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Comment by rhdunn | Hacker News Reader

IIUC, complex numbers are a number system that supports rotations -- one representation is as an angle and a magnitude. As such they work well at describing systems that have rotational components. This makes them useful for working with waves like in QM (light, etc.) and Fourier transformations/analysis (sine waves) which is why they are used in QM.

If you exclude non-real operations and states you are removing part of the system such that it becomes impossible to work with certain cases -- like handling non-real roots of ax^2 + bx + c polynomials.

It is possible to represent complex numbers as 2x2 matrices as those can encode 2D rotations. With the matrix formulation you are not dealing with imaginary numbers -- or you are, but they are not encoded with i = sqrt(-1) but as a 45deg rotation. IIRC, there is a formulation of Dirac's QED (Quantum ElectroDynamics) using matrices.

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Even simpler, complex numbers are really 2D vectors with addition and multiplication defined: a field. There's nothing "imaginary" about that second dimension, very frustrating to see them defined that way because it makes people think of it as an "escape hatch" out of real numbers. When you're working with complex numbers, you are working with a different system: `5 + 0i` is still a complex number because it's really `(5, 0)`.

Today I learned that complex numbers can be represented by matrices... thanks! https://www.youtube.com/watch?v=HbUewIIpl6I

Yeah https://xkcd.com/2028/ hit the nail on the head on this.

But working with complex numbers I hardly if ever write (a, b) for a+ib, while I use the "escape hatches" all the time. They solve equations that have no real solution, they give me paths from x=-1 to x=1 that don't cross the origin, etc. There's only so much to learn about C as a vector space, while the theory tying it to R (and even N) is very deep.

Thing is, there's no such thing as an escape hatch. Either you are working in the reals, or you are working in the complex plane. They don't "solve equations that have no real solution", that equation is either a real number equation or a complex number equation, not both. If you work in the complex plane, that is a different equation describing a different space! It just looks the same in standard notation.

If you don't realize this, then you can draw conclusions that don't make sense in the space you're working with. Take a simple equation like y = -x^2 - 5, representing a thrown ball's trajectory. It never crosses zero, there are no solutions. You can't "pop into the complex numbers and find a solution" because the thing it represents is confined to the reals.

So if you find yourself reaching for complex numbers, you have to understand that the thing you are working with is no longer one-dimensional, even if that second dimension collapses back to 0 at the end.

My mental model is that complex numbers are the first of the basic number systems that no longer has a total ordering. That alone is super useful for it.

Quantum is an odd one, as the name indicates that it deals in quantums. Minimum values that can't be divided. The difficult parts seems more to be in systems that have a probability space more than an analytical model that describes them. Which, fair, it is not a number system.

The loss of ordering is what makes complex numbers unique and useful for describing systems like rotations and probabilities.

Classical probability works with real numbers (probabilities between 0 and 1). Quantum probability involves amplitudes represented by complex numbers. These amplitudes can wave-interfere with each other, leading to superposition and entanglement

The phase space formulation of QM still only uses REAL-valued probabilities, but outside the interval [0,1]. I'm not sure I agree with the rest of your comment either.

A function (which is an isomorphism) from complex numbers a+bi to matrices is a+bi |-> [[a,-b],[b,a]] where the matrix is listed by rows. So i is sent to the matrix R with a 0 in the top left, 1 in the bottom left, 0 in the bottom right and a -1 in the top right. R is a 90 degree rotation, you can check that it sends the unit vector [1,0] on the x-axis to [0,1], and the unit vector [0,1] on the y-axis to [-1,0].

So.. some folks not all crackpots have been looking at octonions as a quantum shibboleth

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