex numbers. If waves exist, complex numbers do too.

From above: let us know what it is we are doing with language, so that we may deliberately do much more of it.

Well, some internet searching reveals this is not a very original way of justifying them after all. So be it. Still, it seems like the right way.