Numbers are just mathematical objects which satisfy a few interesting properties that lead to very rich structures we can study. Some of those structures map onto concepts in nature, which makes them actually useful.
Instead of real and imaginary or complex we could have chosen entitirely different adjectives. They're completely arbitrary. They don't matter.
On the other hand, there is no straightforward intuitive mapping for complex numbers. Geometrically, they arise out of the spiral motion needed to provide a continuous solution to an oscillating function such as y=(-1)^x, but there is no regular everyday situation that mirrors that. Algebraically, they arise in solving certain polynomials, but in practical situations they are generally only a required intermediate calculation, and in practical (engineering) usage it is usually only the real roots that have any meaning.
Complex equations are used greatly for modeling waveforms, but that's mainly just because it's more mathematically convenient than dealing with a bunch of sin() functions. Not because waveforms are inherently complex/imaginary -- they're not.
So the idea that negative numbers are just as unintuitive as complex numbers, is an idea I think should be firmly rejected. Negative numbers make easy, intuitive, real-world sense in a way that complex numbers simply don't.
There's still an ongoing philosophical debate as to the "reality" of complex numbers, and the formulation of QM plays a part in that debate. But it doesn't answer it -- similar to waveforms, we can argue whether the math behind QM is "essentially" complex, or if we use complex representations merely for convenience. It's entirely possible to express QM without complex numbers at all, obviously.
Au contraire! Complex numbers are just rotations, translations, and scalings in the plane. In the same sense that real numbers capture 1D rigid transformations and scaling, complex numbers capture 2D rigid transformations and scaling.
When you treat negative numbers as "distance in an opposite direction", you're more generally treating real numbers as a transformation of the 1D line, i.e. translation (addition) or scaling (multiplication) by some amount. Try thinking about the same but for the 2D plane and complex numbers. Addition is 2D translation and you'll see that multiplication corresponds to a rotation and a scaling.
The first section of Needham's "Visual Complex Analysis" walks you through this perspective in detail, if you're interested and also want lots of good exercises.
When engineers need to handle rotations, translations, and scaling, they don't use complex numbers. They use vectors, usually.
There's no intuitive mapping for the concept of the square root of negative one in real life. Not in the way there is for negative numbers.
And complex numbers are not about 2D geometric representation in general, the way vectors are. They are much more specifically about rotation or spiral motion.
I really think you might be a victim of bad pedagogy. Admittedly, the notation confuses complex numbers as points vs. complex numbers as operations. Let me drive home the point. You can represent complex numbers as matrices:
e := [[1 0] our identity element, i.e. what you typically write as 1.
[0 1]]
i := [[0 -1] our imaginary unit
[1 0]]
<e,i> = [[0 0] the inner product of e and i
[0 0]]
Now, using the above matrix representation uses 4 parameters, but you really only need 2. Indeed, any general matrix in our 2D algebra looks like this
a×e + b×i = [[a -b]
[b a]]
However! Even though these are 2D things, it's very important to not confuse them with the 2D vectors they manipulate. The matrix representation above makes that more manifest. Said another way, it doesn't make sense to compose points in space, but it does make sense to compose operations which operate on that space.
It's an unfortunate and confusing quirk that we often write a+bi to mean a 2D point in space, a 2D translation operation, or a 2D rotation and scaling, despite all these really being completely different things.
If you are trying to factor a polynomial with complex roots, is not a coincidence they appear in pairs which cancel out eventually, if you want to get back to a "real" value implied by the polynomial.
1. Do numbers exist? Have you seen a number in your life?
2. What does exist, can you name one thing? Because everything isn't what it seems.
Can you elaborate more on what you mean?
Why are complex numbers “unexpected”?
So much of contemporary physics is about distillation. You care more about things like consistency and simplicity over readability.
Our current methods of solving these equations, vector calculus, requires certain operations, specifically, square roots and derivatives.
To be able to utilize the well studied methods of linear algebra we need to make ample use of the Pythagorean theorem that ultimately uses a square root. To ensure that our model works in all domains we have to use a form of the square root that allows negative numbers. Complex numbers.
It’s why we use e^(iHt/h_bar) in the Schrödinger equation. Rather than see it as “nature works by using exponentiation!” I see it as “exponentiation is ‘stable’ in the face of complex derivatives”, ie the derivative of the exponential function is the exponential function f(x)=e^x == f’(x)=e^x. So this formulation simplifies our calculations.
These things can be calculated with any coordinate and numbering system, the ones we choose just help make the problems more tractable for us.
In standard quantum mechanics, the state of a quantum system is represented by a complex-valued wave function or a vector in a complex Hilbert space. Observables are represented by self-adjoint operators acting on this complex Hilbert space. The complex nature of the wave function gives rise to the phenomenon of quantum interference and the probabilistic interpretation of quantum mechanics.
However, in this real quantum mechanics, the complex Hilbert space is now replaced by a real Hilbert space, and the complex wave function is replaced by a real-valued wave function or a vector in this real Hilbert space. The observables are represented by self-adjoint operators acting on the real Hilbert space.
One way to achieve this is by using real algebra or real matrix representations. Now we let the complex numbers represented by 2x2 real matrices of the form:
a + ib = [a -b]
[b a]
i = [0 -1]
[1 0]
More importantly (as is pointed out in the lecture), every time-evolution of a quantum state from A to B can be represented as a unitary operator, and if you want to take the square root any such operator, you must in general use complex numbers (or something equivalent). Taking the square root like that is a very simple operation that answers the question "what is the operator for the evolution to the point halfway between A and B?"