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Encyclopedia > Numeral system

A numeral is a symbol or group of symbols, or a word in a natural language that represents a number. Numerals differ from numbers just as words differ from the things they refer to. The symbols "11", "eleven" and "XI" are different numerals, all representing the same number. This article attempts to explain the various systems of numerals. See also number names. In mathematics, a number system is a set of numbers, or number-like objects, together with one or more operations, such as addition or multiplication. ... For other uses, see Word (disambiguation). ... The term natural language is used to distinguish languages spoken and signed (by hand signals and facial expressions) by humans for general-purpose communication from constructs such as writing, computer-programming languages or the languages used in the study of formal logic, especially mathematical logic. ... For other uses, see Number (disambiguation). ... Different cultures have different traditional numeral systems used for writing numbers and for naming large numbers. ...

Numeral systems by culture
Hindu-Arabic numerals
Western Arabic
Eastern Arabic
Khmer
Indian family
Brahmi
Thai
East Asian numerals
Chinese
Counting rods
Korean
Japanese
Alphabetic numerals
Armenian
Cyrillic
Ge'ez
Hebrew
Ionian/Greek
Sanskrit

Other systems
Attic
Etruscan
Urnfield
Roman
Babylonian
Egyptian
Mayan
List of numeral system topics
Positional systems by base
Decimal (10)
2, 4, 8, 16, 32, 64
3, 9, 12, 24, 30, 36, 60, more…
v  d  e

Ideally, a numeral system will:

• Represent a useful set of numbers (e.g. all whole numbers, integers, or real 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 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. The whole numbers are the nonnegative integers (0, 1, 2, 3, ...) The set of all whole numbers is represented by the symbol = {0, 1, 2, 3, ...} Algebraically, the elements of form a commutative monoid under addition (with identity element zero), and under multiplication (with identity element one). ... The integers are commonly denoted by the above symbol. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... For other uses, see Decimal (disambiguation). ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... For other senses of this word, see sequence (disambiguation). ... In mathematics and computer science, a numerical digit is a symbol, e. ... In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will terminate in a defined end-state. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... A recurring or repeating decimal is a number which when expressed as a decimal has a set of final digits which repeat an infinite number of times. ...

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. In mathematics, a number system is a set of numbers, or number-like objects, together with one or more operations, such as addition or multiplication. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... In mathematics, the p-adic number systems were first described by Kurt Hensel in 1897. ...

## Types of numeral systems GA_googleFillSlot("encyclopedia_square");

 Arabic ١ ٢ ٣ ٤ ٥ ٦ ٧ ٨ ٩ ١٠ Devanagari १ २ ३ ४ ५ ६ ७ ८ ९ १० Hebrew א ב ג ד ה ו ז ח ט י Hindu-Arabic 1 2 3 4 5 6 7 8 9 10 Kanji 一 二 三 四 五 六 七 八 九 十 Roman Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ Ⅵ Ⅶ Ⅷ Ⅸ Ⅹ Thai ๑ ๒ ๓ ๔ ๕ ๖ ๗ ๘ ๙ ๑๐

The most commonly used system of numerals is known as Hindu-Arabic numerals, and two great Indian mathematicians could be given credit for developing them. Aryabhatta of Kusumapura who lived during the 5th century developed the place value notation and Brahmagupta a century later introduced the symbol zero. Numerals sans-serif Arabic numerals, known formally as Hindu-Arabic numerals, and also as Indian numerals, Hindu numerals, Western Arabic numerals, European numerals, or Western numerals, are the most common symbolic representation of numbers around the world. ... Aryabhata (&#2310;&#2352;&#2381;&#2351;&#2349;&#2335;) (&#256;ryabha&#7789;a) is the first of the great astronomers of the classical age of India. ... Brahmagupta (à¤¬à¥à¤°à¤¹à¥à¤®à¤—à¥à¤ªà¥à¤¤) ( ) (589â€“668) was an Indian mathematician and astronomer. ...

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. In practice, the unary system is normally only useful for small numbers, although it plays an important role in theoretical computer science. Also, 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 numeral system is the simplest numeral system to represent natural numbers: in order to represent a number N, an arbitrarily chosen symbol is repeated N times. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... A sign warning hikers on the trail to Hanakapiai Beach. ... Computer science (informally, CS or compsci) is, in its most general sense, the study of computation and information processing, both in hardware and in software. ... Elias gamma code is a universal code encoding positive integers. ... â€œSource codingâ€ redirects here. ...

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 number 123 as + - - /// without any need for zero. This is called sign-value notation. The ancient Egyptian system is of this type, and the Roman system is a modification of this idea. In Computers Sign-value notation in computers is the use of the high-order bit (left end) of a binary word to represent the numeric sign: 0 for +, 1 for - followed by a binary number that is an absolute magnitude or a twos complement of an absolute magnitude. ... The system of Egyptian numerals was a numeral system used in ancient Egypt. ... Roman numerals are a numeral system originating in ancient Rome, adapted from Etruscan numerals. ...

