Editing Numeral system (section)

==Main numeral systems==
{{main|List of numeral systems}}
The most commonly used system of numerals is [[decimal]]. [[Indian mathematicians]] are credited with developing the integer version, the [[Hindu–Arabic numeral system]].<ref>{{cite book |author=David Eugene Smith |author2=Louis Charles Karpinski |title=The Hindu–Arabic numerals |url=https://archive.org/details/hinduarabicnume05karpgoog |year=1911 |publisher=Ginn and Company}}</ref> [[Aryabhata]] of [[Patna|Kusumapura]] developed the [[place-value notation]] in the 5th&nbsp;century and a century later [[Brahmagupta]] introduced the symbol for zero. The system slowly spread to other surrounding regions like Arabia due to their commercial and military activities with India. Middle-Eastern mathematicians extended the system to include negative powers of 10 (fractions), as recorded in a treatise by Syrian mathematician [[Abu'l-Hasan al-Uqlidisi]] in 952–953, and the decimal point notation was introduced{{when|date=February 2021}} by [[Sind ibn Ali]], who also wrote the earliest treatise on Arabic numerals. The Hindu–Arabic numeral system then spread to Europe due to merchants trading, and the digits used in Europe are called [[Arabic numerals]], as they learned them from the Arabs.

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 {{mono|/}} is chosen, for example, then the number seven would be represented by {{mono|///////}}. [[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 {{mono|+++ ////}} and the number 123 as {{mono|+ − − ///}} 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 (the number of these abbreviations is sometimes called the ''base'' of the system). 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'' ({{nowrap|60 + 10 + 9}}) and in Welsh is ''pedwar ar bymtheg a thrigain'' ({{nowrap|4 + (5 + 10) + (3 × 20)}}) or (somewhat archaic) ''pedwar ugain namyn un'' ({{nowrap|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 notation|positional system]]'', also known as place-value notation. The positional systems are classified by their ''base'' or ''[[radix]]'', which is the number of symbols called ''[[numerical digit|digit]]s'' used by the system. In base&nbsp;10, ten different digits 0,&nbsp;..., 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 {{nowrap|304 {{=}} 3×100 + 0×10 + 4×1}} or more precisely {{nowrap|3×10<sup>2</sup> + 0×10<sup>1</sup> + 4×10<sup>0</sup>}}. 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&nbsp;10 system.

Arithmetic is much easier in positional systems than in the earlier additive ones; furthermore, additive systems need a large number of different symbols for the different powers of 10; a positional system needs only ten different symbols (assuming that it uses base&nbsp;10).<ref>{{Cite book|last=Chowdhury|first=Arnab|url=https://books.google.com/books?id=WXn-mT3K6dgC&q=Arithmetic+is+much+easier+in+positional+systems+than+in+the+earlier+additive+ones;+furthermore,+additive+systems+need+a+large+number+of+different+symbols+for+the+different+powers+of+10;+a+positional+system+needs+only+ten+different+symbols+(assuming+that+it+uses+base+10).&pg=PA2|title=Design of an Efficient Multiplier using DBNS|publisher=GIAP Journals|isbn=978-93-83006-18-2|language=en}}</ref>

The positional decimal system is presently universally used in human writing. The base&nbsp;1000 is also used (albeit not universally), by grouping the digits and considering a sequence of three decimal digits as a single digit. This is the meaning of the common notation 1,000,234,567 used for very large numbers.

In computers, the main numeral systems are based on the positional system in base&nbsp;2 ([[binary numeral system]]), with two [[binary digit]]s, 0 and 1. Positional systems obtained by grouping binary digits by three ([[octal numeral system]]) or four ([[hexadecimal numeral system]]) are commonly used. For very large integers, bases&nbsp;2<sup>32</sup> or 2<sup>64</sup> (grouping binary digits by 32 or 64, the length of the [[machine word]]) are used, as, for example, in [[GNU Multiple Precision Arithmetic Library|GMP]].

In certain biological systems, the [[unary coding]] system is employed. Unary numerals used in the [[neural circuit]]s responsible for [[birdsong]] production.<ref> Fiete, I. R.; Seung, H. S. (2007). "Neural network models of birdsong production, learning, and coding". In Squire, L.; Albright, T.; Bloom, F.; Gage, F.; Spitzer, N. New Encyclopedia of Neuroscience.</ref> The nucleus in the brain of the songbirds that plays a part in both the learning and the production of bird song is the HVC ([[high vocal center]]). The command signals for different notes in the birdsong emanate from different points in the HVC. This coding works as space coding which is an efficient strategy for biological circuits due to its inherent simplicity and robustness.

The numerals used when writing numbers with digits or symbols can be divided into two types that might be called the [[Arithmetic sequence|arithmetic]] numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and the [[Geometric sequence|geometric]] numerals (1, 10, 100, 1000, 10000 ...), respectively. The sign-value systems use only the geometric numerals and the positional systems use only the arithmetic numerals. A sign-value system does not need arithmetic numerals because they are made by repetition (except for the [[Greek numerals|Ionic system]]), and a positional system does not need geometric numerals because they are made by position. However, the spoken language uses ''both'' arithmetic and geometric numerals.

In some areas of computer science, a modified base ''k'' positional system is used, called [[bijective numeration]], with digits 1, 2,&nbsp;..., ''k'' ({{nowrap|''k'' ≥ 1}}), and zero being represented by an empty string. This establishes a [[bijection]] between the set of all such digit-strings and the set of non-negative integers, avoiding the non-uniqueness caused by leading zeros. Bijective base-''k'' numeration is also called ''k''-adic notation, not to be confused with [[p-adic number|''p''-adic numbers]]. Bijective base&nbsp;1 is the same as unary.