I don't know why this answer was downvoted. There is nothing wrong with it.

Agree. I upvoted it.

sometimes people downvote answer to the questions that are dupes or low-quality

The number 5 has the physical interpretation of being a certain amount compared to the other numbers? Of course agreed that it involves the creation of a second dimension, but what does i^2=-1 identity mean about the relationship between the two dimensions? How are complex numbers different to two normal dimensions?

This fails to answer the question. The basic units of physics are length, time, and mass. Real multiples of the base units have a physical interpretation. Complex multiples seemingly don't. Why do the complex numbers in the theoretical formalism disappear from the predictions? And if they disappear, why were they introduced in the first place? This question has a lot of specific answers in different contexts. "Complex numbers are used in math" is not an answer.

The identity $i^2=-1$ means that if you rotate a plane by $90^\circ$, and then by another $90^\circ$, the resultant effect is the same as if you reflected each point across the center of rotation.

Lee Mosher: Yes! This means physically that there must be two dimensions or properties AND that these dimensions or properties are able to effect each other? Of course, all physicists know this complex numbers are rotations answer... But that is sidestepping the question, giving me another question: What does it mean to say that two dimensions are connected in this way? The conversation below with Agnius Vasiliuskas perhaps shows this confusion many physicists have, perhaps people here can provide some deeper answers to clear it up better?

Having a physical interpretation of number 5 is easy. I have 5 apples. that number represents 5 apples on my desk. Try the same with $i$ apples. Hmmnoo...

To bring it to a simple example of running on a sports field. The person is running from one end to the other, how far do they run? With no imaginary component they can only run down the field, so the length of the field. Introducing an imaginary component is allowing the runner to run across the field as well, increasing the distance they run, they could switch directions or run against a cross wind or even turn their running direction in only one direction (circular), or they could increase their speed less as their angle to the original direction increases (exponential).

@benrg: Twodimensional numbers do not have a physical representation, but their individual dimensions both do. There is no physical representation of (x,y) other than representing x and y separately. The same applies to complex numbers.

There are several group-theoretical responses found elsewhere (e.g., physics.stackexchange.com/questions/32422/… and physics.stackexchange.com/questions/3503/…), but it ultimately comes down to complex numbers being convenient rather than necessary. In terms of physical interpretation: numbers are conceptual rather than physical. They can be used to describe physical things, but you shouldn't look for deep physical insights from the numbers themselves.

@benrg The basic units of physics are time, length, mass, current, temperature, amount of substance and luminous intensity.

@Richter65 The second write up from a quantum mechanics perspective fits well with the idea that complex numbers are used to analyse a multi dimensional system by taking one dimension (the real) as the main dimension, then just looking at the effect of all the other dimensions combined on the main dimension (imaginary). The effects in quantum mechanics being due to there being multiple dimensions that you can go away from the main dimension being studied and you could have taken one or the other or a combination. Does this fit with your understanding?

@OzOz I don't know if I would describe it that way, as a multi-dimensional system with a "main dimension". By construction, complex (with real & imaginary parts) Lagrangians are used in field theory to describe physical interactions (en.wikipedia.org/wiki/Lagrangian_(field_theory) ). The 2nd question was: can you write down a Lagrangian without complex numbers and use it to derive the same physical interactions described by the complex Lagrangian? The short answer is yes. So it was really about whether you really need complex numbers to describe physics in general.

@Flater You can of course define a complex number as a vector with two reals - but the vector would only be and behave like a vector in a complex plane. One of the components in a complex number does not behave as you would expect it to. Take the $\sqrt i$ for instance, it is not as well behaved as $\sqrt 1$. Sure, the geometric calculation of makes sense in the complex plane but not as "just another real in a vector".

@StianYttervik, A way to represent i apples would be to agree on a physical representation relative to 1 (or a unit) apple. One scheme would be to rotate the apple 90° from its standard 'unit' orientation. Thus, 1 apple is oriented (say) to the North, while i apples is one apple oriented to the East. This also extends well to rationals, such as 1/2 an apple, or i / 2 apples.

@DavidRTribble At the onset it looks easy enough. But lets do some maths on it. I would like to construct a square of 2x2 apples. Now, if I do that with the normal apples, I will find I have 4 apples in that square. How peachy. Or appley. However, if I do that with the rotated apples, I not only lost the apples I had, I am now 4 apples indebted. Darn. I should have left them alone.

@OzOz - numbers like the number 5 or 2.5 are inspired by everyday experience like counting things and measuring distances, but they are abstract mathematical concepts, no more real than complex (or poorly named "imaginary") numbers. The whole point of numbers is that they could mean/represent anything that has some aspect that is described well by operations defined for those numbers. 5 - 2 = 3 could be "given 5 apples, if you eat 2, 3 remain", or "5 steps forward and 2 steps back is effectively 3 steps forward" - but only because this models an aspect of those physical processes well.

@OzOz - Same thing with complex numbers. There's nothing mysterious about them, the word "imaginary component" is just a label, it doesn't really have a profound meaning attached to it. The abstract mathematical concept involves a set of properties (operations, behaviors, rules), and some physical phenomena have aspects that can be modeled (represented) using those. In particular, the * operation does a rotation and a scale - arguments (angles) are added, magnitudes are multiplied.

