Number systems

In mathematics, a 'number system' is a set of numbers, (in the broadest sense of the word), together with one or more operations, such as addition or multiplication.

Examples of number systems include: natural numbers, integers, rational numbers, algebraic numbers, real numbers, complex numbers, p-adic numbers, surreal numbers, and hyperreal numbers.

For a history of number systems, see number. For a history of the symbols used to represent numbers, see numeral system.

Natural numbers
Simply put, the natural numbers consist of the set of all whole numbers greater than zero. The set is denoted with a bold face capital N or with the special symbol $$\mathbb{N}$$. (In some books, the natural numbers begin with 0. There is no general agreement on this subject.) Giuseppe Peano developed axioms for the natural numbers, and is considered the founder of axiomatic number theory.

More advanced number systems
The word number has no generally agreed upon mathematical meaning, nor does the word number system. Instead, we have many examples. Thus there is no rule to say what is a number and what is not. Some of the more interesting examples of abstractions that can be considered numbers include the quaternions, the octonions, ordinal numbers, and the transfinite numbers.