r/math • u/Short_Bluebird_3845 • 7d ago
What are your favourite N-dimensional number systems?
From complex numbers, duals, split, some quirky 3D space, quaternions, up to hypercomplex. There are, indeed, many ways to write a vector as a number and play with them, some more obscure or specialized than others. What are your favourite N-dimensional number systems (or, more formally, algebras over the Reals)?
NOTE: Last time I asked, which was for ambiguous notation, you talked about them from spite and hate. I'm not sure if in math the line between pain and love blurs... I just wanted to ask for it as well lol.
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u/AntiProton- 7d ago
Complex numbers - I know it's boring.
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u/DebatedKnowledge Harmonic Analysis 7d ago
Literally the best object in math of all time, don't sell yourself.
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u/NoSuchKotH Engineering 7d ago
I love hyperreal numbers (which are ℝ^ℕ dimensional). They saved my PhD.
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u/DebatedKnowledge Harmonic Analysis 7d ago
What topic was your PhD in? I've never used hyperreals, from my perspective they seem mostly just a curiosity
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u/NoSuchKotH Engineering 6d ago
Part 1 of my thesis was on modeling of noise in atomic clocks. I needed proofs for stuff that every engineer just assumes to be true, mostly because every textbook since the dawn of time says they are true. Just, that I never could find a proof for those. Not a single one. So, in my ignorance and hubris, I sat down to prove these myself. Can't be that difficult, right? Everyone is using those, right? Well... it took me almost a year, and some pretty non-standard math (pun intended)
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u/SuppaDumDum 2d ago
So the proofs were not engineer proofs, but where they physicist proofs or mathematician proofs?
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u/NoSuchKotH Engineering 2d ago
My advisor was a mathematician, so I made sure he was happy with the proofs.
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u/Traditional_Town6475 6d ago
I’m actually curious now. What sort of problem would using the hyperreals be a better approach than standard technique (not so much for ease of proof, but because I imagine the typical analyst would be unfamiliar with nonstandard analysis).
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u/NoSuchKotH Engineering 6d ago
They helped me get a proof of a variant of the dominated convergence theorem that works for functions which are not absolutely integrable.
I do not know whether there is a way to prove the same without hyperreal numbers. At least my feeble engineer's brain could not find one.
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u/Traditional_Town6475 5d ago
Well by Łoś’s theorem, there’s always a way to go back to standard techniques, albeit the standard approach may be “ugly”.
Do you remember the assumptions about the function?
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u/NoSuchKotH Engineering 5d ago
The specific class of functions I was interest in was power limited functions. I.e. for which lim_{T -> infty} 1/T \int_{-T}^T (f(x))^2 dx converges.
The proof works for any "normalized" integrable function (i.e. for an arbitrary sequence of g_k(x) that converge to a function g(x) a.e, with g_k being used as a normalization function \int g_k(x) (f(x).)^2 dx
Currently the proof is only in my thesis, which, if I'm being honest, nobody is going to read. At some point I need to figure out where to publish this result and find the time to write a proper paper about it.
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u/DebatedKnowledge Harmonic Analysis 5d ago
I would love to read the proof. It looks like you're dealing with normalizations of L^2 norms. I wonder if your work was related to almost-periodic functions. Would love to read an exact statement.
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u/SergeAzel 7d ago
Given my comp sci history, quaternions are a classic obviously.
Fav so far gotta be the (finite) rings and fields of polynomials over gf(2).
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u/DebatedKnowledge Harmonic Analysis 7d ago
What interests you in the polynomials over gf(2) and what I assume are the rational functions over gf(2)? Like what problems, etc
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u/SergeAzel 6d ago
What interests me is simple. Basic, trivial one could even say.
I work with computers. Like standard binary, polynomials over gf2 can be perfectly represented by bits.
But despite this, they were entirely absent from my formal education (compsci, not math).
I am aware they have uses in both cryptography and digital communications, etc, however nothing in my education path included that at least.
But even so I find it fascinating that, despite being a practically fundamental operation, polynomial multiplication over gf(2) has no native support in most/popular modern software developments languages, as far as I've seen. Even assembly instructions for it are relatively obscure?
And I just wonder how much software engineering might be missing out on for that.
Anyways.. I'm probably wrong about a lot here so I'll stop.
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u/DebatedKnowledge Harmonic Analysis 5d ago
Ohhhh I see. Polynomials over gf2 can be perfectly interpreted via base 2 integers.