More useful still are systems which employ special abbreviations for repetitions of symbols; for example, using the first nine letters of our alphabet for these abbreviations, with A standing for "one occurrence", B "two occurrences", and so on, we could then write C+ D/ for the number 304. The numeral system of English is of this type ("three hundred [and] four"), as are those of virtually all other spoken languages, regardless of what written systems they have adopted. The English language is a West Germanic language that originates in England. ...

More elegant is a positional system, also known as place-value notation. Again working in base 10, we use ten different digits 0, ..., 9 and use the position of a digit to signify the power of ten that the digit is to be multiplied with, as in 304 = 3×100 + 0×10 + 4×1. 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, borrowed from India, is a positional base 10 system; it is used today throughout the world. A positional notation or place-value notation system is a numeral system in which each position is related to the next by a constant multiplier, a common ratio, called the base or radix of that numeral system. ... For other senses of this word, see zero or 0. ... I like cream cheese, it tastes good on toast. ...

Arithmetic is much easier in positional systems than in the earlier additive ones; furthermore, additive systems have a need for a potentially infinite number of different symbols for the different powers of 10; positional systems need only 10 different symbols (assuming that it uses base 10).

The numerals used when writing numbers with digits or symbols can be divided into two types that might be called the arithmetic numerals 0,1,2,3,4,5,6,7,8,9 and the geometric numerals 1,10,100,1000,10000... respectively. The sign-value systems use only the geometric numerals and the positional system use only the arithmetic numerals. The sign-value system does not need arithmetic numerals because they are made by repetition (except for the Ionic system), and the positional system does not need geometric numerals because they are made by position. However, the spoken language uses both arithmetic and geometric numerals. This is a page about mathematics. ... In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ... Greek numerals are a system of representing numbers using letters of the Greek alphabet. ...

In certain areas of computer science, a modified base-k positional system is used, called bijective numeration, with digits 1, 2, ..., k (k ≥ 1), and zero being represented by the 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. Bijective numeration is any numeral system that establishes a bijection between the set of non-negative integers and the set of finite strings over a finite set of digits. ... A bijective function. ... In mathematics, the p-adic number systems were first described by Kurt Hensel in 1897. ...

## History

Counting aids, especially the use of body parts (counting on fingers), were certainly used in prehistoric times as today. There are many variations. Besides counting 10 fingers, some cultures have counted knuckles, the space between fingers, and toes as well as fingers. The Oksapmin culture of New Guinea uses a system of 27 upper body locations to represent numbers. In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...

To preserve numerical information, Tallies carved in wood, bone, and stone have been used since prehistoric times. Stone age cultures, including ancient American Indian groups, used tallies for gambling, personal services, and trade-goods. A sign warning hikers on the trail to Hanakapiai Beach. ... Native Americans (also Indians, Aboriginal Peoples, American Indians, First Nations, Alaskan Natives, or Indigenous Peoples of America) are the indigenous inhabitants of The Americas prior to the European colonization, and their modern descendants. ...

A method of preserving numeric information in clay was invented by the Sumerians between 8000 and 3500 BC. This was done with small clay tokens of various shapes that were strung like beads on a string. Beginning about 3500 BC clay tokens were gradually replaced by number signs impressed with a round stylus at different angles in clay tablets (originally containers for tokens) which were then baked. About 3100 BC written numbers were dissociated from the things being counted and became abstract numerals. Sumer (or Shumer, Sumeria, Shinar, native ki-en-gir) formed the southern part of Mesopotamia from the time of settlement by the Sumerians until the time of Babylonia. ...

Between 2700 BC and 2000 BC in Sumer, the round stylus was gradually replaced by a reed stylus that was used to press wedge-shaped cuneiform signs in clay. These cuneiform number signs resembled the round number signs they replaced and retained the additive sign-value notation of the round number signs. These systems gradually converged on a common sexagesimal number system; this was a place-value system consisting of only two impressed marks, the vertical wedge and the chevron, which could also represent fractions. This sexagesimal number system was fully developed at the beginning of the Old Babylonia period (about 1950 BC) and became standard in Babylonia. In Computers Sign-value notation in computers is the use of the high-order bit (left end) of a binary word to represent the numeric sign: 0 for +, 1 for - followed by a binary number that is an absolute magnitude or a twos complement of an absolute magnitude. ... The sexagesimal (base-sixty) is a numeral system with sixty as the base. ...

