One way to sharpen the question is to stop asking whether C is "fundamental" and instead ask whether it is forced by mild structural constraints. From that angle, its status looks closer to inevitability than convenience.

Take R as an ordered field with its usual topology and ask for a finite-dimensional, commutative, unital R-algebra that is algebraically closed and admits a compatible notion of differentiation with reasonable spectral behavior. You essentially land in C, up to isomorphism. This is not an accident, but a consequence of how algebraic closure, local analyticity, and linearization interact. Attempts to remain over R tend to externalize the complexity rather than eliminate it, for example by passing to real Jordan forms, doubling dimensions, or encoding rotations as special cases rather than generic elements.

More telling is the rigidity of holomorphicity. The Cauchy-Riemann equations are not a decorative constraint; they encode the compatibility between the algebra structure and the underlying real geometry. The result is that analyticity becomes a global condition rather than a local one, with consequences like identity theorems and strong maximum principles that have no honest analogue over R.

I’m also skeptical of treating the reals as categorically more natural. R is already a completion, already non-algebraic, already defined via exclusion of infinitesimals. In practice, many constructions over R that are taken to be primitive become functorial or even canonical only after base change to C.

So while one can certainly regard C as a technical device, it behaves like a fixed point: impose enough regularity, closure, and stability requirements, and the theory reconstructs it whether you intend to or not. That does not make it metaphysically fundamental, but it does make it mathematically hard to avoid without paying a real structural cost.

I used to feel the same way. I now consider complex numbers just as real as any other number.

The key to seeing the light is not to try convincing yourself that complex number are "real", but to truly understand how ALL numbers are abstractions. This has indeed been a perspective that has broadened my understanding of math as a whole.

Reflect on the fact that negative numbers, fractions, even zero, were once controversial and non-intuitive, the same as complex are to some now.

Even the "natural" numbers are only abstractions: they allow us to categorize by quantity. No one ever saw "two", for example.

Another thing to think about is the very nature of mathematical existence. In a certain perspective, no objects cannot exist in math. If you can think if an object with certain rules constraining it, voila, it exists, independent of whether a certain rule system prohibit its. All that matters is that we adhere to the rule system we have imagined into being. It does not exist in a certain mathematical axiomatic system, but then again axioms are by their very nature chosen.

Now in that vein here is a deep thought: I think free will exists just because we can imagine a math object into being that is neither caused nor random. No need to know how it exists, the important thing is, assuming it exists, what are its properties?

A question I enjoy asking myself when I'm wondering about this stuff is "if there are alien mathematicians in a distant galaxy somewhere, do they know about this?"

For complex numbers my gut feeling is yes, they do.

For me, the complex numbers arise as the quotients of 2-dimensional vectors (which arise as translations of the 2-dimensional affine space). This means that complex numbers are equivalence classes of pairs of vectors is a 2-dimesional vector space, like 2-dimensional vectors are equivalence classes of pairs of points in a 2-dimensional affine space or rational numbers are equivalence classes of pairs of integers, or integers are equivalence classes of pairs of natural numbers, which are equivalence classes of equipotent sets.

When you divide 2 collinear 2-dimensional vectors, their quotient is a real number a.k.a. scalar. When the vectors are not collinear, then the quotient is a complex number.

Multiplying a 2-dimensional vector with a complex number changes both its magnitude and its direction. Multiplying by +i rotates a vector by a right angle. Multiplying by -i does the same thing but in the opposite sense of rotation, hence the difference between them, which is the difference between clockwise and counterclockwise. Rotating twice by a right angle arrives in the opposite direction, regardless of the sense of rotation, therefore i*i = (-i))*(-i) = -1.

Both 2-dimensional vectors and complex numbers are included in the 2-dimensional geometric algebra, whose members have 2^2 = 4 components, which are the 2 components of a 2-dimensional vector together with the 2 components of a complex number. Unlike the complex numbers, the 2-dimensional vectors are not a field, because if you multiply 2 vectors the result is not a vector. All the properties of complex numbers can be deduced from those of the 2-dimensional vectors, if the complex numbers are defined as quotients, much in the same way how the properties of rational numbers are deduced from the properties of integers.

A similar relationship like that between 2-dimensional vectors and complex numbers exists between 3-dimensional vectors and quaternions. Unfortunately the discoverer of the quaternions, Hamilton, has been confused by the fact that both vectors and quaternions have multiple components and he believed that vectors and quaternions are the same thing. In reality, vectors and quaternions are distinct things and the operations that can be done with them are very different. This confusion has prevented for many years during the 19th century the correct use of quaternions and vectors in physics (like also the confusion between "polar" vectors and "axial" vectors a.k.a. pseudovectors).

We have too much mental baggage about what a "number" is.

Real numbers function as magnitudes or objects, while complex numbers function as coordinatizations - a way of packaging structure that exists independently of them, e.g. rotations in SO(2) together with scaling). Complex numbers are a choice of coordinates on structure that exists independently of them. They are bookkeeping (a la double‑entry accounting) not money

In my view nonnegative real numbers have good physical representations: amount, size, distance, position. Even negative integers don't have this types of models for them. Negative numbers arise mostly as a tool for accounting, position on a directed axis, things that cancel out each other (charge). But in each case it is the structure of <R,+> and not <R,+,*> and the positive and negative values are just a convention. Money could be negative, and debt could be positive, everything would be the same. Same for electrons and protons.

