Are imaginary numbers real?

[Rigil Kent]

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?

Mintaka

[AcinonyxScepticus]

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.

James

[bluegray]

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

[Mefiante]

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 (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| = √(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…

'Luthon64

[Mefiante]

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.

'Luthon64

[patchwork]

best thing I ever read on this and some other wierd mathematical concepts was "Zero - the history of a dangerous idea"
some of the stuff makes no sense to me, but we wouldnt be able to do things like computersand space travel etc without maths using complexe numbers etc.

[Rigil Kent]

I'm always amazed at the wealth of information you guys can post off the cuff!

Perhaps it's all about the meaning of the word "imaginary" that put me off the concept and lead to this excessive scrutiny

and

The trouble is an interpretational one: nobody knows exactly what is described by the two components (real and imaginary) of such a solution.

Exactly. Irrational numbers with their infinite number of decimals, and to a lesser extent negative numbers, still makes some sense because they are at least imaginable. One can imagine infinity, and a bank overdraft. Imaginary numbers, however, is anything but imaginable.

Perhaps because I'm no expert and don't ever use them, I find imaginary numbers have the same incredible aura surrounding them as, say, astrology or claims of holographic universes. I guess it just shows how careful one must be before dismissing strange ideas. Sometimes we just have to trust authority!

Luthon - holy smoke. I'm going to need some more time with your post to come to terms with it. Wll study it carefully tonight. Many thanks.


Mintaka

[Spike]

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

Luthon - holy smoke.

Luthon, are you real?

[Mefiante]

Exactly. … Imaginary numbers, however, is anything but imaginable.

Well, my “The trouble is…” explanation that you cited was with particular reference to the QM wave function problem.  In other problems, for example resonance in mechanical systems and fluid or electrical circuits, the meaning of the imaginary terms is clear: they represent phase shifts or angular displacements.  However, it is still the case that it is hard to fathom a geometrical sense for them.  It is often helpful to think of complex numbers as two-component vectors, and i as an operator (like “+”) rather than as a number, and in this way complex numbers become more of a notational convenience than abstractly perplexing quantities.

Mostly, scientists simply use complex numbers because of their power and utility, and because these numbers are a scientific staple.  They only start worrying about the meaning of complex numbers when they arise essentially unbidden in the treatment of a problem or when, as in the QM case, it is entirely obscure what physical quantity a phase displacement applies to, as well as how.  In some ways, it is comparable to a reversal of our everyday use of arithmetical notions like multiplication and division.  Some of the profoundest mysteries about natural numbers (and primes in particular) find shelter there, yet everyone familiar with arithmetic will confidently assert that they properly understand those ideas.  The interested reader is invited to investigate the so-called Riemann hypothesis to establish just how deep the connections go.

Luthon, are you real?

I’m certainly not imaginary, but I’ve been called “complex” at various times.  I suppose that means I inhabit a hybrid, phase-shifted shadowland in a plane that covers the real and the imaginary…

'Luthon64

[Tweefo]

I've tried to explain to my bank manager that my overdraft is imaginary but he doesn't want to believe me.

[Rigil Kent]

 

I've tried to explain to my bank manager that my overdraft is imaginary but he doesn't want to believe me.

  ROFL 

[Mefiante]

Here is a very worthwhile and easy-to-follow introduction to the topic of complex numbers and how they find practical application.  The geometrical interpretation that is given is especially lucid, and none of the material requires anything more than high school algebra and a bit of trigonometry.

The Wikipedia entry on the “imaginary unit” i is also helpful, though a bit more challenging because it is necessarily brief and barebones.

There is an analogy with the real numbers.  What is the physical meaning of a negative number?  You can’t, for example, have –3.7 apples (which is a convenient notation that indicates that you owe 3.7 apples, but –3.7 apples simply don’t exist physically), and nor can you divide them among –5.21 people, yet we are happy to accept and perform such mathematical constructs and operations without worrying too much about their correspondences with reality.

From a mathematical perspective, the set of complex numbers is just the most compact set that has the complete set of real numbers as a (proper) subset, and which also includes a solution x = ±i to the equation x²+1 = 0.

There are in fact infinitely many algebraic structures (“number fields” and “rings”) in which the equation x²+1 = 0 has a solution – in other words, where there is at least one x so that x² = –1.  For example, in modular arithmetic with a modulus of 5, x²+1 = 0 is solved by x = 2 or x = 3.  With a modulus of 73, x = 27 or x = 46 solve the equation (^* — see below for explanations).  (Can you spot the relationship between the two solutions in each example?)  In this sense, –1 can be said to have square roots in certain modular rings, but it should be noted that not all moduli admit a solution and the moduli need not be prime numbers.

'Luthon64


^* 2²+1 = 5 ≡ 0 (mod 5) because 5 leaves a remainder of 0 when divided by 5.  Similarly, 3²+1 = 10 ≡ 0 (mod 5) because 10 leaves a remainder of 0 when divided by 5; 27²+1 = 730 ≡ 0 (mod 73) because 730 leaves a remainder of 0 when divided by 73, and 46²+1 = 2,117 ≡ 0 (mod 26) because 2,117 leaves a remainder of 0 when divided by 73.

[Rigil Kent]

(Can you spot the relationship between the two solutions in each example?)

the modulus value = sum of its solutions?

Thanks for the handy links Luthon. I'm learning a lot today! The New Zealand site is particularly lucid.

Mintaka

[Mefiante]

the modulus value = sum of its solutions?

Yes, exactly — well done.  The proof of this general property of modular arithmetic is almost trivial, i.e. that if a² ≡ b (mod n), then (n – a)² ≡ b (mod n) as  well, and thus a² ≡ (n – a)² (mod n).  Another way of looking at it is that –a ≡ (n – a) (mod n), meaning that, as is the case in usual arithmetic, √b ≡ a or –a (mod n).  Bearing the focus of this thread and the basics of modular arithmetic in mind, we have merely restricted ourselves to b ≡ –1 ≡ n – 1 (mod n).

All of this is merely to show that √–1 depends on some context and is not as straightforward as a first glance would indicate.  Also, I hope that this captivating topic is not a source of boredom because, in truth, we have barely scratched the surface.

ETA: It occurs to me that perhaps I am taking too much for granted, but I do hope otherwise.  In particular, I assume that it is understood that the square root of any negative integer is reducible to the problem of √–1.  To illustrate the point, √–7 = √7∙√–1, where √7 should present little conceptual difficulty.  Moreover, I take it for granted that the reader can see how Euler’s formula facilitates the extraction of any r^th root of any real or complex number if r is a rational number, i.e. extressible as a ratio of two integers.

'Luthon64

[Rigil Kent]

Came across this podcast on imaginary numbers. See the one dated 23 Sep 2010. Some nice history in it.


http://www.bbc.co.uk/podcasts/series/iots/all