Integer addition is just bitwise XOR and multiplication (apparently it's encoded via the instruction
pclmulqdqon x86 processors https://codegolf.stackexchange.com/questions/50240/xor-multiplication). Also you might find something like that in say python packages for working with finite fields / algebra. I think numpy can deal with other fields for example too.
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u/Darian123_ 7d ago
My favourite infinite dimensional numbers are the real numbers
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u/Miss-Quiz-Mis 6d ago
Explain?
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u/Darian123_ 6d ago
The real numbers ar are a field extension of the rational numbers Q. You can easily check yourself that this means that you can interpret the real numbers as a Q - vector space. Since it is also very easy to confirm that as a Q vector space the reals dont have a finite basis, that means that the real numbers are an infinite dimemsional Q vector space. Hence my statement
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u/Darian123_ 6d ago
Its similiar to how the complex numbers are an R vector space, with the added complexity of a lack of a basis
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u/hmmstdvent 7d ago
I’m a fan of the uhh, let’s say square numbers. A square number is a real linear combination of n^2 imaginary units {e_ij}, indexed by a pair of indices i,j with values from 1 to n, the multiplication table for these units is given by e_ij * e_kl = δ_jk e_il.
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u/RingularCirc 4d ago
Did you know square numbers are actually endomorphisms of an object n in a particular semiadditive category? (The particular category Mat_ℝ contains various kinds of square and rectangular numbers as morphisms between arbitrary natural numbers m → n, and allows them to mingle.)
Taking other semiadditive categories allow to generalize those kinds of rectangular numbers to weird depths, like, to binary relations, weighted graphs, rings, groups etc..
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u/DebatedKnowledge Harmonic Analysis 7d ago
These are called matrices!
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u/hmmstdvent 6d ago
Shhh, they're numbers, in a number system :)
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u/DebatedKnowledge Harmonic Analysis 5d ago
What's wrong with calling matrices numbers? It's an informal question after all
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u/Gilded-Phoenix 7d ago
This actually brings up a good question: when does it stop being numbers and start being some other, more complicated object? In Pythagoras' and Euclid's days, it was anything beyond natural numbers. Today, we accept real numbers, and even complex numbers. However, are quaternions numbers, or are they a more deeply structured vector? Octonions? etc. What makes a number "numberish?"
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u/DebatedKnowledge Harmonic Analysis 7d ago
I think it's vibes based, just answer based off that, not everything needs a formal description :)
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u/andimai 6d ago
Numbers are for counting, measuring, comparing and ordering. Anything that cannot be used for that should not be called a number. That was a historical mistake, that still causes a lot of confusion and unnecessary complexity (pun intended) in math.
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u/RingularCirc 4d ago
That's subjective, though, ain't it? People want to call different collections of algebraic/order structures "numbers", and we indeed see they do, and the question "what's a number" is ever reiterated. So far we haven't seemed to converge on it!
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u/andimai 3d ago
„A number is a mathematical object used to count, measure, order and label.“
This is the first sentence in https://en.wikipedia.org/wiki/Number .
Of course you can always make up your own definitions. But then nobody will understand you.
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u/edderiofer Algebraic Topology 3d ago
Please count sqrt(2) apples for me. If you can't, I guess sqrt(2) isn't a number, by that definition.
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u/andimai 2d ago
But you can measure the diagonal of a square with side of length 1 and you will get sqrt(2). And you can order that and say this is less than the diagonal of a square with sides of length 2. I think the minimum you need to be able to do with a number is measuring and ordering by size.
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u/edderiofer Algebraic Topology 2d ago
No, you are making up your own definition. As you argued earlier, a number is a mathematical object used to count, measure, order and label. If you can't do all four, then it's not a number.
Please count sqrt(2) apples for me. If you can't, then sqrt(2) isn't a number by that definition.
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u/RingularCirc 16h ago edited 16h ago
Of course you can always make up your own definitions. But then nobody will understand you.
Please give some references assuring us that the definition from Wikipedia is actually used by mathematicians, at least in a sense that they will mostly converge on it without having conspired beforehand.
I mean no offence to the chain of edits that ended up making the article as you found it, but I wouldn't treat it as a definition hard enough and universal enough as to be followed by all, compared to just being a handwavy raison d'être, a heuristic for understanding, a first approximation to the idea that is itself seemingly vague enough and not deserving enough standardization that it remains as it is: we definitely call some algebraic structures "numbers", definitely don't call some others and there's a lot of grey area inbetween. And frankly that's not a problem in itself.
