Comment by clintonc

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?

> I have a real philosophical problem with complex numbers

> I believe real numbers to be completely natural

I have to say I find this perspective interesting but completely alien.

We need to have a way to find x such that x^2-2 = 0, and Q won’t cut it so we have R. (Or if you want, we need a complete ordered field so we have R)

We need to have a way to find x such that x^2+2 = 0, and R won’t cut it so we have C. (Or if you want, we need algebraic closure of R by the fundamental theorem of algebra so we need C)

I don’t really think any numbers (even “natural” numbers) are any more natural than any other kind of numbers. If you start to distinguish, where do you stop. Negative numbers are ok or not? What about zero? Is that “natural”? Mathematicians disagree about whether 0 is in N at least.

It reminds me of the famous quote from Gauss:

   That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question.

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.

The real numbers have some very unreal properties. Especially, their uncountable infinite cardinality is mind boggling.

A person can have a finite number of thoughts in his live. The number of persons that have and will ever live is countably infinite, as they can be arranged in a family tree (graph). This means that the total thoughts that all of mankind ever had and will have is countably infinite. For nearly all real numbers, humankind will never have thought of them.

You can do a similar argument with the subset of real numbers than can be described in any way. With description, I do not just mean writing down digits. Sentences of the form "the limit of sequence X", "the number fulfilling equation Y", etc are also descriptions. There are a countably infinite descriptions, as at the end every description is text, yet there are uncountably many real numbers. This means that nearly no real number can even be described.

I find it hard to consider something "real" when it is not possible to describe most of it. I find equally hard when nearly no real number has been used (thought of) by humankind.

The complex extension of the rational numbers, on the other hand, feel very natural to me when I look at them as vectors in a plane.

I think the main thing people stumble over when grasping complex numbers is the term "number". Colloquially, numbers are used to order stuff. The primary function of the natural numbers is counting after all. We think of numbers as advanced counting, i.e., ordering. The complex "numbers" are not ordered though (in the sense of an ordered field). I really think that calling them "numbers" is therefore a misnomer. Numbers are for counting. Complex "numbers" cannot count, and are thus no numbers. However, they make darn good vectors.

All of logic and math is a convincence tool. There are no, circles, quantities. Reality just is. We created these tools because they're a convinent way to cope with complexity of reality. There are no "objects" in a sense that chair is just atoms arranged chair-like. And atoms are just smaller particles arranged atom-like and yet physics operate in these objects treating them as something that exist.

So, now we have created these mental tools called mathematics that are heavily constrained. Then we create models that are approximately map 1:1 to some patterns that exist in reality (IE patterns that are roughly local so that we can call them objects). Due to the fact that our mental tools have heavy constrains and that we iteratively adjust these models to fit reality at focal points, we can approximately predict reality, because we already mapped the constrains into the model. But we shouldn't mistake model for the reality. Map is not territory.

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).

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.

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".

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

> I believe real numbers to be completely natural

You can teach middle school children how to define complex numbers, given real numbers as a starting point. You can't necessarily even teach college students or adults how to define real numbers, given rational numbers as a starting point.

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 doubt anyone could make a reply to this comment that would make me feel any better about it.

I am also a complex number skeptic. The position I've landed on is this.

1) complex numbers are probably used for far more purposes across math than they "ought" to be, because people don't have the toolbox to talk about geometry on R^2 but they do know C so they just use C. In particular, many of the interesting things about complex analysis are probably just the n=2 case of more general constructions that can be done by locating R inside of larger-dimensional algebras.

2) The C that shows up in quantum mechanics is likely an example of this--it's a case of physics having a a circular symmetry embedded in it (the phase of the wave functions) and everyone getting attached to their favorite way of writing it. (Ish. I'm not sure how the square the fact that wave functions add in superposition. but anyway it's not going to be like "physics NEEDS C", but rather, physics uses C because C models the algebra of the thing physics is describing.

3) C is definitely intrinsic in a certain sense: once you have polynomials in R, a natural thing to do is to add a sqrt(-1). This is not all that different conceptually from adding sqrt(2), and likely any aliens we ever run into will also have done the same thing.