So in our everyday reality I think -1 and i exist the same way. I also think that complex numbers are fundamental/central in math, and in our world. They just have so many properties and connections to everything.

I have MS in math and came to a conclusion that C is not any more "imaginary" than R. Both are convenient abstractions, neither is particularly "natural".

A long time ago on HN, I said that I didn't like complex numbers, and people jumped all over my case. Today I don't think that there's anything wrong with them, I just get a code smell from them because I don't know if there's a more fundamental way of handling placeholder variables.

I get the same feeling when I think about monads, futures/promises, reactive programming that doesn't seem to actually watch variables (React.. cough), Rust's borrow checker existing when we have copy-on-write, that there's no realtime garbage collection algorithm that's been proven to be fundamental (like Paxos and Raft were for distributed consensus), having so many types of interprocess communication instead of just optimizing streams and state transfer, having a myriad of GPU frameworks like Vulkan/Metal/DirectX without MIMD multicore processors to provide bare-metal access to the underlying SIMD matrix math, I could go on forever.

I can talk about why tau is superior to pi (and what a tragedy it is that it's too late to rewrite textbooks) but I have nothing to offer in place of i. I can, and have, said a lot about the unfortunate state of computer science though: that internet lottery winners pulled up the ladder behind them rather than fixing fundamental problems to alleviate struggle.

I wonder if any of this is at play in mathematics. It sure seems like a lot of innovation comes from people effectively living in their parents' basements, while institutions have seemingly unlimited budgets to reinforce the status quo..

I don't know if this will help, but I believe that all of mathematics arises from an underlying fundamental structure to the universe and that this results in it both being "discoverable" (rather than invented) and "useful" (as in helpful for describing, expressing and calculating things).

> I believe real numbers to be completely natural,

Most of real numbers are not even computable. Doesn't that give you a pause?

1. Algebra: Let's say we have a linear operator T on a real vector space V. When trying to analyze a linear operator, a key technique is to determine the T-invariant subspaces (these are subspaces W such that TW is a subset of W). The smallest non-trivial T-invariant subspaces are always 1- or 2-dimensional(!). The first case corresponds to eigenvectors, and T acts by scaling by a real number. In the second case, there's always a basis where T acts by scaling and rotation. The set of all such 2D scaling/rotation transformations are closed under addition, multiplication, and the nonzero ones are invertible. This is the complex numbers! (Correspondence: use C with 1 and i as the basis vectors, then T:C->C is determined by the value of T(1).)

2. Topology: The fact the complex numbers are 2D is essential to their fundamentality. One way I think about it is that, from the perspective of the real numbers, multiplication by -1 is a reflection through 0. But, from an "outside" perspective, you can rotate the real line by 180 degrees, through some ambient space. Having a 2D ambient space is sufficient. (And rotating through an ambient space feels more physically "real" than reflecting through 0.) Adding or multiplying by nonzero complex numbers can always be performed as a continuous transformation inside the complex numbers. And, given a number system that's 2D, you get a key topological invariant of closed paths that avoid the origin: winding number. This gives a 2D version of the Intermediate Value Theorem: If you have a continuous path between two closed loops with different winding numbers, then one of the intermediate closed loops must pass through 0. A consequence to this is the fundamental theorem of algebra, since for a degree-n polynomial f, when r is large enough then f(r*e^(i*t)) traces out for 0<=t<=2*pi a loop with winding number n, and when r=0 either f(0)=0 or f(r*e^(i*t)) traces out a loop with winding number 0, so if n>0 there's some intermediate r for which there's some t such that f(r*e^(i*t))=0.

So, I think the point is that 2D rotations and going around things are natural concepts, and very physical. Going around things lets you ensnare them. A side effect is that (complex) polynomials have (complex) roots.

The author mentioned that the theory of the complex field is categorical, but I didn't see them directly mention that the theory of the real field isn't - for every cardinal there are many models of the real field of that size. My own, far less qualified, interpretation, is that even if the complex field is just a convenient tool for organizing information, for algebraic purposes it is as safe an abstraction as we could really hope for - and actually much more so than the real field.

Complex numbers are just a field over 2D vectors, no? When you find "complex solutions to an equation", you're not working with a real equation anymore, you're working in C. I hate when people talk about complex zeroes like they're a "secret solution", because you're literally not talking about the same equation anymore.

There's this lack of rigor where people casually move "between" R and C as if a complex number without an imaginary component suddenly becomes a real number, and it's all because of this terrible "a + bi" notation. It's more like (a, b). You can't ever discard that second component, it's always there.

As a math enjoyer who got burnt out on higher math relatively young, I have over time wondered if complex numbers aren’t just a way to represent an n-dimensional concept in n-1 dimensions.

