This article is not about numeral systems, which are systems of symbols for representing numbers.

A number system is a set of numbers or number-like objects together with some arithmetic operations that can be performed on them.

Examples of number systems are: the real numbers, the rational numbers, the algebraic numbers, the complex numbers, the p-adic numbers, the surreal numbers, the hyperreal numbers.

The phrase number-like objects appearing above is vague, as is, therefore the very concept of number systems. Whether, for example, the quaternions should be considered to comprise a number system is perhaps only a matter of convention. Rings of matrices would perhaps not usually be considered number systems.