What's next?

Well, for some of the more alien parts of math we can extend this standard number system with some exotic types of numbers. To name a few:

• Cardinals and ordinals
  Both are numbers in ZF set theory [Enderton77, Henle86, Hrbacek84] and so they are sets as well. Cardinals are numbers that represent the sizes of sets, and ordinals are numbers that represent well ordered sets. Finite cardinals and ordinals are the same as the natural numbers. Cardinals, ordinals, and their arithmetic get interesting and ``tricky'' in the case of infinite sets.
• Hyperreals
  These numbers are constructed by means of ultrafilters [Henle86] and they are used in non-standard analysis. With hyperreals you can treat numbers like Leibnitz and Newton did by using infinitesimals.
• Quaternions and octonions
  Normally these are constructed by algebraic means (like the alternative C definition that uses ideals) [Shapiro75, Dixon94]. Quaternions are used to model rotations in 3 dimensions. Octonions, a.k.a. Cayley numbers, are just esoteric artifacts :-). Well, if you know where they are used for, feel free to contribute to the FAQ.
• Miscellaneous
  Just to name some others: algebraic numbers [Shapiro75], p-adic numbers [Shapiro75], and surreal numbers (a.k.a. Conway numbers) [Conway76].

Cardinals and ordinals are commonly used in math. Most mortals won't encounter (let alone use) hyperreals, quaternions, and octonions.

References

J.H. Conway. On Numbers and Games, L.M.S. Monographs, vol. 6. Academic Press, 1976.

H.B. Enderton. Elements of Set Theory. Academic Press, 1977.

G.M. Dixon. Division Algebras; Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics. Kluwer Academic, 1994.

J.M. Henle. An Outline of Set Theory. Springer Verlag, 1986.

K. Hrbacek and T. Jech. Introduction to Set Theory. M. Dekker Inc., 1984.

L. Shapiro. Introduction to Abstract Algebra. McGraw-Hill, 1975.

This subsection of the FAQ is Copyright (c) 1994, 1995 Hans de Vreught. Send comments and or corrections relating to this part to J.P.M.deVreught@cs.tudelft.nl