# New set of numbers possible?

If complex numbers were invented in order to solve equations like x^2 + 1 = 0, then is it possible for one to come up with a completely new set of numbers to solve certain unsolved set of problems? Will there be a day when all possible sets of numbers needed will have been invented?

• This would actually be a better question for math.SE Commented Oct 30, 2013 at 1:04
• Oh okay, I initially asked another question about numbers on the math one but people told me to ask it here, so then for this question I thought this was a more appropriate place to ask Commented Oct 30, 2013 at 1:07
• Using your own logic you already have your answer: the "day when all possible sets of numbers needed will have been invented" there will be no "unsolved set of problems", and a world without "unsolved set of problems" seems not human/not rational. Commented Nov 30, 2022 at 7:33

In one sense, the answer to your question is "no, complex numbers are the end of the line", and that is in the sense of being an algebraically, closed field. The next step in the sequence of algebraic field extensions is to ask, "Do all complex algebraic equations have at least one complex root?" Turns out they do (this is known as the Fundamental Theorem of Algebra, if that rings a bell from your high school days). So it appears we finally both have all the numbers we need (at least for these purposes) and have exhausted this method as resource for constructing new numbers. Time to get more creative if we'd like more toys to play with.

One well known route is the Cayley-Dickson construction which generates a sequence of algebras over the real numbers. Each step produces and real algebra with twice the dimension of the previous step. While the previous kind of extensions centered on the idea of broadening the variety of operations we may perform on numbers (subtraction, division, infinite summations, algebraic roots), the Cayley-Dickson extensions revolve around generalizing the variety of structures numbers may represent (from the number line, to the complex plane, and then higher dimensional spaces after that). After the real and complex numbers, there are quaternions, octonions, and sedenions. Now our chain of number sets and their subsets is up to:

ℕ ⊆ ℤ ⊆ ℚ ⊆ ℝ ⊆ ℂ ⊆ ℍ ⊆ 𝕆 ⊆ 𝕊.

And it wouldn't stop there. This process actually can be carried out ad infinitum. And are precursors of vector spaces and related structures of modern linear algebra.

There are many ways to answer this, but let me give you a progression.

1. ℕ natural numbers: 1, 2, 3, … (some contest that 0 ∉ ℕ)
2. 0 non-negative integers: 0, ℕ
3. ℤ integers: … -2, -1, ℕ0
4. ℚ rational numbers: ℤ/ℕ
5. ℝ real numbers: infinite sums of ℚ (e.g. rearrangements of alternating harmonic series)
6. ℂ complex numbers: solutions to polynomials with real (and complex) coefficients
7. quaternions
8. 𝕆 octonions
9. 𝕊 sedenions

I'm cheating a bit, because from #6 on, I'm really giving specific instances of algebras. Basically, this means that there are a possibly(?) infinite set of combinations of axioms which can be used to generate rules for manipulating numbers. An example would be Clifford algebras. So your question turns on: what is a number? If a number is merely an element of an algebra (roughtly: a set of symbols with rules of how they can be combined to get other symbols), I will answer a tentative Yes. We aren't guaranteed that reality has finite complexity, so it may be that we could be forever discovering new algebras which match new models of nature, and they would bring in new numbers.

As an example of how ℍ is useful, consider orientations in three dimensions. An easy way to visualize them is as a location on a sphere, like latitude/longitude. What is your latitude at the North Pole? There is no wrong answer, which means one runs into computational difficulties when one needs to "move through that point". If you fly directly over the North Pole, you are going North one instant, and South the next. Sudden 'flips' like this tend to be very bad, because we very much prefer continuous changes. ℍ avoids such singularities with orientations—pretend that you add "which way you're facing" to latitude/longitude.

For fun, our progression gains properties up through #5, then starts losing properties:

1. has total ordering
2. ? having zero is important
3. now a ring
4. now a field
5. Dedekind-complete ordering
6. linear ordering is lost
7. commutativity is lost
8. associativity is lost
9. alternativity is lost
• Great minds think alike =p. You raised an important point that I really should have included: even though the 2N-ion progression continues indefinitely, their usefulness does not. Hence, if you continue the search for alternative useful ways to generalize numbers, you are forced branch off of this chain of number sets. The result is a flowering, ever-branching "bush" of subset relations, instead of just a 1-dimensional chain. Commented Oct 29, 2013 at 21:03
• @DavidH, :-) I have intuitive and ideological reasons for thinking that we can't just go off in a single direction forever. (1) Even the ordinal numbers have a biggest number, ε0. Intuitively, this says that we just haven't yet figured out how to describe bigger infinities. (2) I think the nature of reality is too complex to allow simple enumeration (e.g. past sedenions); I think reality is infinitely complex in infinitely surprising ways. Stated alternatively: I think mathematicians will always have fascinating jobs. :-) Commented Oct 29, 2013 at 21:22
• @labreuer Late to the party, but $\epsilon_0$ is not the biggest ordinal number in any sense. In fact, it's even countable! There is no largest ordinal number at all. Commented Nov 30, 2022 at 5:39

The complex numbers form a complete algebraically closed field, but treating them as numbers makes it difficult to define what numbers are1. As other answers indicated, should we consider quaternions, octonions or sedenions as numbers? What about p-adic numbers?

What immediately comes to my mind is that you can't tell "i" from "-i", even if you would find a real world manifestation of complex numbers. For quaternions, you can't tell "i" from any unreal quaternion of length one.

Is there a similar problem for p-adic numbers? I don't know, but I could check. However, I wrote an answer to this question, because nobody has mentioned the surreal numbers yet. They are the largest possible ordered field, and being an ordered field should be sufficient for the label "number".

1. Complex numbers are nevertheless named correctly, according to the red herring principle in mathematics. It stresses that a “red herring” need not, in general, be either red or a herring. Don't laugh. Look it up and think about it, then you will understand why.

• I just tried to look it up, and it seems that there is no such problem for p-adic numbers, i.e. the identity is the only automorphism of the field. So it looks like the p-adic numbers are also entitled to the label "number". But I will have to read a bit more about it, before I edit my answer. Commented Oct 30, 2013 at 1:07

It has been done quite a lot of times. Complex numbers for example, as you said yourself.

But also every other number set. Once, people found it weird, that they couldn't express "1/2" as one single number, so they just invented 0.5 for that. And then they had debts, so they created -10 as a possible number and so on. The only rule is, that it cannot collide with any of the accepted axioms of math, I think.

I, myself not being a mathematician, cannot imagine any problems in actual math that could be solved by creating another set of numbers, but there might be some and if so, maybe someone will someday create another set, conforming to the actual math rules.

I'd just like add one point aside to the present answers, which I really like.

Mathematics is a creative process. (Think of Fermat's Last Theorem and Wiles approach to it.) So, it is not only possible that there will be new sets of numbers, but it is very likely.

As @labreuer states, the OP of course breaks down to the question, what a number is. And from a viewpoint that numbers reflect real world concepts, which in general is not true for mathematics, it is hard to believe that we will have another set of numbers of this kind, as the constructions

ℕ ⊆ ℤ ⊆ ℚ ⊆ ℝ

not only are explainable with real world concepts, but they also somehow reflect what people do with numbers (see @PerikOnti's answer).

What we actually know from looking back is that mathematicians do actually invent new kinds of numbers. Apart from the complex numbers and friends there are also Hyperreal numbers, studied in non-standard analysis.

Yes, there is, hyperreals solves the problem of finding a positive number less then all positive reals. They can be extended for ever in similar fashion. You also have the surreal numbers.

You can do pure abstract thinking that the bicomplex numbers are a field, but you can't make it actually work and truly become a field.