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I know about the concept of the complex plane, but is it possible to move to the third dimension? What about arbitrary many dimensions?

Edit: could you please give me some examples of 3D numbers?

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The properties of the number $i$ don't "break" the properties of the real numbers, they expand on them in a way that allows for square roots of negative numbers to be actual values. – Envious Page Sep 22 '12 at 17:01
@EnviousPage, of course, that's why I put it in the quotation signs... – jcora Sep 22 '12 at 17:09
Well, it does break the total ordering property: Unlike for real numbers, there's no total ordering of the complex numbers which is compatible with its algebraic structure. – celtschk Sep 22 '12 at 17:40
up vote 12 down vote accepted

This was a popular pursuit in the 19th century, called hypercomplex numbers, based on the successful example of quaternions and the more esoteric case of octonions. Hypercomplex numbers are now called "finite dimensional associative algebras" but of such algebras, only quaternions and octonions retain features that resemble complex numbers. Matrices form another algebra similar to complex numbers in important ways, and have their own geometric interpretation different from that of complex numbers and quaternions.

Quaternions are related to 3-dimensional rotations and this aspect is generalized to all higher dimensions by Clifford algebras and the representation theory of the $n$-dimensional rotation group.

To answer the EDIT, there are no nontrivial 3-dimensional examples but you can write down trivial examples by listing the multiplication table. Take $1$ and $i$ with the multiplication law the same as in the complex numbers, and add a new element $N$ with some rule for how to multiply it by $1$ and by $i$, such as $N 1 = 1 N = iN = Ni = N$. This gives an associative commutative multiplication law on "3-d numbers" of the form $a.1 + b.i + c.N$, which can be thought of as a rule for combining triples $(a,b,c)$ and $(a',b',c')$ into a third triple. Examples like this are easy to write down, but there is no rule of combination for triples that goes beyond a minor variations on real or complex numbers. Another example of a trivial variation is to multiply the first two components as complex numbers and the third as real numbers. This is called the "direct product" or "direct sum" of the real and complex number system but there is no new structure there.

In dimension 4 there appear new and interesting examples, the quaternions and the algebra of 2x2 matrices.

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Octonions are not associative, so are excluded from the later interpretation of hypercomplex numbers as finite dimensional associative algebras, but are part of the 19th century project to look for additional quaternion-like multiplication rules that could be related to geometry. – zyx Sep 22 '12 at 21:04

You might want to look at the Wikipedia article on quaternions. This gets us to dimension $4$, and is useful (and used) for computer graphics. Beyond dimension $4$, there are the octonions, which are not too badly behaved, though we lose associativity of multiplication. Beyond octonions, we lose too many of the algebraic properties of "numbers."

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What sort of properties do we begin to lose? I know with quaternions multiplication doesn't commute. What else do we begin to lose? – Michael Dyrud Sep 22 '12 at 17:09
Associativity, for one. – André Nicolas Sep 22 '12 at 17:10
How can you loose those? Wow, I really need to research into this... – jcora Sep 22 '12 at 17:12
For losing associativity, perhaps look at octonions (Cayley numbers). But maybe I was a little too hard on octonions. – André Nicolas Sep 22 '12 at 17:19
At least, we still have power-associativity! – M Turgeon Sep 22 '12 at 17:21

The Cayley–Dickson construction is less than the whole truth, but it's there. It's an infinite sequence beginning with the reals; the second step is the complex field; the third is the quaternions; the fourth is the octonions.

Quaternion multiplication is not commutative; octonion multiplication is not associative.

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But there is nothing known to be interesting after the fourth or fifth step, and it is provably uninteresting in well-defined senses, such as there not being a bilinear sum-of-squares identity past dimension 8. – zyx Sep 22 '12 at 18:11
Is it possible that even though no individual case is interesting, the sequence as a whole is interesting? – Michael Hardy Sep 22 '12 at 18:34
Is that equivalent to the existence of an interesting statement about a finite or infinite subset of the sequence, where at least one of the individuals in the subset has dimension higher than 16? – zyx Sep 22 '12 at 20:59

Historically, people have accepted as "numbers" only a limited number of systems that "break" the properties we are familiar with for real numbers. But this is really only a matter of terminology. If you focus on generalizations of the transformation rules that rotations and dilations of the plane satisfy, you start to glimpse a vast part of the landscape of advanced mathematics. Linear algebra, Lie group theory, operator theory and dynamical systems, just for a start, are aimed at describing useful kinds of transformations on complicated kinds of data, and the rules that govern them.

A simple example in 3D is Euler's rotation theorem: the composition of two rotations in 3D, about possibly different axes, is another rotation, whose axis can be computed. Note that the order matters: 3D rotations do not necessarily commute!

So yes, there are many ways to generalize 2D rotations and get objects with fantastic and useful rules. It's just that we don't call them "numbers."

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It is very simple to go to three dimensions, recall that the complex plane is more or less crafted from the fact that $i^2 = -1$. For $\mathbb{R}^3$ you simply define a number $j$ that has the property $j^3 = -1$.

You get a perfect number system (even a perfect complex analysis thing) but it is not a field because there are so called divisors of zero to be found.

Analog to the complex plane $\mathbb{C}$ where it is usual to identify every point $(x, y)$ with a complex number $z = x + iy$, in $\mathbb{R}^3$ you indentify $(x, y, z)$ with $X = x + jy + j^2z$.

An amazing fact is that the equation $X^2 = -1$ has no solution inside this $\mathbb{R}^3$ number system. But if you think about it, that is good because otherwise going to 3 dimensions adds nothing new...

(Yeah yeah, think about that.)

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For R3 you simply define a number j that has the property j^3 =− 1. - isn't that -1? – jcora May 10 '14 at 21:14

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