And also believe me I'm not a fan of somebody diluting good definitions of terms that do exist: I often like to get as specific as possible myself. But this is just not the case; the world of humans is like that.
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u/Traditional_Town6475 6d ago
l^infinity(N). It most certainly is an algebra over the reals and has all sorts of fun properties.
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u/RingularCirc 4d ago
Clifford algebras (maybe including over degenerate quadratic forms, but those are a pain...), octonions (I don't know them much, but they're both very important, quirky and not a Clifford algebra, unlike splits, ℂ and ℍ) and some of those homebrew algebras to describe affine similarities, though those should all be (even) Clifford as well.
But I don't see much sense in the question, except something of a sentimentality, so my answer is probably pretty useless!
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u/DebatedKnowledge Harmonic Analysis 7d ago
Probably the ring of adeles, if you can consider them to be infinite dimensional.
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u/forgetfulfunctor1 7d ago
Can you well-define what is "number system"?
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u/AndreasDasos 7d ago edited 4d ago
From a purely descriptivist perspective, the well/known structures whose elements to be called ‘numbers’, due originally I suppose to ‘vibes’, seem to include the following:
Cardinals and ordinals in ZFC.
Anything that’s typically presented as a subset of (has a traditional, canonical, often suppressed injection into) R or C, as well as integers modulo N (often still identified with actual integers). This covers N, Z, Q, algebraic closure of Q, random irregular subsets, etc.
Extensions of R to Cayley-Dickson algebras, and variants: H, O, sedenions etc., as well as split complex numbers and dual numbers. For some reason not extending to Clifford or Grassmann algebras beyond the simplest cases.
p-adic numbers and similar constructions.
Extensions of R and C that handle infinitesimals for non-standard analysis: hyperreals, superreals, surreals.
For some reason not the likes of SU(2), etc.
I don’t know if anyone has quite tried to unify a definition that matches the vibe of what qualifies as a ‘number’ and what doesn’t in this way, but suspect this isn’t worthwhile and a lot of it is simply that we’ve decided quaternions are ‘numbers’ but elements of SO(3) are not.
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u/RingularCirc 4d ago
Great list!
I think projective lines over "numbers" can as well be called numbers somewhere (that means mostly adding the sole projective ∞). Not sure
For some reason not extending to Clifford or Grassmann algebras beyond the simplest cases.
I have very much seen somebody talking about "Clifford numbers" in an older book about Clifford algebras. I don't remember who exactly. And I'm not sure if those were all elements of the algebra or only specific ones (say, scalar multiples of products of vectors).
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u/DebatedKnowledge Harmonic Analysis 7d ago
Why would you do that? It's up to your interpretation what it should be. Not everything requires a formal definition. Do not let formal definitions rob you from your thinking. Doing math post-rigorously is realizing that math is whatever you want it to be, and that analogies and visualizations are worth a thousand half-assed definitions.
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u/RingularCirc 4d ago
True but personally I don't see what's the use in understanding differently which entities are "numbers" and which aren't. This seems almost the purest case of naming conventions that don't add essence, like, intuitions.
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u/Breki_ 7d ago
A number system is a set of numbers. A number is anything that resembles other objects that are called numbers
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u/DebatedKnowledge Harmonic Analysis 7d ago
This basically, the original question is pure undergrad one-upper energy.
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u/Short_Bluebird_3845 7d ago
"a tensor is something that transforms like a tensor" ahh response
but yeah, number system = numbers.
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u/Short_Bluebird_3845 7d ago
idk, any set such that each element is a sum of a scalar and a unit and has component-wise addition and well-defined multiplication will do.
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u/TheNukex Graduate Student 7d ago
You seem to be mixing fields and vector spaces. Scalars and component wise addition belongs to vectorspaces. Unit does have a vector space meaning, but addition of scalar and units are not defined. Unit also has a field meaning, in which case it would make sense with addition of scalars, however then it's a very weird condition. Furthermore multiplication is not usually defined on vector spaces, that is again a field property.
You need to be more clear if you want people to understand your idea.
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u/DebatedKnowledge Harmonic Analysis 7d ago
I don't think being clear is necessary. You can just interpret the question loosely and give your answer based off of vibes, this isn't an exam.
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u/Short_Bluebird_3845 7d ago
scalar generally means real numbers, and component-wise addition happens in (common?) complex numbers and quaternions, 1 is technically a unit, and number systems do have multiplication.
alright then, how would you formally define it? hypercomplex numbers does not cover all N-dimensional numbers.