> I believe real numbers to be completely natural,

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

One nice way of seeing the inevitability of the complex numbers is to view them as a metric completion of an algebraic closure rather than a closure of a completion.

Taking the algebraic closure of Q gives us algebraic numbers, which are a very natural object to consider. If we lived in an alternative timeline where analysis was never invented and we only thought about polynomials with rational coefficients, you’d still end up inventing them.

If you then take the metric completion of algebraic numbers, you get the complex numbers.

This is sort of a surprising fact if you think about it! the usual construction of complex numbers adds in a bunch of limit points and then solutions to polynomial equations involving those limit points, which at first glance seems like it could give a different result then adding those limit points after solutions.

Also a PhD in math, where complex numbers are fundamental, and also part of large swaths of similar structures that are also fundamental. They fit in nicely among a ton of other similar structures and concepts, so they seem about as fundamental as sets or addition or groups or fields (and there it is).

They also seem fundamental to physical reality in a way most math concepts do not: they're required (in structure) for quantum mechanics, in many equations that seem to be part of the universe. The behavior of subatomic particles (and more precisely, QFTs), require the waveforms to evolve as complex valued functions, where the probability of an event is the magnitude of the complex value.

This has been tested between theory and experiment to about 14 decimal digits precision for QED.

I'd guess they should be considered as real as radio waves (which we don't see), as the fact things we think are solid are mostly empty space (which we don't feel), or that time flows at different rates under different situations (which we also don't experience). Yet all those things are more real than stuff our limited senses experiences.

There's some string of research on if/how fundamental complex numbers are to QM, e.g., https://www.scientificamerican.com/article/quantum-physics-f...

> 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.

---

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.

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.

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..

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.

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.

I think you would enjoy (and possibly have your mind blown) this series of videos by the “Rebel Mathematician” Prof Norman Wildburger. https://youtu.be/XoTeTHSQSMU

He constructs “true” complex numbers, generalises them over finite and unbounded fields, and demonstrates how they somewhat naturally arise from 2x2 matrices in linear algebra.

I wonder off and on if in good fiction of "when we meet aliens and start communicating using math"- should the aliens be okay with complex residue theorems? I used to feel the same about "would they have analytic functions as a separate class" until I realized how many properties of polynomials analytic functions imitate (such as no nontrivial bounded ones).

I always wondered in the higher levels of maths, theoretical physics etc how much of it reflects a "real" thing and how much of is hand-wavey "try not to think about it too much but the equations work".

EG complex numbers, extra dimensions, string theory, weird particles, whatever electrons do, possibly even dark matter/energy.

> 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.

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.

Now algebraic closure, but better:

  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.

Next, general theory of fields:

  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.

I think maybe differential geometry can provide some help here.

  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.

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.

Unsure this would help, but maybe thinking in English Prime could be an interesting exercice. https://en.wikipedia.org/wiki/E-Prime

I don't understand what it means for something to feel "natural". You can formally define the real numbers in multiple ways which are all isomorphic and coherent. These definitions are usually more complicated than people expect which nicely show that the real set is not a very intuitive object. Same thing for C.

There is not evidence for C. It's a construction. Obviously it shows up in physics models. They are built using mathematical formalism.

If multiple definitions turn out to be isomorphic, that's generally because there is an underlying structure linking the properties together.

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.

> I believe real numbers to be completely natural

Are real numbers not just "a convenience" in a sense? I do not see anything "fundamental" or "natural" about dedekind cuts or any other construction of the real numbers. If anything real numbers, to me, are more built out of the convenience of having a complete field extension of the rational numbers. We could do just fine with computable numbers and avoid a lot of problems that this line of convenience leads to.

I guess I don't even really understand the objection. That's how ALL mathematics works. You specify some axioms or a construction and then reason about objects that satisfy those constraints. Some of them like the complex numbers turn out to be particularly useful.

But it's not fundamentally any different than what we do with the natural numbers. Those just feel more familiar to you.

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

I suspect, as you may as while, that this quote is at the core of the matter. Identifying what you find the difference between real and complex numbers are. You are inclined to split them into separate categories. I suspect you must identify the platonic (Or HTW, if that is your metaphor) property of the real numbers which the complex lack.