Sexagesimal numerals were a mixed radix system that retained the alternating base 10 and base 6 in a sequence of cuneiform vertical wedges and chevrons. By 1950 BC this was a positional notation system. Sexagesimal numerals came to be widely used in commerce, but were also used in astronomical and other calculations. This system was exported from Babylonia and used throughout Mesopotamia, and by every Mediterranean nation that used standard Babylonian units of measure and counting, including the Greeks, Romans and Egyptians. Babylonian-style sexagesimal numeration is still used in modern societies to measure time (minutes per hour) and angles (degrees). The sexagesimal (base-sixty) is a numeral system with sixty as the base. ... Mixed radix numeral systems are more general than the usual ones in that the numerical base may vary from position to position. ... A positional notation or place-value notation system is a numeral system in which each position is related to the next by a constant multiplier, a common ratio, called the base or radix of that numeral system. ... Look up time in Wiktionary, the free dictionary. ... This article is about angles in geometry. ...

In China, armies and provisions were counted using modular tallies of prime numbers. Unique numbers of troops and measures of rice appear as unique combinations of these tallies. A great convenience of modular arithmetic is that it is easy to multiply, though quite difficult to add. This makes use of modular arithmetic for provisions especially attractive. Conventional tallies are quite difficult to multiply and divide. In modern times modular arithmetic is sometimes used in Digital signal processing. In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value â€” the modulus. ... Digital signal processing (DSP) is the study of signals in a digital representation and the processing methods of these signals. ...

The oldest Greek system was the that of the Attic numerals, but in the 4th century BC they began to use a quasidecimal alphabetic system (see Greek numerals). Jews began using a similar system (Hebrew numerals), with the oldest examples known being coins from around 100 BC. Attic numerals were used by ancient Greeks, possibly from the 7th century BC. They were also known as Herodianic numerals because they were first described in a 2nd century manuscript by Herodian. ... The 4th century BC started the first day of 400 BC and ended the last day of 301 BC. It is considered part of the Classical era, epoch, or historical period. ... Greek numerals are a system of representing numbers using letters of the Greek alphabet. ... The system of Hebrew numerals is a quasi-decimal alphabetic numeral system using the letters of the Hebrew alphabet. ...

The Roman empire used tallies written on wax, papyrus and stone, and roughly followed the Greek custom of assigning letters to various numbers. The Roman numerals system remained in common use in Europe until positional notation came into common use in the 1500s. Roman numerals are a numeral system originating in ancient Rome, adapted from Etruscan numerals. ... A positional notation or place-value notation system is a numeral system in which each position is related to the next by a constant multiplier, a common ratio, called the base or radix of that numeral system. ... The decade of years from 1500 to 1509, inclusive. ...

The Maya of Central America used a mixed base 18[citation needed] and base 20 system, possibly inherited from the Olmec, including advanced features such as positional notation and a zero. They used this system to make advanced astronomical calculations, including highly accurate calculations of the length of the solar year and the orbit of Venus. Mayan numerals. ... Monument 1, one of the four Olmec colossal heads at La Venta. ... For other senses of this word, see zero or 0. ... (*min temperature refers to cloud tops only) Atmospheric characteristics Atmospheric pressure 9. ...

Some authorities believe that positional arithmetic began with the wide use of the abacus in China. The earliest written positional records seem to be tallies of abacus results in China around 400. In particular, zero was correctly described by Chinese mathematicians around 932, and seems to have originated as a circle around a spot containing no beads. It has been suggested that Abax be merged into this article or section. ... Events First invasion of Italy by Alaric (probable date). ... Events Foundation of the St. ...

The modern positional Hindu-Arabic numeral system was developed by mathematicians in India, and passed on to Muslim mathematicians, along with astronomical tables brought to Baghdad by an Indian ambassador around 773. I like cream cheese, it tastes good on toast. ... This article is under construction. ... Islamic mathematics is the profession of Muslim Mathematicians. ... Baghdad (Arabic: ) is the capital of Iraq and of Baghdad Governorate. ... Events Charlemagne crosses the Alps and invades the kingdom of the Lombards. ...

From India, the thriving trade between Islamic sultans and Africa carried the concept to Cairo. Arabic mathematicians extended the system to include decimal fractions, and Muḥammad ibn Mūsā al-Ḵwārizmī wrote an important work about it in the 9th century. The modern Arabic numerals were introduced to Europe with the translation of this work in the 12th century in Spain and Leonardo of Pisa's Liber Abaci of 1201. In Europe, the complete Indian system with the zero was derived from the Arabs in the 12th century. Nickname: Egypt: Site of Cairo (top center) Coordinates: , Government  - Governor Dr. Abdul Azim Wazir Area  - City 214 kmÂ²  (82. ... For other uses, see Decimal (disambiguation). ... Khwarizmi comemmorated on this Soviet stamp. ... As a means of recording the passage of time the 9th century was the century that lasted from 801 to 900. ... For other uses, see Arabic numerals (disambiguation). ... (11th century - 12th century - 13th century - other centuries) As a means of recording the passage of time, the 12th century was that century which lasted from 1101 to 1200. ... Drawing of Leonardo Pisano Leonardo of Pisa or Leonardo Pisano (Pisa, c. ... // The town of Riga was chartered as a city. ...

