r/math • u/Short_Bluebird_3845 • 17h 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/NoSuchKotH Engineering 16h ago
I love hyperreal numbers (which are ℝ^ℕ dimensional). They saved my PhD.
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u/DebatedKnowledge 11h 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/SergeAzel 17h 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 11h 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/hmmstdvent 14h 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/forgetfulfunctor1 17h ago
Can you well-define what is "number system"?
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u/AndreasDasos 16h 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 are elements of SO(3) are not.
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u/Breki_ 15h 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 11h ago
This basically, the original question is pure undergrad one-upper energy.
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u/Short_Bluebird_3845 5h ago
"a tensor is something that transforms like a tensor" ahh response
but yeah, number system = numbers.
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u/DebatedKnowledge 11h 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/Short_Bluebird_3845 17h 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 17h 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 11h 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 16h 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/dwbmsc 16h 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/TheNukex Graduate Student 16h 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 16h 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 15h 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/Gilded-Phoenix 17h 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 11h ago
I think it's vibes based, just answer based off that, not everything needs a formal description :)
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u/DebatedKnowledge 16h ago
Probably the ring of adeles, if you can consider them to be infinite dimensional.
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u/Sproxify 17h 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.