My naive take is we discovered it as a math tool first but later on rediscovered it in nature when we discovered the electromagnetic field.

The electromagnetic field is naturally a single complex valued object(Riemann/Silberstein F = E + i cB), and of course Maxwell's equations collapse into a single equation for this complex field. The symmetry group of electromagnetism and more specifically, the duality rotation between E and B is U(1), which is also the unit circle in the complex plane.

It's not like I have a real answer, of course, but something flipped inside of me after hearing the following story by Aaronson. He is asking[0], why quantum amplitudes would have to be complex. I.e., can we imagine a universe, where it's not the case?

> Why did God go with the complex numbers and not the real numbers?

> Years ago, at Berkeley, I was hanging out with some math grad students -- I fell in with the wrong crowd -- and I asked them that exact question. The mathematicians just snickered. "Give us a break -- the complex numbers are algebraically closed!" To them it wasn't a mystery at all.

Apparently, you weren't one of these math grad students, and, to be fair, Aaronson is starting with the question that is somewhat opposite to yours, but still, doesn't it intuitively make sense somehow? We are modeling something. In the process of modeling something we discover functions, and algebra, and find out that we'd like to use square roots all over the place. And just that alone leads us naturally to complex numbers! We didn't start with them, we only imagined an algebra that allows us to describe some process we'd like to describe, and suddenly there's no way around complex numbers! To me, thinking this way makes it almost obvious that ℂ-numbers are "real" somehow, they are indeed the fundamental building block of some complex-enough model, while ℝ are not.

Now, I must admit, that of course it doesn't reveal to me what the fuck they actually are, how to "imagine" them in the real world. I suppose, it's the same with you. But at least it makes me quite sure that indeed this is "the shadow of something natural that we just couldn't see", and I just don't know what. I believe it to be the consequence of us currently representing all numbers somehow "wrong". Similarly to how ancient Babylonian fraction representations were preventing ancient Babylonians from asking the right questions about them.

P.S. I think I must admit, that I do NOT believe real numbers to be natural in any sense whatsoever. But this is completely besides the point.

[0] https://www.scottaaronson.com/democritus/lec9.html

Ok... How about this? All (human) models of the universe are "Ptolemaic" to some degree. That is, they work but don't necessarily describe the true underlying structure ().

So it is a mistake to assume that any model is actually true.

Therefore complex numbers are just another modeling language, useful in certain contexts. All mathematics is just a modeling language.

() If you doubt this, ask yourself the question: Will the science of particle physics have changed in 100 years?

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/

Given that you have a Ph.D. in mathematics, this might seem hopelessly elementary, but who knows--I found it intuitive and insightful: https://news.ycombinator.com/item?id=18310788

Perhaps of your interest might be this work https://arxiv.org/abs/2101.10873v1 on why quantum physics needs complex numbers to work. Interesting noting though that as for solving polynomials, quantum physics might be also considered a “convenience” within the Copenhagen interpretation

> I doubt anyone could make a reply to this comment that would make me feel any better about it.

You may be right, but just to have said it : the Fast Fourier Transform requires complex numbers. One can write a version that avoids complex numbers, but (a) its ugliness gives away what's missing, and (b) it's significantly slower in execution.

Oh -- also --

e^(i Ⲡ) + 1 = 0

Nevertheless, you may be right.

If you view all of math as just a set of logic games with the axioms as the basic rules, then there's nothing unnatural about complex numbers. Various mathematical constructs describe various phenomena in the real world well. It just so happens that many physical systems behave in a way that can be very naturally described using complex numbers.

Stepping out of pure maths and into engineering we find complex numbers indispensable for describing physical systems and predicting system change over time.

I don’t have a list to hand, but there are so many areas of physics and engineering where complex numbers are the best representation of how we perceive the universe to work.

People thought negative numbers were weird until the 1800s or so, they arose in much the same way as a way to solve algebraic equations (or even just to balance the books, literally).

Complex numbers were always going to show up just so we could diagonalise matrices, which is an important part of solving (linear) differential equations.

