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Кстати, а кто-нибудь может мне ответить на терминологический вопрос «что такое число»? Я некоторое время назад искал ответ и не нашёл.
Почему вещественные, комплексные и даже p-адические числа называют числами, а например матрицы числами не называют? Где проходит граница?
Upd: кажется, нашёл ответ, который меня устраивает:
In my observation, the term number is used almost exclusively in the following cases:
• The natural numbers, basically the origin of the term.
• Any system which extends a system already called “numbers” by adding numbers that are in some sense “missing” That is, there are some constructions which sometimes give a number, but sometimes don’t, although is “looks like” such a number should exist.
The actual construction of the extension is then different from just adding those “missing” numbers just in order to make sure that what you do makes sense. However the goal is always to add the “missing” numbers, and any further numbers that are implied by their existence, but nothing else.
The classic sequence of number sets is exactly of this type:
• The natural numbers allow to solve equations like 2+x=6, but not equations like 5+x=2. Adding solutions to the latter gives the integers (which, of course, are considered to be numbers).
• The integers allow to solve equations like 2x=6, but not equations like 5x=2. Adding solutions to the latter gives the rational numbers.
• In the rational numbers, all convergent sequences are Cauchy, but not all Cauchy sequences converge. But they look like they should converge. Thus the missing limits are added. (Note that there are other constructions of the real numbers, based on other constructions where numbers are found “missing”).
• In the real numbers, equations like x^2=2 can be solved, but equations like x^2=−2 cannot. Adding those numbers then gives the complex numbers.
But is also is true for most other systems which are generally called numbers. For example:
• Finite sets always have an integer number of elements. For infinite sets, there’s no natural number describing the set’s size. The cardinal numbers add those numbers describing the size of infinite sets.
• Similarly, when considering well-orderings, you need a generalization of the terms “first”, “second”, … to infinite size orderings. The ordinal numbers provide those missing numbers.
• When informally talking about differentiation and integration, one often uses the concept of “infinitesimal quantities”. Those don’t exist in the real or complex numbers. Non-standard analysis adds those “missing” infinitesimals (and everything implied by them), arriving at the hyperreal numbers.
• When looking at the sequence r_n of remainders modulo p^n, p prime, every integer gives an unique sequence. But not every sequence gives a number. The p-adic numbers add those “missing” numbers.
Indeed, off the top of my head, I can only think of two constructions called “numbers” that fall outside this scheme:
• The quaternions are not constructed as extension, but explicitly as “numbers” that allow to describe three-dimensional geometry, after seeing that complex numbers describe two-dimensional geometry. It turned out that you need a fourth dimension to get something meaningful. However not everyone considers the quaternions to be numbers.
• The surreal numbers are not constructed as extension to anything, but arise independently from combinatoric game theory. However since not only behave very much like numbers do, but in addition contain subsets isomorphic to several other number sets or classes (you’ll find subsets isomorphic to the real numbers, to all systems of hyperreal numbers, and to the ordinal numbers under natural arithmetic). So while the surreal numbers are not constructed as extension, they can be viewed in some sense as such, justifying the term “number”.