/sci/ - Science & Math

in mathematics, the term "number" refers to a variety of concepts. In addition to your standard [math]\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}[/math], you also have constructs like the quaternions, the cardinal numbers, the ordinal numbers, hyperreal numbers, surreal numbers, p-adic numbers and others. At the same time, there are concepts which are generally not considered to be numbers, such as vectors, matrices, arbitrary groups and arbitrary fields.
Is there a simple rule (in English) for what is and what is not a number?