# Is there a third dimension of numbers?

Is there a third dimension of numbers like real numbers, imaginary numbers, [blank] numbers?

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Alas, there are no algebraically coherent “triplexes”. The next step in the construction as has been said already are “quaternions” with 4 dimensions.

Many young aspiring mathematicians have tried to find them since Hamilton in the 19th century. This impossibility links geometric dimensionality, fundamental properties of polynomial equations, algebraic systems and many other aspects of mathematics. It is really worth studying.

A quite recent book by modern mathematicians which details all this for advanced college undergraduates is Numbers by Ebbinghaus, Hermes, Hirzebruch, Koecher, Mainzer, Neukirch, Prestel, Remmert, and Ewing.

However, the set of quaternions with zero real part is an interesting system of dimension 3 with very interesting properties, linked to the composition of rotations in space.

You may also find of interest some more general results besides the mentioned Frobenius Theorem. Weierstrass (1884) and Dedekind (1885) showed that every finite dimensional commutative extension ring of $\mathbb R$ without nilpotents ($x^n = 0\,\Rightarrow\, x = 0$) is isomorphic as a ring to a direct sum of copies of $\rm\:\mathbb R\:$ and $\rm\:\mathbb C\:.\:$
Wedderburn and Artin proved a generalization that every finite-dimensional associative algebra without nilpotent elements over a field $\rm\:F\:$ is a finite direct sum of fields.

Such structure theoretic results greatly simplify classifying such rings when they arise in the wild. For example, I applied a special case of these results last week to prove that a finite ring is a field if its units $\cup\ \{0\}$ comprise a field of characteristic $\ne 2\:.\:$ For another example, a sci.math reader once proposed an extension of the real numbers with multiple “signs”. This turns out to be a very simple case of the above results. Below is my 2009.6.16 sci.math post on these “PolySign” numbers.

The results in Eitzen’s paper Understanding PolySign Numbers the Standard Way, characterizing Tim Golden’s so-called PolySign numbers as ring direct sums of $\mathbb R$ and $\mathbb C$, have been known for over a
century and a half. Namely that $\rm\:P_n =\: \mathbb R[x]/(1+x+x^2+\:\cdots\: + x^{n-1})\$ is isomorphic to a certain ring direct sum of $\:\mathbb R$ and $\:\mathbb C\:,\:$ is just a
special case of more general results due to Weierstrass and Dedekind
in the 1860s. These classic results are so well-known that you will find
them mentioned even in many elementary textbooks on number systems and
their generalizations. For example, in Numbers by Ebbinghaus et.al. p.120:

Weierstrass (1884) and Dedekind (1885) showed that every finite
dimensional commutative ring extension of R with unit element but
without nilpotent elements, is isomorphic to a ring direct sum of
copies of R and C.

Ditto for historical expositions, e.g. Bourbaki’s Elements of the
History of Mathematics
, p. 119:

By 1861, Weierstrass, making precise a remark of Gauss, had, in his
lectures, characterized commutative algebras without nilpotent elements
over R or C as direct products of fields (isomorphic to R or C);
Dedekind had on his side reached the same conclusions around 1870, in connection with his “hypercomplex” conception of the theory of commutative fields, their proofs were published in 1884-85 [1,2]. […] These methods rely
above all on the consideration of the characteristic polynomial of an
element of the algebra relative to its regular representation (a polynomial
already met in the work of Weierstrass and Dedekind quoted earlier) and
on the decomposition of the polynomial into irreducible factors.

Nowadays these fundamental results are merely special cases of more
general structure theories for algebras that are part of any first course
on algebras (but not always met in a first course on abstract algebra).
A web search turns up more on the subsequent history, e.g. excerpted from

