transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
Throughout mathematics there are various entities referred to as “numbers”; in modern mathematics it would be more accurate to refer to any one of various types called “number systems”, and simply define a number to be a term of such a type.
A numeral on the other hand is a syntactic representation of a number, part of a system of numeration.
It is interesting to try to describe the general conditions under which something comes to be designated as a “number”. After all, such number systems tend to form commutative rings or at least commutative rigs, but not all commutative rigs are considered to consist of “numbers”, or at least that is not how the language is used in practice.
The root notion, known to the great ancient civilizations and particularly ancient Hellenic civilization, is that of (the system of) natural numbers, and in some way or other each of the various number systems are extensions of that one. If we may attempt a broad classification, qualitatively there are two broad types of number systems:
Roughly speaking, transfinite numbers tend to be quantities that measure sizes of elements in ZF-type structures (as in algebraic set theory, or in the study of games in the sense of Conway), sometimes arising through a process of decategorification.
In algebraic number theory, there seem to be two distinct general trends in the creation of new number systems from old (starting from the natural numbers):
For example, one passes from the rig of natural numbers to the ring of integers through additive group completion, or from the integral domain of integers to the field of rationals through a field of fractions completion. Or, one constructs number fields as finite extensions of the field of rational numbers (similarly, algebraic integers $\alpha$ inside such a number field $K/\mathbb{Q}$ as those elements such that $\mathbb{Z}[\alpha]$ is a finite rank extension of $\mathbb{Z}$). Or, one constructs various local field completions of number fields, based on the valuations one can define on them; these include the real numbers and complex numbers and $p$-adic numbers. Or, one has various finite-dimensional algebra extensions such systems, including for example the quaternions (aka Hamiltonian numbers) and octonions (aka Cayley numbers), and various $p$-adic relatives of these.
Sometimes one singles out subtypes of such types; for example, the field of all algebraic numbers is the union of all number fields considered within the type of complex numbers. Sometimes numbers are designated according to what they are not, for example one speaks of irrational (= non-rational) numbers and of transcendental (= non-algebraic complex) numbers.
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