Why do imaginary numbers work (somewhat philosophical question)?

Question

Asking as a layman, I've always puzzled over imaginary numbers and how they can be used to solve problems involving real numbers or quantities only (e.g. contour integration methods or Fourier analysis obtaining the frequency domain). It seems strange how using an unreal number conjoined with the normal rules of arithmetic can sometimes produce results which are difficult or impossible using real variables/numbers alone. I don't know of any imaginary units of any physical quantity in the real world.

Could someone direct me to a reference (perhaps a philosophy of mathematics text) which explains how these complex analysis methods work? Maybe all these methods somehow concisely represent or parse certain lengthy or complicated real-number operations 'behind the scenes'? Another useful reference would be one which shows how one can obtain with real numbers only, any result obtainable using methods involving imaginary numbers.

Answer 1

When you are a little kid you learn to count and add whole numbers. In first grade, a problem like $3-5$ is said to have no answer, because you can't take $5$ apples from a pile of $3$. This makes total sense and you're ok with that. A little later you learn about negative numbers and at first you're like "whoa" but before long negative numbers are every bit as legitimate as positive ones.

Then you learn about multiplication and division and at first $13\div5$ doesn't have an answer because $5$ "doesn't go into" $13$. This makes perfect sense and you're ok with that. Then you start saying it's $2$ with remainder $3$ and finally you're told of fractions and $13\div5$ is a perfectly fine number called $2 \frac{3}{5}$ or $\frac{13}{5}$. It's a little strange at first but at some point you wonder how you could've ever gotten along without these fractional numbers.

Sometime later you run into problems that seem like they should have an answer like what's the length across the diagonal of a square but no fraction fits the bill. You're in good company as this bothered some pretty smart people over the ages. But we overcome the problem and add new numbers like $\sqrt{2}$ into the mix. It's a number whose square is $2$. This makes perfect sense and you're ok with that. These numbers solve equations like $x^2=2$ or any other equation you can make with integers and all the usual operations of arithmetic.

The $\sqrt{2}$ thing takes a little getting used to at first because it's 1.41421356237... and on and on infinitely and randomly and for the first time you're questioning the "reality" of such numbers. Yet surely the length of the diagonal is real so you accept it.

Around the same time you learn about $\pi$, another number with an infinite decimal, which is slightly more mysterious because it's got a greek name. Probably you're not told it doesn't solve any equation with integers and arithmetic operations but if you study math you find out it really is a bit weirder, and that weirdness has a name, transcendental. But you accept all these transcendentals like $\pi$ and $\sin 1$ and $\log 2$ because they seem to have a value somewhere on the number line.

At this point you think you're done because what else could there be? The whole line is accounted for. Equations like $x^2+1=0$ have no solution but you're ok with that.

You're introduced to complex numbers which solve equations like $x^2+1=0$ but they are presented as some kind of trick or device. Turns out they have all kinds of uses but they're usually not considered part of reality the way all the other numbers are. Why not?

It's often brought up that you don't need complex numbers, that you could do all the same calculations and analysis using real numbers. But that's like saying rational aren't necessary. Anything you can do with rational numbers you can do by carrying around pairs of integers.

To me complex numbers are as "real" as all the others, only more so. It just takes a little longer to accept them than it did for fractions, say. Fractions made you stretch your imagination a little. Complex numbers stretch your imagination more. That's part of makes them beautiful.

Answer 2

There may be dissenting comments on this but there is a sense in which complex numbers (and quaternions and octonions) are built into ordinary geometry and are not something that is just tacked on. This is done through Grassmann algebra, which may be the best axiomatic description of geometry. It is a graded algebra with scalars, points, vectors, bivectors, etc. and various types of products: exterior, regressive, interior, Clifford along with a complement.

Just to give a taste of this, here is an interior product of a bivector and a vector (both are basis vectors of the Grassmann algebra of the plane). The product is then simplified using the Euclidean metric and built-in rules of the algebra. The interior product is symbolized by a CircleMinus, which looks a little like a capital theta.

The result is a 90 degree counterclockwise rotation, so the bivector acts just as i does in the imaginary plane. Here it is applied four times to rotate around the plane.

The point is that these elements and products were designed for a complete algebra for ordinary geometry and yet they have built in operations that act like complex arithmetic. This makes complex numbers seem less artificial. It is possible to add generalized Grassmann products, hypercomplex products and then more formally build complex, quaternion and octonion algebras.

Some taste for this can be obtained in Grassmann Algebra Volume 1 by John Browne. (But the generalized Grassmann products will appear in Volume 2.) or books such as Geometric Algebra for Physicists by Chris Doran & Anthony Lasenby.