Y. M. Ryabukhin, Algebras without nilpotent elements, I,
Algebra i Logika, Vol. 8, No. 2, pp. 181-214, March-April, 1969

Algebras without nilpotent elements have been studied long ago. So,
Weierstrass characterized in his lectures in 1861 finite-dimensional
associative-commutative algebras without nilpotent elements over the field
of real or complex numbers as finite direct sums of fields. To be exact,
some nonessential restrictions have there been imposed. In 1870 Dedekind
removed those nonessential restrictions. The following theorem of
Weierstrass-Dedekind is now considered as a classical one: every
finite-dimensional associative-commutative algebra without nilpotent
elements over a field F is a finite direct sum of fields. The results of
Weierstrass and Dedekind (for the case when F is the field of complex or
real numbers) have been published in [1,2]. The results of works of
Molien, Cartan, Wedderburn and Artin [3-6] imply that Dedekind’s theorem
holds for any field F. Moreover, the following theorem of Wedderburn-Artin
holds: every finite-dimensional associative algebra without nilpotent
elements over a field F is a finite direct sum of fields.” […]

1. K. Weierstrass, “Zur Theorie der aus n Haupteinheiten gebildeten
complexen Grossen,” Gott. Nachr. (1884).
2. R. Dedekind, “Zur Theorie der aus n Haupteinheiten gebildeten complexen
Grossen,” Gott. Nachr. (1885).
3. F. Molien, “Ueber Systeme hoherer complexer Zahlen,” Math. Ann., XLI,
83-156 (1893).
4. E. Cartan, “Les groupes bilineaires et les systemes de nombres complexes,”
Ann. Fac. Sci., Toulouse (1898).
5. J. Wedderburn, “On hypercomplex numbers,” Proc. London Math. Soc. (2),
VI, 349-352 (1908).
6. E. Artin, “Zur Theorie der hyperkomplexen Zahlen,” Abh. Math. Sere.
Univ. Hamburg, 5, 251-260 (1927).

and excerpted from its sequel

Y.M. Ryabukhin, Algebras without nilpotent elements, II,
Algebra i Logika, Vol. 8, No. 2, pp. 215-240, March-April, 1969

In [1] we proved structural theorems on the decomposition of algebras
without nilpotent elements into direct sums of division algebras;
certain chain conditions were imposed on these algebras.

Yet it is possible to prove structural theorems also without imposing any
chain conditions. In this case the direct sums are replaced by subdirect
sums and instead of division algebras we shall consider algebras without
zero divisors.

The first structural theorem of this kind is apparently the classical
theorem of Krull [2]:

Any associative-commutative ring without nilpotent elements can be
represented by a subdirect sum of rings without zero divisors.
Krull’s theorem was subsequently extended to the case of any associative
ring. This was done by various authors and in various directions. In [3],
Thierrin came very close to a final generalization of Krull’s theorem to
the associative, but not commutative case. The final result was obtained
in [4]:

Any associative ring without nilpotent elements can be represented by a
subdirect sum of rings without zero divisors.
At the Ninth All-Union Conference on General Algebra (held at Gomel’),
I. V. L’vov reported an even stronger result:

Any alternative ring without nilpotent elements can be represented by a
subdirect sum of rings without zero divisors.

It could be assumed that the theorem on decomposition into a subdirect sum
of algebras without zero divisors holds for any ring. Yet this assumption
is erroneous (see [1]), since there exists a finite-dimensional simple,
special Jordan algebra without nilpotent elements that has zero divisors
and cannot therefore be decomposed into a subdirect sum of algebras (or
rings) without zero divisors.

There naturally arises the following question: what conditions must
a ring without nilpotent elements satisfy to permit its representation
by a subdirect sum of rings without zero divisors?

In this paper we answer this question:

An algebra R over an associative-commutative ring F with unity can be
represented by a subdirect sum of rings without zero divisors, iff
it is a conditionally associative algebra without nilpotent elements.

Let us recall that an algebra R is said to be conditionally associative,
iff we have in R the conditional identity x(yz) = 0 iff (xy)z = 0.

We say a (not necessarily associative) algebra R does not have nilpotent
elements, iff in R we have the conditional identity x^2 = 0 iff x = 0.

From this theorem we easily obtain the above-mentioned results of [2-4],
as well as the result of L’vov (it suffices to take as the ring F
the ring Z of integers). […]

1. Yu. M. Ryabukhin, “Algebras without nilpotent elements,I,” this issue,
pp. 215-240.
2. W. Krull, “Subdirect representations of sums of integral domains,”
Math. Z., 52, 810-823 (1950).
3. J. Thierrin, “Completely simple ideals of a ring,” Acad. Belg. Bull.
C1. Sci., 5 N 43, 124-132 (1957}.
4. V. A. Andrunakievich and Yu. M. Ryabukhin, “Rings without nilpotent
elements in completely simple ideals,” DAN SSSR, 180, No. 1, 9 (1968).