Which makes me wonder if complex numbers that show up in physics are a sign there are dimensions we can’t or haven’t detected.

I saw a demo one time of a projection of a kind of fractal into an additional dimension, as well as projections of Sierpinski cubes into two dimensions. Both blew my mind.

Given that you have a Ph.D. in mathematics, this might seem hopelessly elementary, but who knows--I found it intuitive and insightful: https://betterexplained.com/articles/a-visual-intuitive-guid...

Related: https://news.ycombinator.com/item?id=18310788

> Is this the shadow of something natural that we just couldn't see, or just a convenience?

They originally arose as tool, but complex numbers are fundamental to quantum physics. The wave function is complex, the Schrödinger equation does not make sense without them. They are the best description of reality we have.

How does your question differ from the classic question more normally applied to maths in general - does it exist outside the mind (eg platonism) or no (eg. nominalism)?

If it doesn't differ, you are in the good company of great minds who have been unable to settle this over thousands of years and should therefore feel better!

More at SEP:

https://plato.stanford.edu/entries/philosophy-mathematics/

Even the counting numbers arose historically as a tool, right?

Even negative numbers and zero were objected to until a few hundred years ago, no?

> I believe real numbers to be completely natural, but far greater mathematicians than I found them objectionable only a hundred years ago

I believe even negative numbers had their detractors

I like to think of complex numbers as “just” the even subset of the two dimensional geometric algebra.

Almost every other intuition, application, and quirk of them just pops right out of that statement. The extensions to the quarternions, etc… all end up described by a single consistent algebra.

It’s as if computer graphics was the first and only application of vector and matrix algebra and people kept writing articles about “what makes vectors of three real numbers so special?” while being blithely unaware of the vast space that they’re a tiny subspace of.

Maybe the bottom ~1/3, starting at "The complex field as a problem for singular terms", would be helpful to you. It gives a philosophical view of what we mean when we talk about things like the complex numbers, grounded in mathematical practice.

I'm presuming this is old news to you, but what helped me get comfortable with ℂ was learning that it's just the algebraic closure of ℝ.

Personally, no number is natural. They are probably a human construct. Mathematics does not come naturally to a human. Nowadays, it seems like every child should be able to do addition, but it was not the case in the past. The integers, rationals, and real numbers are a convenience, just like the complex numbers.

A better way to understand my point is: we need mental gymnastics to convert problems into equations. The imaginary unit, just like numbers, are a by-product of trying to fit problems onto paper. A notable example is Schrodinger's equation.

The complex numbers is just the ring such that there is an element where the element multiplied by itself is the inverse of the multiplicative identity. There are many such structures in the universe.

For example, reflections and chiral chemical structures. Rotations as well.

It turns out all things that rotate behave the same, which is what the complex numbers can describe.

Polynomial equations happen to be something where a rotation in an orthogonal dimension leaves new answers.

> In particular, they arose historically as a tool for solving polynomial equations.

That is how they started, but mathematics becomes remarkable "better" and more consistent with complex numbers.

As you say, The Fundamental Theorem of Algebra relies on complex numbers.

Cauchy's Integral Theorem (and Residue Theorem) is a beautiful complex-only result.

As is the Maximum Modulus Principle.

The Open Mapping Theorem is true for complex functions, not real functions.

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Are complex numbers really worse than real numbers? Transcendentals? Hippasus was downed for the irrationals.

I'm not sure any numbers outside the naturals exist. And maybe not even those.

I've been thinking about this myself.

First, let's try differential equations, which are also the point of calculus:

  Idea 1: The general study of PDEs uses Newton(-Kantorovich)'s method, which leads to solving only the linear PDEs,
  which can be held to have constant coefficients over small regions, which can be made into homogeneous PDEs,
  which are often of order 2, which are either equivalent to Laplace's equation, the heat equation,
  or the wave equation. Solutions to Laplace's equation in 2D are the same as holomorphic functions.
  So complex numbers again.

  Idea 2: Infinitary algebraic closure. Algebraic closure can be interpeted as saying that any rational functions can be factorised into monomials.
  We can think of the Mittag-Leffler Theorem and Weierstrass Factorisation Theorem as asserting that this is true also for meromorphic functions,
  which behave like rational functions in some infinitary sense. So the algebraic closure property of C holds in an infinitary sense as well.
  This makes sense since C has a natural metric and a nice topology.

  Idea 3: Fields of characteristic 0. Every algebraically closed field of characteristic 0 is isomorphic to R[√-1] for some real-closed field R.
  The Tarski-Seidenberg Theorem says that every FOL statement featuring only the functions {+, -, ×, ÷} which is true over the reals is
  also true over every real-closed field.

  Idea 4: Conformal geometry in 2D. A conformal manifold in 2D is locally biholomorphic to the unit disk in the complex numbers.

  Idea 5: This one I'm not 100% sure about. Take a smooth manifold M with a smoothly varying bilinear form B \in T\*M ⊗ T\*M.
  When B is broken into its symmetric part and skew-symmetric part, if we assume that both parts are never zero, B can then be seen as an almost
  complex structure, which in turn naturally identifies the manifold M as one over C.