Numeral system

A numeral system 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 eleven in the decimal or base-10 numeral system, the number three in the binary or base-2 numeral system, and the number two in the unary numeral system.
The number the numeral represents is called its value. Additionally, not all number systems can represent the same set of numbers; for example, Roman, Greek, and Egyptian numerals don't have an official representation of the number zero.
Ideally, a numeral system will:

Represent a useful set of numbers
Give every number represented a unique representation
Reflect the algebraic and arithmetic structure of the numbers.

For example, the usual decimal representation gives every nonzero natural number a unique representation as a finite sequence of digits, beginning with a non-zero digit.
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, various hypercomplex number systems, the system of p-adic numbers, etc. Such systems are, however, not the topic of this article.

History

Western Arabic	0	1	2	3	4	5	6	7	8	9
Eastern Arabic	٠	١	٢	٣	٤	٥	٦	٧	٨	٩
Persian	۰	۱	۲	۳	۴	۵	۶	۷	۸	۹
Devanagari	०	१	२	३	४	५	६	७	८	९

Early numeral systems varied across civilizations, with the Babylonians using a base-60 system, the Egyptians developing hieroglyphic numerals, and the Chinese employing rod numerals. The Mayans independently created a vigesimal system that included a symbol for zero. Indian mathematicians, such as Brahmagupta in the 7th century, played a crucial role in formalizing arithmetic rules and the concept of zero, which was later refined by scholars like Al-Khwarizmi in the Islamic world. As these numeral systems evolved, the efficiency of positional notation and the inclusion of zero helped shape modern numerical representation, influencing global commerce, science, and technology. The first true written positional numeral system is considered to be the Hindu–Arabic numeral system. This system was established by the 7th century in India, but was not yet in its modern form because the use of the digit zero had not yet been widely accepted. Instead of a zero sometimes the digits were marked with dots to indicate their significance, or a space was used as a placeholder. The first widely acknowledged use of zero was in 876. The original numerals were very similar to the modern ones, even down to the glyphs used to represent digits.
Image:Maya.svg|thumb|left|The digits of the Maya numeral system
By the 13th century, Western Arabic numerals were accepted in European mathematical circles. Initially met with resistance, Hindu–Arabic numerals gained wider acceptance in Europe due to their efficiency in arithmetic operations, particularly in banking and trade. The invention of the printing press in the 15th century helped standardize their use, as printed mathematical texts favored this system over Roman numerals. They began to enter common use in the 15th century. By the 17th century, the system had become dominant in scientific works, influencing mathematical advancements by figures like Isaac Newton and René Descartes. In the 19th and 20th centuries, the widespread adoption of Arabic numerals facilitated global finance, engineering, and technological developments, forming the foundation for modern computing and digital data representation. By the end of the 20th century virtually all non-computerized calculations in the world were done with Arabic numerals, which have replaced native numeral systems in most cultures.

Other historical numeral systems using digits

The exact age of the Maya numerals is unclear, but it is possible that it is older than the Hindu–Arabic system. The system was vigesimal, so it has twenty digits. The Mayas used a shell symbol to represent zero. Numerals were written vertically, with the ones place at the bottom. The Mayas had no equivalent of the modern decimal separator, so their system could not represent fractions.
The Thai numeral system is identical to the Hindu–Arabic numeral system except for the symbols used to represent digits. The use of these digits is less common in Thailand than it once was, but they are still used alongside Arabic numerals.
The rod numerals, the written forms of counting rods once used by Chinese and Japanese mathematicians, are a decimal positional system used for performing decimal calculations. Rods were placed on a counting board and slid forwards or backwards to change the decimal place. The Sūnzĭ Suànjīng, a mathematical treatise dated to between the 3rd and 5th centuries AD, provides detailed instructions for the system, which is thought to have been in use since at least the 4th century BC. Zero was not initially treated as a number, but as a vacant position. Later sources introduced conventions for the expression of zero and negative numbers. The use of a round symbol 〇 for zero is first attested in the Mathematical Treatise in Nine Sections of 1247 AD. The origin of this symbol is unknown; it may have been produced by modifying a square symbol. The Suzhou numerals, a descendant of rod numerals, are still used today for some commercial purposes.

0	1	2	3	4	5	6	7	8	9
Image:Counting rod 0.png	Image:Counting rod v1.png	Image:Counting rod v2.png	Image:Counting rod v3.png	Image:Counting rod v4.png	Image:Counting rod v5.png	Image:Counting rod v6.png	Image:Counting rod v7.png	Image:Counting rod v8.png	Image:Counting rod v9.png
−0	−1	−2	−3	−4	−5	−6	−7	−8	−9
Image:Counting rod -0.png	Image:Counting rod v-1.png	Image:Counting rod v-2.png	Image:Counting rod v-3.png	Image:Counting rod v-4.png	Image:Counting rod v-5.png	Image:Counting rod v-6.png	Image:Counting rod v-7.png	Image:Counting rod v-8.png	Image:Counting rod v-9.png

Main 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. Aryabhata of Kusumapura developed the place-value notation in the 5th 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, or fractions, as recorded in a treatise by Syrian mathematician Abu'l-Hasan al-Uqlidisi in 952–953, and the decimal point notation was introduced 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 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 typically reserved for small numbers, although it plays an important role in theoretical computer science. Elias gamma coding, which is commonly used in data compression, expresses arbitrarily 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 using a 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.
Other systems 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, such as "three hundred four", as are those of other spoken languages, regardless of what written systems they have adopted. Many languages use mixtures of bases, and other features, for instance 79 in French is soixante dix-neuf and in Welsh is pedwar ar bymtheg a thrigain or the somewhat archaic pedwar ugain namyn un. In English, "four score less one", as in the famous Gettysburg Address representing "87 years ago" as "four score and seven years ago".
A positional system, also known as place-value notation, is classified by its base or radix, which is the number of symbols called digits used by the system. 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 or more precisely. Zero, which is not used in the other systems, is used in this system, 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.
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 ten different symbols, if it uses base 10.
The positional decimal system is universally used in human writing. The base 1000 is also used in many systems 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 a binary numeral system, or base 2, with two binary digits, 0 and 1. Positional systems obtained by grouping binary digits by three or four are commonly used. For very large integers, for example, the GNU Multiple Precision Arithmetic Library uses bases 2³² or 2⁶⁴—grouping binary digits by 32 or 64, the length of the machine word—are used.
In certain biological systems, the unary coding system is employed. Unary numerals used in the neural circuits responsible for birdsong production. The nucleus in the brain of the songbirds that plays a part in both the learning and the production of bird song is the 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 a 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 numerals and the geometric numerals, 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 use arithmetic numerals because they are made by repetition—except for the Ionic system—and a positional system does not use geometric numerals because they are made by position. 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, ..., k, 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 numbers. Bijective base 1 is the same as unary.