# Are there any numbers more fundamental than Complex numbers?

Are there any numbers other than Complex Numbers which contain something more that Real and Imaginary numbers?

• There are also dual quaternions, but they may be equivalent to one of the number systems Tim.Ratigan cites. – Stefan Smith Nov 20 '13 at 23:13
• The real issue with defining number sets and their properties is to do it in a non paradoxical way. Simply by defining a set $S$ and declaring that $\mathbb C \subset S$, we can be fairly sure it is a consistent definition. However, even with complex numbers, you have to be careful how you define things like exponentiation or the definitions will not be consistent. – DanielV Nov 20 '13 at 23:33
• Surreal Numbers...en.wikipedia.org/wiki/Surreal_number – Henry Nov 21 '13 at 0:54
• What counts as "number" for you? – Torsten Schoeneberg May 20 '20 at 15:03

Yes. There is in fact a well-structured set of numbers of dimension $2^n$ for all $n$, but after a certain point it simply becomes pointless and unwieldy. The next set of numbers is called the quaternions of the form $a+bi+cj+dk$, where $a,b,c,d\in\mathbb{R}$ and $i,j,k$ are all square roots of $-1$. In the quaternions, $ij=k, jk=i, ki=j$ but $ji=-k, kj=-i, ik=-j$ (that's right, they're non-commutative). The complex numbers $\mathbb{C}$, though, are considered more fundamental than the quaternions $\mathbb{H}$ or the sets that follow (the octonions, sedenions, etc.), since the complex numbers are an abelian field with algebraic closure (algebraic closure of a field $F$ means that all polynomials $\sum a_ix^i$ with $a_i\in F$ have all of their roots in $F$. This cannot be said of $\mathbb{R}$, since $x^2+1$ has real coefficients but imaginary roots.), giving it the greatest structure of all of these number sets. Funny things start to happen after the quaternions, as the octonions are no longer associative and the sedenions can have zero divisors (which means two non-zero elements $a$ and $b$ can have the property $ab=0$).

• Sedenions. Just a minor detail. :) – Tito Piezas III Nov 20 '13 at 23:43
• Woops. Thanks for the correction. – Tim Ratigan Nov 20 '13 at 23:51
• I was told that there were no higher order number systems outside of sedenions, where can I read more about an arbitrary $2^n$ system? – R R Nov 21 '13 at 0:41
• The construction used to generate these algebras is something called the Cayley-Dickson construction which can simply be reapplied at each dimension to obtain the next algebra. Further reading can be found here or here or with a quick Google search. – Tim Ratigan Nov 21 '13 at 4:20

The Quaternions $\mathbb H$ maybe? They are non-commutative, though.

The Cayley Dickson construction might be a useful related remark.

Going off in a slightly different direction than sets containing the complex numbers are the hyperreals ${}^\ast\mathbb{R}$, which is used in non-standard analysis. It can be thought of all adding infinitesimals (numbers that are unequal to zero but smaller in magnitude then any non-zero real number) and infinitely large numbers (larger in magnitude than any number). The are the rigorous formulation of the intuitions using infinitesimals and indivisibles before the $\epsilon-\delta$-definitions were used to bypass the limited rigor employed at the time. What's nice about this is that it remains a field, and is in a sense equivalent to the reals through the "Transfer principle", which relates first-order sentences in ${}^\ast\mathbb{R}$ to those in $\mathbb{R}$.

• Do you have a good reference for the novice, hopefully one that is rigorous but does not require mastery of logic or set theory? – Stefan Smith Nov 20 '13 at 23:11
• @StefanSmith A first shot might be wikipedia / Hyperreal number as well (I know that the german article is of some quality to get you introduced...) – AlexR Nov 20 '13 at 23:12
• For a rigorous treatment, see "Foundations of Infinitesimal Calculus" by Keisler. For a light introduction, look at "Calculus, an Infinitesimal Approach" by Keisler (again). Wikipedia is a nice intro, perhaps. In order to truly understand it, ultrafilers and ultraproducts are necessary though. – Hayden Nov 20 '13 at 23:24
• @Hayden : from what I have heard, attempts to teach beginning calculus to freshmen using nonstandard analysis have been a disaster. You can correct me if I'm wrong. It is my impression that understanding infinitesimal calculus rigorously is a lot more difficult than understanding traditional, epsilon - delta calculus rigorously. It is also my impression that anything you can do with infinitesimal calculus can be done with traditional calculus. You can correct me if any of these impressions are incorrect. However, I am still curious about infinitesimal calculus. – Stefan Smith Nov 21 '13 at 20:58
• @Hayden : Are there any advantages to using nonstandard analysis as opposed to traditional analysis? – Stefan Smith Nov 21 '13 at 20:59