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Note: this is not a realism-esque question on the reality of numbers. Also, to not be confused with this question, I'm not questioning the usage of these numbers.

As far as I know, every numerical field has a correspondence to reality - e.g. the natural numbers can be represented by counting apples (1, 2, 3 apples), and even the irrational numbers correspond to phenomena we find in nature (the ratios that pi and phi represent correspond to real phenomena).

But what would the imaginary numbers correspond to in nature? Can we find any phenomena that represents them? I'd like to distinguish between the complex numbers that unities the imaginary part with a real part, thus perhaps providing it a real correspondence, e.g. in certain representations of QM (although perhaps even complex numbers are problematic in this sense). A complex number, represented as the combination of the real part (a number from the ℝ field) and an imaginary part (i). My emphasis is on the numbers that appear in the imaginary part (the so-called "imaginary numbers", square root of -1).

Furthermore, if these kind of numbers cannot be represented in nature, does it mean they have an expiration date in our usage of them outside of math? Are they simply place-holders for the next mathematical discovery that does correspond to nature and that would replace them? (i.e. isn't it an aspiration of the natural sciences to prefer correspondence to nature?)

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    Already asked yesterday – Mauro ALLEGRANZA May 3 at 9:02
  • I am curious as to what π corresponds to "in nature", the ratio of the circumference to the diameter of platonic circles? In that case, complex numbers correspond to the amplitudes in AC circuits and wave functions of electrons. Or rather to their platonic versions that we put into our math. If the distinction in the middle paragraph (it is very murky) is between complex numbers and real pairs then it is really complex numbers that are used their, not real pairs, the algebra depends on it. And there is no expiration date on any math, corresponding to nature is not its job. – Conifold May 3 at 9:22
  • @MauroALLEGRANZA I have not seen that question, thank you. But I do want to emphasize on the distinction I made between complex and "merely" imaginary, which is not apparent in that question; hence I'd like to keep this question open (unless you'd suggest me to edit this into the other question which I think won't be a very good idea). – Yechiam Weiss May 3 at 9:37
  • @Conifold I'll attempt to better describe the distinction, which would answer your first sentence. Regarding the last sentence, I'm talking about sciences like physics, chemistry etc; not mathematics. – Yechiam Weiss May 3 at 9:38
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    I also meant math in natural science. It is a system of modeling tools used in them, like alphabet, only more complex. There was a time when negatives and irrationals were "absurd" and "impossible", de Morgan denounced negatives as late as mid 19th century. Now they are "real" because we are used to them enough to look through to what they are used to model. The power of habit, the difference is purely psychological. AC amplitudes and wave functions take purely imaginary values just as well, push them down the curriculum enough and they will "correspond to nature" too. – Conifold May 3 at 10:39
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Complex numbers are very commonly used in the field of electrical engineering as a computational tool for solving problems in alternating-current circuits. In this field, the complex number formalism allows convenient representation of impedance, which is frequency-dependent resistance to the flow of electric power.

Although complex numbers do not see common use in everyday life, their application in AC circuit analysis is as "real" as anything is in the world of engineering.

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The complex numbers are just the regular pair of real numbers in ℝ², but with a certain rule for multiplication and division. There are 3 reasons why they don't come up often in nature:

  1. They are a pair of numbers. We usually operate with numbers in ℝ like length, magnitude, and counting, but numerical considerations of 2D space are not as common.

  2. Finding the relation √(−1) = i in a natural situation requires a situation for calculating powers/roots, which is not common in everyday tasks let alone for 2D space.

  3. The complex numbers arise naturally when you multiply/divide vectors in ℝ² in such a way that their magnitudes are multiplied/divided and their angles with respect to a reference are added/subtracted respectively. I'm struggling to find an example of this in typical highschool/undergraduate mechanics. Usually we multiply magnitudes (i.e. lengths to get an area) or a vector times a scalar (velocity multiplied by time). In the few cases where 2 vectors are multiplied like in torque, the multiplication is a special case not following the rules of above.

You could definitely come up with a super artificial example that boils down to complex numbers though as long as it satisifies the rules in (3).

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