Does this seemingly-amazing 2swap math exposition video offer any extra perspective? https://youtu.be/9HIy5dJE-zQ

I ask as someone who doesn't understand as much as you, but who is charmed by such visual explanations :)

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.

How are there real numbers real? They're certainly not physical in a finite universe with quantised fundamental fields. I would say that natural numbers are there only physically represented ones and everything else is convenience.

HN once recommended a book

https://press.princeton.edu/books/paperback/9780691133911/ne...

> Is this the shadow of something natural that we just couldn't see

In special relativity there are solutions that allow FTL if you use imaginary numbers. But evidence suggests that this doesn’t happen.

> 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'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.

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?

Maybe it is a notation issue.

What is a negative number? What is multiplication? What is a complex "number"? Complex are not even orderable. Is complex addition an overloading of the addition operator. Same with multiplication?

What i squared is -1 ? What does -1 even mean? Is the sign, a kind of operator?

The geometric interpretation help. These are transformations. Instead of 1 + i, we could/should write (1,i)

The AI might be clearer: https://gemini.google.com/share/6e00fab74749

A lot of math is not very clear because it is not very well taught. The notations are unclear. For instance, another example is: what is the difference between a matrix and a tensor? But that is another debate for anyone who wants to think about it. The definition found in books is often kind of wrong making a distinction that shouldn't really exist more often than not.

[Obligatory: Engineering background. Not an expert]

I've always found it a bit odd that we DO define "i" to help us express complex numbers, with the convenient assumption that "i = sqrt(-1)"... but we DON'T have any such symbols to map between more than 2 dimensions.

I felt a bit better when I found out about - (nth) roots of unity (to explore other "i"-like definitions, including things like roots of unity modulo n, and hidden abelian subgroup problems which feel a bit to me like dealing with orthogonal dimensions) - tensors (e.g. in physics, when we need a better way to discuss more than 2 dimensions, and often establish syntactic sugar for (x,y,z,t))

IDK if that helps at all (or worse, simply betrays some misunderstanding of mine. If so, please complain- I'd appreciate the correction!)

I am with you on this (the challenge, not (yet) the phd), however, I myself have a far greater problem.

I do not see what’s the deal about prime numbers which seems to be more of a limitation on our end, similar to our shortage in understanding to a point we call e, π, √2 etc Irrationals.

We simply did not get the actual mathematical structure of the universe and we came up with something “good enough” that helps moving forward.

In the universe the perfect circle has perfect symmetry, hence perfect ratio, hence well-defined sweet heaven balanced harmonic entity.

Exponentials are natural phenomena. The very fact that e is its own derivative tells us we are all wrong here.

We are in an infinite escape that no matter how long we will play, and how many riddles we will solve, we will never get the entire picture.

Yes, primes are nice structure when you deal with us humans counting potatoes. But e, just e, let alone √2 or π are far more fascinating to me.

The e point cuts deep. e being its own derivative isn’t a curiosity. It’s saying that there’s a growth process so fundamental that its rate is indistinguishable from its state. That’s not a number — it’s a signature of how change works. And yet: π, e, √2 — we only name them, define them, catch them using the integers. π is the ratio of circumference to diameter. Ratio of what to what? e is lim(1 + 1/n)^n. The integers sneak in. Is that just our access route? Or is discreteness also woven into the fabric, alongside continuity?

My intuition led me to the following: we think our counting units (1, 2, 3, …) and fractions are the “numbers”, and when we want to refer to multi-dimensional phenomena, we use vectors or matrices or any other logical structure.

However, this is a very superficial aspect of the business, since the actual math is multi-dimensional inherently. The natural math is not linear, nor is it a plane. It is simply a multi-dimensional number system (imagine our complex numbers, but many other dimensions). Perhaps tensors or even more. This is why we experience quantum mechanics as statistical states, results of specific measurements. We think in units, and we don’t understand things are happening in parallel across all directions. Once we figure this out, we will understand why e, π and others are as natural as it gets, while our natural numbers are barely a dot, a point in the real math universe.

Sorry for the length but you triggered me with a long time pain point.

Thanks for your comment.

