Can numbers exist outside of the number space?

I was unsure about how to actually write the question.

I know that in the real number axis we have the naturals, integers, rationals, irrationals and reals; and in the imaginary number axis we have the imaginary numbers, together forming the complex number plane. I also know that adding dimensions to that plane we can have numbers of dimension $2^n$ (quaternions, octonions, etc).

What I want to know is: Are more numbers outside of that? Has a type of numbers ever been created from scratch just to satisfy or demonstrate a given property, even if they are not used outside of that? Are there numbers that are not numbers by the previous definition?

Any reference or links to lists of number types other than those I have mentioned would be greatly appreciated.

• What's your definition of "number"? Does it include integers mod m, polynomials, polynomials mod f, power series, matrices, tuples of numbers with pointwise operations? – Bill Dubuque Mar 8 '14 at 20:02
• Aside: you say "quaternions, octonions etc." but actually that's pretty much it, at least for the division algebras. You can keep iterating the construction after that but division by any nonzero element stops being possible. – Ben Millwood Mar 8 '14 at 20:03
• @Achifaifa I think it is necessary to make that definition more precise in order to get a useful answer. – augurar Mar 8 '14 at 20:45
• Figure out what a number is, you say? We have a question for that. – Rahul Mar 9 '14 at 5:43
• Ordinals and cardinals are extensions of natural numbers but not integers. – starblue Mar 9 '14 at 17:00

It depends what you mean by a "number." If you mean a number system over which we can perform operations we're used to, namely division, we can only have 3 finite-dimensional associative division algebras over the real numbers, $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$, as proven by Frobenius.

An algebra is a vector space equipped with a bilinear product (in Layman's terms, a space where we have vectors and can multiply numbers), so a real division algebra is an algebra over the reals where we can perform division.

But the more we relax these axioms, the more number systems we can have. If we give up associativity, we can have octonions, and if we give up division, there are sedenions. We also have M-algebras, Musean hypernumbers.

All in all, we can make up whatever we want by dropping more and more of our field axioms, but as we do so, the number system we make becomes less and less useful and well-defined.

• Interesting. Only three division algebras, you say? What about all those hyperreal fields, all those non-standard real-closed fields, all their algebraic closures? I thought that division is always well-defined in a field. – Asaf Karagila Mar 9 '14 at 19:59
• @AsafKaragila Some algebras have multiplicative inverses but are not division algebras! See the Cayley Dickinson Construction: math.ucr.edu/home/baez/octonions/node5.html#cayley-dickson. I also edited my post to be more precise. – William Chang Mar 9 '14 at 20:02
• That's better. Thank you. (Am I right that mathematicians can be the most annoying people ever? :-D (also :-()) – Asaf Karagila Mar 9 '14 at 20:04
• Yes, haha! But definitions are extremely important and it's worth nitpicking sometimes! – William Chang Mar 9 '14 at 20:06

There are also the $p$-adic numbers. They arise from completing $\mathbb{Q}$ with respect to the $p$-adic metric, which is based on divisibility.

The numbers you've listed are the "hypercomplex" numbers; there are a few that you're missing there. There are also "transfinite numbers".

Yes, most of these numbers are used to create a system wherein certain manipulations are permissible. Yes, some of these "numbers" are not numbers per se by other definitions. Making new numbers isn't necessarily the only way to express these ideas, but it is often an intuitive and even elegant way to do so.

There are objects called supernumbers which are formed from an infinite series of Grassmann generators. Generally $z = z_o + z_i \eta^i + z_{ij}\eta^i\eta^j + \cdots$ where $\eta_i \eta_j = -\eta_j \eta_i$ and $z_I \in \mathbb{C}$. If you replace numbers with supernumbers then you obtain supermath which can be used to set up superfields. It is one possible formalism to set-up supersymmetric physics and/or classical field theories where anticommuting variables are needed for the path-integral formalism. That said, there are other approaches where the modification of math is made via sheaf-theoretic arguments... anyway, my point here is that supernumbers exist and there is a somewhat vast literature concerning them from about 1975-present. To be more honest, there are actually many different sorts of supernumbers, I just give a broad comment here.

Are [there] more numbers outside of [real division algebras]?

Nowadays, mathematicians don't really speak of "numbers", but of algebraic structures, non-empty sets equipped with one or more binary operations satisfying axioms. The binary operations are typically viewed as generalizing "addition" and "multiplication"; the axioms include the associative axiom, the existence of a neutral element and possibly of inverses, the commutative axiom, a distributive axiom for multiplication with respect to addition, and so forth.

Depending on the axioms satisfied, the resulting structure (non-empty set plus one or more operations) may be called a group, a ring, a field, or any of a plethora of similar terms.

If a collection of axioms is fixed, one can attempt to classify (up to isomorphism) the structures satisfying the axioms. Only relatively rarely is there a structure characterized completely by abstract properties. One famous example is the ordered field of real numbers, which is the unique complete, ordered field.

There are more types of algebraic structure, and more examples of such structures, than you can shake a stick at.