[mjfisher]

I used to wonder if imaginary numbers were "real" (in the common sense, rather than the mathematical one). This only intensified after I learnt that they're required in quantum mechanics to explain any physical behaviour at all.

Now I consider them to be just as "real" as the integers - which is not at all. Both just human-invented concepts with no fundamental physical basis.

As you point out - useful, though!

[dotnet00]

I think "real" in the common sense is just too vague to really apply to things as abstract as numbers. 'Real' numbers seem real in a common sense because counting is a universal human experience. Adding in the imaginary plane allows for "counting" systems which have 2 interconnected components like waves, which also turn out to occur very often when you look at nature a bit more closely. Thus why you can derive all of trigonometry from Euler's formula. It just isn't a universal human experience to look at nature that closely yet.

[adrian_b]

I do not believe that it is possible to claim that integer numbers or imaginary numbers do not have a fundamental physical basis. The small integer numbers are not invented by humans, because many other animals can count up to some small number (like 5 or 6).

Both are abstractions. That means that they are properties of real physical objects, which are obtained by ignoring all the other properties of those physical objects that are irrelevant in the context of the application.

Therefore an abstract property is an equivalence class of physical objects, where all their other properties are ignored, so they are equivalent if they have the same value for the property of interest.

Non-negative integer numbers are equivalence classes of collections of physical objects, integer numbers are equivalence classes of pairs of such collections.

The imaginary unit is the equivalence class of all rotations by a right angle in the 2-dimensional space. Humans, like many animals, have an innate ability to recognize right angles, like also certain small numbers, so looking around you can perceive as easily all imaginary units like all numbers 3.

The complex numbers are the equivalence classes of all geometric transformations of the 2-dimensional space that can be decomposed in rotations and similarities (a subset of the affine transformations). In contrast, the 2-dimensional vectors are the equivalence classes of all translations of the 2-dimensional space (another subset of the affine transformations).

All the things that are equivalent from the point of view of an integer number or a complex number, so they are the basis from which such numbers are abstracted, are things that you can see with your own eyes in the physical world (similarity transformations appear in optical projections, e.g. in the shadows of physical objects, and the eyes are based on them).

[diedyesterday]

I don't think complex numbers are entirely without "fundamental physical basis"; Distinct antiparticles (antimatter) could be said to be the manifestation of complex numbers in the physical world. The important thing to keep in mind about complex numbers is that they have 2 degrees-of-freedom (DoFs); Analogously if a quantum field has two DoFs (that couple in a certain way) it is said to be a complex field and it will have two kinds of distinct particles.

Real-valued (operator) fields (like the photon field) do not have distinct particles (a photon is its own antiparticle)

I can elaborate more if needed.

[Arech]

Much earlier than quantum mechanics, they are extremely useful in electrical engineering. And even before that (historically) - their 4D generalization, quaternions, are extremely useful to describe 3D rotations.

[77pt77]

Complex numbers are also useful to describe rotations on the plane, and conformal transformations more generally.

[77pt77]

> I learnt that they're required in quantum mechanics to explain any physical behaviour at all.

You can do QM without complex numbers as people are used to use them.

But it gets really awkward really fast.

[diedyesterday]

And even then it's only a disguise. The two degrees-of-freedom which a complex numbers has, are still there (and QM needs it for interference, etc.). There is not way getting around the number of degrees-of-freedom of a system or a theory. It's as "real" as it gets.