@OzOz - "'the resultant effect is the same as if you reflected each point across the center of rotation'; What does it mean to say that two dimensions are connected in this way?" - it just means that the physical object in question has a certain symmetry and behaves in a certain way, but that has everything to do with the object itself, and complex numbers are just a mathematically convenient way to describe that (you could avoid complex numbers and create an equivalent, but possibly clunkier mathematical model without them).

If they are just the same as a 2d vector then surely just using vectors would he the same? Symmetry is more interesting... What symmetry is set when you introduce an imaginary component?

@OzOz You're assuming vectors are simpler than complex numbers. Why?

@StianYttervik There's no such thing as "5 apples" in nature, is there? That's a human abstraction, itself built around our perception interpreting an apple as a distinct object. It happened to be convenient in the business of genes spreading themselves more efficiently, but it doesn't really mean anything physically. All you're saying is really "natural numbers are intuitive, complex numbers aren't". But human intuition is wrong. We know that. It works good enough for the kinds of problems we had in our ancestral environment. It breaks down when a lot outside of that scope.

@StianYttervik The same way, fractions don't really work for apples either. When you say you have three quarters of an apple, you don't actually mean a 3/4th of an apple - you mean "I split an apple into four parts, hopefully close to identical, and took three of those". That's very far from the mathematical sense of 3/4ths. And of course, nature doesn't (seem to) have real numbers either - heck, it doesn't even have arbitrary fractions, even in principle. You cannot divide "things" infinitely. In the end, the question is always the same - is this useful or not?

Complex numbers have an extra i, if it is irrelevant why invent complex numbers at all? Furthermore, often systems that i is used in have multiple dimensions, complex numbers only describe 2 dimensions? But they can be applied to higher dimension systems, I think Ive even seen them applied to one dimensional systems where the concept of it being 90 degrees loses all meaning?

The more this discussion has gone one the more sure I am that complex numbers are a different way of describing a multi dimensional system.... But clearly in complex numbers, all, all the other dimensions are described in a single dimension, the imaginary side is how the combination of all the other dimensions effect the real dimension being studied?

@Luaan I feel our discussion sliding into philosophy, but I am OK with that... The number 5 was conceived because there was a need to count apples (or whatever). That is why there is a number 5 in maths and why there are natural numbers. They certainly have a physical interpretation, because that is how they came to be. It is hard (to the point of futility) to try to frame some physical explanation for complex numbers because they are defined and they exist to do things real numbers cannot. Solve problems natural numbers cannot. (or takes a lot of work to do with naturals).

@Stian Is it that hard? Clearly complex numbers are a type of vector? Clearly they are a special type of vector that can only be 2d? They are a mathematical tool for dealing with vector systems? Now, all we need to answer is how they are different from normal vectors, apart from the fact that they are limited to 2D? which is a big hint?

@OzOz Sure, they can be seen as vectors, but only if the plane you put them in is the complex plane. You'll run into problems if you just accept them as vectors, because there are serious differences between $1$ and $i$. Consider $x^n$. For $x=1$ this is always $1$. For $x=i$ it is a repeating series $1,i,-1,-i$. But the multiplication rules are the same as for vectors, so if it is helpful for you to consider them as vectors - please do so =)

@Stian: Great, we are getting close to an answer now... If we can answer exactly what the difference that i makes then we are near to an answer. The other big difference is that complex numbers are limited to 2D. When we understand what this means, and I think we will soon, then we can actually understand complex numbers!

@Stian: I really hate when mathematicians say that there is no physical interpretation... Mathematics is just a precise language to describe things that happen in the world, and so must have a physical interpretation! Like any language, you can describe things which cannot happen, but when something has been as widely used as complex numbers have, it must be close to being how the world is and have a physical interpretation?

@OzOz Complex numbers aren't inherently 2D. Quaternions are 3D complex numbers. In fact, they came first as the way to describe 3D coordinates in physics; they have been displaced by vector analysis later. You could say they are an extension ("hypercomplex", instead of just i, you have i, j and k), but it's the same principle as with complex numbers. I think one of the very important points you're missing is that vectors are a much newer concept than complex numbers (and quaternions etc.). For some purposes, vectors turned out to be simpler, and became more popular in physics.

@Ozoz Myself I have accepted that many things have no physical interpretation. It took a while. Many of the most useful methods in mathematics rely on the fact that even if a problem is unsolveable, we can transform it into a problem that is. In this transformed state, it has no physical interpretation. This can be methods like using complex numbers, fourier transforms, laplace, etc. We can solve the problems there, and then transform the solution back. Many of these methods can be quite arcane from a "what is the interpretation" POV but as long as you follow the rules you get where you want.

@Luaan: You sound like you have good knowledge, I have got many not so good answers and still confused... What is the difference between complex numbers and normal vectors? And, in many e

@OzOz: "It's like English, if you are not describing things that have an interpretation then it doesn't mean anything."- You can describe dragons and sorcerers and magic in English, and yet, there's no real-world counterpart to those; there is a meaning/interpretation associated with those concepts, though.

@OzOz: Same with math; it doesn't necessarily represent reality. And, while some people may argue that it can express any possible reality, they can't argue against the notion that it's not restricted to describing our particular reality; in that sense, it studies its own abstract constructs.

"By definition, if you can describe a physical hapoening with maths then it must have a physical interpretation." - there is no such definition. Also, that's a logical fallacy. If some of math can describe physical phenomena, it doesn't follow that all of math does.

@FilipMilovanović: Dragons and sorcerers do have physical representations, they are not real ones but still they are a physical thing that you can describe. The language describing something real in the world is entirely different to it having a physical interpretation.