Every finite-dimensional division algebra over $\mathbb{R}$ is one of $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$. This is what is called the Frobenius Theorem. You may refer to here for details.

You might look up quaternions.

In addition to complex numbers and quaternions, you might want to look up Clifford Algebras which encapsulate both and extend to arbitrary dimension. Complex and quaternios are sub-algebras of the Clifford Algebras over $\mathbb{R}^2$ and $\mathbb{R}^3$ respectively.

A lot of responses bring up the fact the next closed algebraic set are quaternions. But these aren’t perfect since there is no problem to which Quaternions naturally arise as a solution:

For example we know $i$ naturally is formed as the solution to the previously unsolvable $\sqrt{-1}$ that being said all polynomials have roots in the complex plane so we are guaranteed no other such new algebraic unit (like) $i$ will be naturally formed.

Now I personally don’t know of a theorem that says this CANNOT happen again. For example it could be that:

$$x^x = i$$
Or (if we define ${}^xx$ to mean tetration)

$${}^xx = i$$

Etc… following the pattern, may not have a solution in C. In that case we are now free to generate a new elementary unit $j$ however this elementary unit will be quite strange since it is associated with a higher operator so expressions such as:

$$j, j^2, j^3 …$$

Are all unique and simplified, leading us to a now infinite dimensional system of numbers

So 3D is not possible, but I believe infinite dimensional is still possible.

Unless you like the unnatural creations that are the quaternions etc… (I only dislike them because of their lack of natural formation)

There are of course, examples of R3, in the form X, Y, Z, which behave as three separate real numbers, with an invarient when X -> Y -> Z -> X applies.

Such numbers turn up in the study of the heptagon and enneagon. The integer systems are discrete points forming a lattice in this space, the invariants cycle through the solutions to the heptagonal ($x^3-x^2+2x-1=0$) and enneagonal equations, which transforms eg {7} to {7/3} to {7/2}.

Such a projection converts an infinitely dense arrangemet of points onto a sparse lattice in three dimensions. That is, in $x,y,z$ form, there is a sphere of size where there are no more than one legitimate value. So if you evaluate a value even approximately in the three coordinates, it is possible to find the exact value.

One can, if one considers the span of numbers like $1.801937736$, $-1.2469796037$ and $0.445041867912$, which serve for the heptagon, the same role that $1.618033$ and $-0.618033$ do for the pentagon, then the cyclic rotation of this list will preserve multiplication relations, etc. eg suppose that the order is $p, q, r$ Now $p^2=2-q$ and $q^2=2-r$, and $r^2=2-p$. One notes that $p+q+r=p*q*r=1$. These $p$, $q$, $r$ are the solutions to the cubic $x^3-x^2-2x+1=0$ in much the same way that 1.618 and -0,618 solve $x^2-x-1=0$.

The product, for example of $ap+bq+cr$ and its transforms ($aq+br+cp$ and $ar+bp+cq$), is always an integer, speciffically the product of any of these: 7, cubes, and of primes of the form $7n+1$, $7n-1$.

Since one can suppose that there is a real hypercomplex plane, of the form of numbers $a+bj$, where $j=\sqrt{1}$. The conjucate is $a-jb$. The pentagonal numbers belong here. It features unit hyperbolae, and two ‘real’ axies, where the values which the value and its conjucates project onto.

And if one can do it meaningfully with two conjucates, it’s possible with three, four, five, &c. I use such systems extensively when wrangling with polygons.

If you think of the dimensions for numbers as going real numbers (1st dimension), fuzzy numbers (2nd dimension), then the 3rd dimension ends up fuzzy numbers of dimension two. For more details see A. Kaufmann and M. M. Gupta’s Introduction to Fuzzy Arithmetic or George and Maria Bojadziev’s Fuzzy Sets, Fuzzy Logic, Applications.