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u/TheNukex Graduate Student 7d ago
Scalar generally means any number from your base field. If you want to view complex and quarternion addition as component-wise, then you are essentially treating them as R^2 and R^4, so again we are back to vector spaces. All non-zero real numbers are units. Whether "number systems" have multiplication, depends on how you define it.
There is no consensus on what a "number system" is, but to most mathematicians, it would either be a ring or just some integer base system (like base 10). However we tend to view both of those as 1-dimensional. With that said dimensions depend on what you think of them with respect to. Real numbers (and complex, quarternions and so on) are infinite dimensional with respect to rationals.
Since you brought up dimensions, scalars and sums of units it seemed like you were thinking of vector spaces (with orhtonormal basis), but upon reading both your comments i think i know what you are thinking of.
You mean the set {a_0+a_1j_1+...+a_Nj_N | a_i∈R} where addition is component wise, and there is some defined multiplication on it, with some relation defining products of different j_i.
This would yield complex numbers as {a+bi | a,b∈R} for N=2 and quarternions {a+bi+cj+dk | a,b,c,d∈R} for N=4. Is this what you were thinking?
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u/Short_Bluebird_3845 7d ago
Bingo!
(also, isn't that what all N-dimensional numbers are? like, conisder Shlomo Jacobi's 3D number system)3
u/TheNukex Graduate Student 7d ago
The answer is a bit complicated. a+bi is really just a number, it's a complex number, and you need not think of it as having multiple dimensions (the number of dimensions depend on what you take it with respect to).
The real numbers are a field (a ring with really nice properties), and as it turns out complex numbers are useful because they remain a field AND they solve a major problem of the real numbers. Then others, like you, thought to themselves if we could keep going, and just add these "extra dimensions". However you run into problems very quickly. I am not super familiar with Shlomo Jacobi's system, however a quick google search says that not all non-zero elements have a multiplicative inverse. In other words division no longer behaves nicely. So we lose a property that is so inherent to our idea of numbers you can multiply. Quarternions don't have that problem, however they lose commutativity, so a*b need not equal b*a. Then moving on from there you lose associativity and more. So it turns out that trying to work in higher dimensions like this loses so many properties, that it turns useless for a vast majority of topics.
All those problems arise from the behaviour of multiplication, so this is where vector spaces fix the issue. For vector spaces you only require componen-wise addition, however you define only multiplication by scaling your vector. That way you get to keep many nice properties, and you can go to as many dimensions as you want, while keeping everything the same.
That's the reason why people don't think of those structures when you say N-dimensional numbers, because it turns out, they are just not that useful compared to the vector space counterpart. It's a case of it seems neat, until you work with it and realize that it's not as nice as you had hoped, so you abandon it and go a different way.
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u/dwbmsc 7d ago
The complex numbers and quaternions are algebras over the field of real numbers.
https://en.wikipedia.org/wiki/Algebra_over_a_field
So I think you mean an algebra over a field. People are complaining because term number system seems ambiguous.
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u/andimai 6d ago
None! Numbers are for counting and comparing things. Everything which isn't an ordered field shouldn't be called a number.
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u/DebatedKnowledge Harmonic Analysis 5d ago
This is such a strong take, I personally disagree! Everything that gives you the motivation of a number is a number
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u/SubjectAddress5180 6d ago
The rationals. They are so complicated, especially the integer subset thereof. The continuous we do slowly; the discrete takes infinite time.
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u/nulvoid000 5d ago
Quaternions. Fucking goated and gives you dirac equation and many sort of new “spin” invariants.
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u/daley972 4d ago
Formal power series are cool. You can do some cool stuff with Riordan arrays. You're often only dealing with series with integer coefficients but it still counts to me. Lol
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u/Zeteticon 1d ago
The infinite levels of the Cayley-Dickson algebras. They are defined for all dimensions a power of 2. 1,2,4,8,…
Their main algebraic property is that multiplication is defined: Rn*Rn -> Rn.
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u/Sproxify 7d ago
Gotta be polynomials (infinite dimensional). After that it's square matrices (n2 dimensions)
I'm interpreting "N-dimensional number system" as n dimensional algebra over the reals as that seems to be the type of object most in line with OP's examples.
Polynomials and matrices are elementary enough that you learn them before abstract algebra because they are so important and deep.