C is the only way to make a field out of pairs of reals. Also (or rather just another facet of the same phenomenon) we might be interested in polynomials with integer coefficients, but some of those will have non integral roots. And we might be interested in polynomials with rational coeffs but some will not have rational roots. Same with the reals but the buck stops with the complex numbers. They are definitely not accidental they are the natural (so to speak) completion of our number system. That they exist physically in some sense is "unreasonable effectiveness" territory.

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.

That's how I see complex numbers:

In mathematics and physics, complex numbers aren't just "imaginary" values—they are the secret language of 2D rotation. While real numbers live on a 1D line, complex numbers inhabit a 2D plane, and multiplying them acts as a bridge between dimensions. 1. The Geometry of i To understand how we switch dimensions, look at the imaginary unit i. In a standard real-number system, you only move left or right. Adding i introduces a vertical axis. * The 90-degree turn: Multiplying a real number by i is geometrically equivalent to a 90° counter-clockwise rotation. * The Dimension Switch: If you start at 1 (on the x-axis) and multiply by i, you land at i (on the y-axis). You have effectively "switched" your direction from horizontal to vertical. 2. Rotation via Euler’s Formula The most elegant link between complex numbers and rotation is Euler’s Formula: This formula places any complex number on a unit circle in the complex plane. When you multiply a vector by e^{i\theta}, you aren't changing its length; you are simply rotating it by the angle \theta. Why this matters: * Algebraic Simplicity: Instead of using messy rotation matrices (which involve four separate multiplications and additions), you can rotate a point by simply multiplying two complex numbers. * Phase in Physics: This is why complex numbers are used in electrical engineering and quantum mechanics. A "phase shift" in a wave is just a rotation in the complex plane. 3. Beyond 2D: Quaternions If complex numbers (a + bi) handle 2D rotations by adding one imaginary dimension, what happens if we want to rotate in 3D? To handle 3D space without hitting "Gimbal Lock" (where two axes align and you lose a degree of freedom), mathematicians use Quaternions. These extend the concept to three imaginary units: i, j, and k. > The Rule of Four: Interestingly, to rotate smoothly in three dimensions, you actually need a four-dimensional number system. > Summary Table | Number System | Dimensions | Primary Use in Rotation | |---|---|---| | Real Numbers | 1D | Scaling (stretching/shrinking) | | Complex Numbers | 2D | Planar rotation, oscillations, AC circuits | | Quaternions | 4D | 3D computer graphics, aerospace navigation |

They can be treated as vectors, but they have "superpowers" that standard vectors do not. 1. The Similarities (The 2D Map) In a purely visual or structural sense, a complex number z = a + bi behaves exactly like a 2D vector \vec{v} = (a, b). * Addition: Adding two complex numbers is identical to "tip-to-tail" vector addition. * Magnitude: The "absolute value" (modulus) of a complex number |z| = \sqrt{a^2 + b^2} is the same as the length of a vector. * Coordinates: Both represent a point on a 2D plane. 2. The Difference: Multiplication This is where complex numbers leave standard 2D vectors in the dust. In standard vector algebra (like what you'd use in an introductory physics class), there isn't a single, clean way to "multiply" two 2D vectors to get another 2D vector. You have the Dot Product (which gives you a single number/scalar) and the Cross Product (which actually points out of the 2D plane into the 3D world). Complex numbers, however, can be multiplied together to produce another complex number. The "Rotation" Secret When you multiply two complex numbers, the math automatically handles two things at once: * Scaling: The lengths are multiplied. * Rotation: The angles are added. Standard vectors cannot do this on their own; you would need to bring in a "Rotation Matrix" to force a vector to turn. A complex number just "knows" how to turn naturally through its imaginary component. 3. When to use which? Mathematically, complex numbers form a Field, while vectors form a Vector Space. * Use Vectors when you are dealing with forces, velocities, or any dimension higher than 2 (like 3D space). * Use Complex Numbers when you are dealing with things that rotate, vibrate, or oscillate (like radio waves, electricity, or quantum particles). > The Peer-to-Peer Truth: Think of a complex number as a vector with an attitude. It lives in the same 2D house, but it knows how to spin and transform itself algebraically in ways a simple (x, y) coordinate cannot. >