There is no precise mathematical definition of the word ‘number.’
There are however precise definitions of the terms ‘natural number’, ‘rational number’, and so on. When a mathematician speaks about numbers she always has one of these specific cases in mind and she should, at the outset, make it clear to which type of number she is referring.
The naïve, inborn concept of number that is shared to some degree by all humans is a matter for philosophical rather than strictly mathematical inquiry, and it may be noted that there has historically been strong opposition to the introduction of new generalizations of established concepts of number.
The kinds of numbers mathematicians study include:
- natural numbers
- whole numbers
- rational numbers
- real numbers
- complex numbers
- algebraic numbers
- transcendental numbers
- ordinals (including transfinite), and
- cardinals (including transfinite)
Other less commonly used numbers include the quaternions, Gaussian integers, and many others.
Abstractly, numbers are elements of a set with some of kind of algebraic structure on it, such as a group or ring structure. The number sets used in most practical applications are Euclidean domains or fields.