I was a bit sceptical to hear from a colleague that imaginary numbers,i.e. the even root of a negative number, are actually used in some branches of science. Whats going on here?

Where to begin? And how much technical detail to include?

Yes, imaginary numbers (more accurately, “complex numbers”) occur frequently in science, particularly in physics. The general form of a complex number

**z** is

**z** =

**x**+

**i**∙

**y** where

**x** (the “real component”) and

**y** (the “imaginary component”) are themselves ordinary real numbers, and

**i**^{2} = –1 (NB: this is not the same thing as saying

**i** = √–1, a point that often confuses people at first). One can plot such complex numbers on an ordinary

Cartesian (**x**–**y**) plane using the

**x**-axis for the real part and the

**y**-axis for the imaginary part, resulting in a so-called “

Argand diagram.” In this view, multiplication by

**i** corresponds to an anticlockwise rotation through 90°. This is perhaps clearer when it is pointed out that four such rotations equate to 360°, which leaves the rotated thing unchanged – but

**i**∙

**i**∙

**i**∙

**i** =

**i**^{4} = +1, i.e. the identity operation (multiplication by +1).

Together with the

Euler identity (

**e**^{i∙θ} = cos

**θ**+

**i**∙sin

**θ**), the above allows a more general notion of the so-called

Fourier transform. Fourier transforms are very useful tools for analysing all manner of wave-like phenomena, including sound, light and quantum state functions (more on this last anon). It is also worth noting that complex numbers are not just even roots of negative numbers. For example, –8 has three cube roots (–2, 1+

**i**∙√3 & 1–

**i**∙√3); +32 has five fifth roots (2, 0.618+

**i**∙1.902, –1.618+

**i**∙1.176, –1.618–

**i**∙1.176 & 0.618–

**i**∙1.902), and in general, any number has

**n**,

**n**^{th} roots.

Technically speaking, Fourier transforms allow (in addition to a few other magic-like tricks) physical problems to be ported between the

time-domain (our ordinary clock-based conception of reality) and the

frequency-domain (where frequency distributions and spectra hold sway). The frequency content of a particular problem often reveals subtleties that are not otherwise (easily) detectable. Also, the mathematical formulations of a vast array of physical problems involve assorted differential equations. General and/or

analytic solutions are in many cases facilitated by the use of various transforms, usually of the Fourier kind. Moreover, complex numbers arise naturally in the

general solutions to several classes of

differential equation even without applying any transforms.

The general state equation of a quantum mechanical system is the solution to

Schrödinger’s equation, which already invokes complex numbers in order to provide a full-phase description. A solution to the Schrödinger equation will, except in the most trivial cases, involve complex quantities. The trouble is an interpretational one: nobody knows exactly what is described by the two components (real and imaginary) of such a solution. The

magnitude (= |**z**| = √(**x**^{2}+**y**^{2})) is usually interpreted as a

probability density function, which seems to accord with observation, but it is not at all clear that that is really what it means. Nor is it clear what the “phase” part (i.e.

**θ**) designates. These particular problems await the next Einstein for their proper solution and/or contextualisation.

In summary, complex numbers are very helpful tools, even indispensable ones for shedding light on assorted physical phenomena, although the interpretation thereof can in some cases be problematic.

I hope that the above hasn’t left the reader more confused than before…

'Luthon64