Are imaginary numbers real?

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Rigil Kent (March 19, 2009, 08:49:37 AM):
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?

Surely something like (-9)1/2has no real meaning?


AcinonyxScepticus (March 19, 2009, 10:02:55 AM):
There is a compelling argument that only Whole Numbers (you know; 1, 2, 3, etcetera, excluding infinity) are the only numbers that truly exist. If we look at counting rocks, there are either rocks to be counted for which we can use whole numbers, or there is no point to counting. We as human beings invented the zero and thus the Natural Number system because there are a lot of interesting things we can do with zero. We then went on to invent negative numbers, if you think about it in real terms, there is no way that the world needs negatives (what would -6 rocks mean?).

Once we realised that certain concepts cannot be expressed in discrete terms on the discrete number line, we needed to use Rational numbers too (those rational numbers which are not Natural Numbers). Although there is no such thing as half a rock, if we look at dividing 2 kilograms of biltong between 7 people we know that we need to represent that concept in a way that doesn't use whole numbers (for the mass of biltong each person gets). If we look at irrational numbers, we adopt the concept so easily even if it is a slightly squirmy concept. In our limited decimal notation there is no way to accurately represent (2)1/2 and yet we accept its reality so easily. If you think about a square drawn on a page which is one unit by one unit in dimensions, the diagonal of that square cannot be represented accurately (using the same units) using decimal notation. We can see the length of the line, it definitely has a beginning and an end, and yet we cannot write down the length of that line using decimal notation. Why do we accept the square root of 2 so readily?

It amazes me how humans have so naturally taken to these concepts (zero, negative numbers, rational numbers, fractions, irrational numbers) and some of us have a problem with imaginary numbers. I too have struggled for a long time with the concept in practical terms (and managed somehow to pass pre-calculus mathematics) but only on reflecting on the other mathematical concepts mentioned above did I realise that I shouldn't resist it as much as I have in the past.

Perhaps it's all about the meaning of the word "imaginary" that put me off the concept and lead to this excessive scrutiny (that was not applied to the other numbers), had the numbers been called something else, perhaps I would have accepted it much more easily? But that's a moot point because we can't go back in time, rename the number system and wait to see if this discussion would still have arisen.

bluegray (March 19, 2009, 11:06:27 AM):
I agree, the term imaginary is a bit unfortunate. All numbers are just a handy concept to be able to do calculations and understand the relationship between real world phenomena. AC current power calculation makes use of complex numbers, where the reactive power is the imaginary part of the complex number. This doesn't make reactive power any less real than 'Real Power' ;) See Real, reactive, and apparent power for more.

Just like there are no real 1's and 0's on a hard drive, it is still useful to represent the magnetic fluctuations as 1's and 0's.
Or take the ascii value of a character. The number 0x5A in hexadecimal and the number 90 in decimal all represent the same ascii character 'Z'. Depending on the application, it is sometimes easier to work with hexadecimal than decimal, but they still represent the same 'Z'.

The mathematical concept of imaginary numbers might not be intuitive, but it allows us to model and predict how the real world will behave.
So yes, they are useful and real ;D
Mefiante (March 19, 2009, 11:12:00 AM):
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 i2 = –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 = i4 = +1, i.e. the identity operation (multiplication by +1).

Together with the Euler identity (ei∙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, nth 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| = √(x2+y2)) 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… ;)

Mefiante (March 19, 2009, 11:25:23 AM):
There is considerably more mindbending lying in wait for a casual contemplation of quaternions and their role in newish complex algebras that also find practical use in physics, particularly cosmology and – again – quantum physics.



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