A
numeral system (or
system of numeration) is a
mathematical notation for representing numbers of a given set by symbols in a consistent manner. It can be seen as the
context that allows the numeral "11" to be interpreted as the
binary numeral for
three, the
decimal numeral for
eleven, or other numbers in different
bases.
Ideally, a numeral system will
For example, the usual decimal representation of whole numbers gives every whole number a unique representation as a finite sequence of digits, with the operations of arithmetic (addition, subtraction, multiplication and division) being present as the standard algorithms of arithmetic. However, when decimal representation is used for the rational or real numbers, the representation is no longer unique many rational numbers have two numerals, a standard one that terminates, such as 2.31, and another that recurs, such as 2.309999999... . Numerals which terminate have no non-zero digits after a given position. For example, numerals like 2.31 and 2.310 are taken to be the same, except in the experimental sciences, where greater precision is denoted by the trailing zero.
Numeral systems are sometimes called number systems, but that name is misleading, as it could refer to different systems of numbers, such as the system of real numbers, the system of complex numbers, the system of p-adic numbers, etc. Such systems are not the topic of this article.
