**Number system**

A **numeral system** (or **system of numeration**) is a __writing system__ for expressing numbers; that is, a __mathematical notation__ for representing __numbers__ of a given set, using __digits__ or other symbols in a consistent manner. The same sequence of symbols may represent different numbers in different numeral systems. For example, “11” represents the number *three* in the __binary numeral system__ (used in __computers__) and the number *eleven* in the __decimal numeral system__ (used in common life); in the __unary numeral__ system, the number will represent “two”.

The number the numeral represents is called its value.

Ideally, a numeral system will:

- Represent a useful set of numbers (e.g. all
__integers__, or__rational numbers__) - Give every number represented a unique representation (or at least a standard representation)
- Reflect the algebraic and arithmetic structure of the numbers.

For example, the usual __decimal__ representation of whole numbers gives every nonzero whole number a unique representation as a __finitesequence__ of __digits__, beginning with a non-zero digit. However, when decimal representation is used for the __rational__ or real numbers, such numbers in general have an infinite number of representations, for example 2.31 can also be written as 2.310, 2.3100000, 2.309999999…, etc., all of which have the same meaning except for some scientific and other contexts where greater precision is implied by a larger number of figures shown.

Numeral systems are sometimes called * number systems*, but that name is ambiguous, 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, however, not the topic of this article.

__Sources____7 External links__

**Main numeral systems:**

The most commonly used system of numerals is the __Hindu–Arabic numeral system__ Two __Indian mathematicians__ are credited with developing it. __Aryabhata__ of __Kusumapura__ developed the place-value notation in the 5th century and a century later __Brahmagupta__ introduced the symbol for __zero__. The numeral system and the zero concept, developed by the Hindus in India, slowly spread to other surrounding countries due to their commercial and military activities with India. The Arabs adopted and modified it. Even today, the Arabs call the numerals which they use “Rakam Al-Hind” or the Hindu numeral system. The Arabs translated Hindu texts on numerology and spread them to the western world due to their trade links with them. The Western world modified them and called them the Arabic numerals, as they learned them from the Arabs. Hence the current western numeral system is the modified version of the Hindu numeral system developed in India. It also exhibits a great similarity to the Sanskrit–Devanagari notation, which is still used in India and neighbouring Nepal.

The simplest numeral system is the __unary numeral system__, in which every __natural number__ is represented by a corresponding number of symbols. If the symbol / is chosen, for example, then the number seven would be represented by ///////. __Tally marks__ represent one such system still in common use. The unary system is only useful for small numbers, although it plays an important role in __theoretical computer science__. __Elias gamma coding__, which is commonly used in __data compression__, expresses arbitrary-sized numbers by using unary to indicate the length of a binary numeral.

The unary notation can be abbreviated by introducing different symbols for certain new values. Very commonly, these values are powers of 10; so for instance, if / stands for one, − for ten and + for 100, then the number 304 can be compactly represented as +++ //// and the number 123 as + − − /// without any need for zero. This is called __sign-value notation__. The ancient __Egyptian numeral system__ was of this type, and the __Roman numeral system__ was a modification of this idea.

More useful still are systems which employ special abbreviations for repetitions of symbols; for example, using the first nine letters of the alphabet for these abbreviations, with A standing for “one occurrence”, B “two occurrences”, and so on, one could then write C+ D/ for the number 304. This system is used when writing __Chinese numerals__ and other East Asian numerals based on Chinese. The number system of the __English language__ is of this type (“three hundred [and] four”), as are those of other spoken __languages__, regardless of what written systems they have adopted. However, many languages use mixtures of bases, and other features, for instance 79 in French is *soixante dix-neuf* (60 + 10 + 9) and in Welsh is *pedwararbymtheg a thrigain* (4 + (5 + 10) + (3 × 20)) or (somewhat archaic) *pedwarugainnamyn un* (4 × 20 − 1). In English, one could say “four score less one”, as in the famous __Gettysburg Address__ representing “87 years ago” as “four score and seven years ago”.

More elegant is a * positional system*, also known as place-value notation. Again working in base 10, ten different digits 0, …, 9 are used and the position of a digit is used to signify the power of ten that the digit is to be multiplied with, as in 304 = 3×100 + 0×10 + 4×1 or more precisely 3×10

^{2}+ 0×10

^{1}+ 4×10

^{0}. Note that zero, which is not needed in the other systems, is of crucial importance here, in order to be able to “skip” a power. The Hindu–Arabic numeral system, which originated in India and is now used throughout the world, is a positional base 10 system.