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 work in applied probability, so I'm forced to use many different tools depending on the application. My colleagues and I would consider ourselves lucky if what we're doing allows for an application of some properties of C, as the maths will tend to fall out so beautifully.
Pure probability focuses on developing fundamental tools to work with random elements. It's applied in the sense that it usually draws upon techniques found in other traditionally pure mathematical areas, but is less applied than "applied probability", which is the development and analysis of probabilistic models, typically for real-world phenomena. It's a bit like statistics, but with more focus on the consequences of modelling assumptions rather than relying on data (although allowing for data fitting is becoming important, so I'm not sure how useful this distinction is anymore).
At the moment, using probabilistic techniques to investigate the operation of stochastic optimisers and other random elements in the training and deployment of neural networks is pretty popular, and that gets funding. But business as usual is typically looking at ecological models involving the interaction of many species, epidemiological models investigating the spread of disease, social network models, climate models, telecommunication and financial models, etc. Branching processes, Markov models, stochastic differential equations, point processes, random matrices, random graph networks; these are all the common objects used. Actually figuring out their behaviour can require all kinds of assorted techniques though, you get to pull from just about anything in mathematics to "get the job done".
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.
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?
For complex numbers my gut feeling is yes, they do.
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).
In special relativity there are solutions that allow FTL if you use imaginary numbers. But evidence suggests that this doesn’t happen.
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
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'm not a physicist, but do we actually know if distance and time can vary continuously or is there a smallest unit of distance or time? A physics equation might tell you a particle moves Pi meters in sqrt(2) seconds but are those even possible physical quantities? I'm not sure if we even know for sure whether the universe's size is infinite or finite?
In my view, that isn’t even true for nonnegative integers. What’s the physical representation of the relatively tiny (compared to ‘most integers’) Graham’s number (https://en.wikipedia.org/wiki/Graham's_number)?
Back to the reals: in your view, do reals that cannot be computed have good physical representations?
I think these questions mostly only matter when one tries to understand their own relation to these concepts, as GP asked.
Otherwise we have a random universe, which does not seem to be the case.
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..
Most of real numbers are not even computable. Doesn't that give you a pause?
Sorry, what do you mean?
The real numbers are uncountable. (If you're talking about constructivism, I guess it's more complicated. There's some discussion at https://mathoverflow.net/questions/30643/are-real-numbers-co... . But that is very niche.)
The set of things we can compute is, for any reasonable definition of computability, countable.
In this case, to actually prove the statement internally that "not every real number is computable", you'd need some non-constructive principle (usually added to the logical system rather than the theory itself). But, the absence of that proof doesn't make its negation provable either ("every real number is computable"). While some schools of constructivism want the negation, others prefer to live in the ambiguity.
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.
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.
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.
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.
Complex numbers are just two dimensional numbers, lol
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:
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.
Even negative numbers and zero were objected to until a few hundred years ago, no?
I believe even negative numbers had their detractors
(I have a math degree, so I don't have any issues with C, but this is the kind of question that would have troubled me in high school.)
It's the birthright of every field.
Complex numbers offers that resolution.
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.
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.
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.
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.Starting with <inc, [? inc], (inc -> comp)*>:
1. a directed relation (increment)
2. The composition of increments, and
3. The repetition composition, i.e. a composition repeatedly performed for each increment traversed in a composition of increments,
...we inevitably get R, complex numbers and algebraic numbers, as follows (listed in one possible logical, not historical, progression):
• "Natural numbers" the elements generated by repeated compositions of increment.
• "Decrement", new notation, not a new relation, for the directed relation of increment from x to y, when written from y to x.
• "Zero", as the inevitable generated element from composing an increment with a decrement. Expands our elements to "Whole numbers".
• "Addition", c = a + b, a generator composed of increment traversal, applying increments. c = a + b, is: a, with as many increment compositions as are increments traversable in b.
• "Subtraction", as with decrement, the same relationship as addition, with opposite element ordering (a + b - b = a).
• "Negative" numbers, the inevitable generated elements of subtracting a longer/larger whole number from a shorter/lesser whole number. This expands "Whole numbers" to "Integers".
• "Multiplication", a new generator composed by adding a number, for each increment traversed in another number. I.e. c = a * b, where c is zero, with an addition of a, for each increment in b.
• "Division", the term for the reversely ordered multiplication (a * b / b = a). Together with multiplication, we have expanded our elements to "Rational numbers"
IMAGINARY NUMBERS
• "Imaginary" numbers by defining i, as the solution to the expression ±i = sqrt(-1). (Note all square roots are ±y = sqrt(x), and require a choice to get a unique root.) This new element is irreducible to previous elements, so expands numbers again to complex numbers.
Unlike the "choice" problem described in the article, "i" itself (in my view/definition) is not signed. The square root has two solutions, but the choice is not associated with "i", but with the choice of sign for any square root. So there is no need to choose +i or -i, and i is uniquely determined. Of course, having defined i, we know based on multiplicative identity that +i = (+1)*i = i. (And -i = (-1)*i, is not reducible beyond that.)
So, given i itself is unsigned, it is the unique solution to the constraint that defines it, and there is only the unique identity isomorphism.
ALGEBRAIC
• "Exponent" and "Log" are new generators, composed from multiplications of one number, applied repeatedly according to the traversal of another number. And now we have "Algebraic numbers".
CONCLUSION
And at this point, we have exhausted the ability of increment, increment composition, and composition via increment traversal, to generate any more elements. We have closure.
Regardless of the order of exploration, closure consists of one unique set of elements, and their unique structure.
LAST NOTE
None of this requires or implies un-constructible reals. Which have gotten arbitrarily shoe-horned into discussions and classes for a lot of unrelated math. Primarily due to a lack of a pithy term for un-constructible reals.