The binary system (base 2), was propagated in the 17th century by Gottfried Leibniz. Leibniz had developed the concept early in his career, and had revisited it when he reviewed a copy of the I ching from China. Binary numbers came into common use in the 20th century because of computer applications. The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ... (16th century - 17th century - 18th century - more centuries) As a means of recording the passage of time, the 17th century was that century which lasted from 1601-1700. ... â€œLeibnizâ€ redirects here. ... Alternative meaning: I Ching (monk) The I Ching (Traditional Chinese: &#26131;&#32147;, pinyin y j&#299;ng; Cantonese IPA: j&#618;k6g&#618;&#331;1; Cantonese Jyutping: jik6ging1; alternative romanizations include I Jing, Yi Ching, Yi King) is the oldest of the Chinese classic texts. ... (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s As a means of recording the passage of time, the 20th century was that century which lasted from 1901&#8211;2000 in the sense of the Gregorian calendar (1900&#8211;1999...

## Bases used

### In computing

Switches, mimicked by their electronic successors built of vacuum tubes, have only two possible states: "open" and "closed". Substituting open=1 and closed=0 (or the other way around) yields the entire set of binary digits. This base-2 system (binary) is the basis for digital computers. It is used to perform integer arithmetic in almost all digital computers; some exotic base-3 (ternary) and base-10 computers have also been built, but those designs were discarded early in the history of computing hardware. Structure of a vacuum tube diode Structure of a vacuum tube triode In electronics, a vacuum tube, electron tube, or (outside North America) thermionic valve or just valve, is a device used to amplify, switch or modify a signal by controlling the movement of electrons in an evacuated space. ... The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ... For other uses, see Digital (disambiguation). ... Ternary or trinary is the base-3 numeral system. ... Computing hardware has been an important component of the process of calculation and computer data storage since it became useful for numerical values to be processed and shared. ...

Modern computers use transistors that represent two states with either high or low voltages. The smallest unit of memory for this binary state is called a bit. Bits are arranged in groups to aid in processing, and to make the binary numbers shorter and more manageable for humans. More recently these groups of bits, such as bytes and words, are sized in multiples of four. Thus base 16 (hexadecimal) is commonly used as shorthand. Base 8 (octal) has also been used for this purpose. This article is about the machine. ... Assorted discrete transistors A transistor is a semiconductor device, commonly used as an amplifier or an electrically controlled switch. ... In computer science a byte (pronounced bite) is a unit of measurement of information storage, most often consisting of eight bits. ... In mathematics and computer science, hexadecimal, base-16, or simply hex, is a numeral system with a radix, or base, of 16, usually written using the symbols 0â€“9 and Aâ€“F, or aâ€“f. ...

A computer does not treat all of its data as numerical. For instance, some of it may be treated as program instructions or data such as text. However, arithmetic and Boolean logic constitute most internal operations. Whole numbers are represented exactly, as integers. Real numbers, allowing fractional values, are usually approximated as floating point numbers. The computer uses different methods to do arithmetic with these two kinds of numbers. Boolean logic is a complete system for logical operations. ... In computer science, the term integer is used to refer to any data type which can represent some subset of the mathematical integers. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer. ... Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word Î±ÏÎ¹Î¸Î¼ÏŒÏ‚ = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. ...

### Five

A base-5 system (quinary) has been used in many cultures for counting. Plainly it is based on the number of fingers on a human hand. It may also be regarded as a sub-base of other bases, such as base 10 and base 60. Quinary (base-5) is a numeral system with five as the base. ...

### Eight

A base-8 system (octal) was devised by the Yuki of Northern California, who used the spaces between the fingers to count. Zero to seven are the only possibly digits. There is also linguistic evidence which suggests that the Bronze Age Proto-Indo Europeans (from whom most European and Indic languages descend) might have replaced a base 8 system (or a system which could only count up to 8) with a base 10 system. The evidence is that the word for 9, newm, is suggested by some to derive from the word for 'new', newo-, suggesting that the number 9 had been recently invented and called the 'new number' (Mallory & Adams 1997). The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. ... [[Image:YukiTribe. ... See Pie (disambiguation) for other uses of PIE. The Proto-Indo-European language (PIE) is the hypothetical common ancestor of the Indo-European languages. ...

### Ten

The base-10 system (decimal) is the one most commonly used today. It is assumed to have originated because humans have ten fingers. These systems often use a larger superimposed base. See Decimal superbase. For other uses, see Decimal (disambiguation). ... This article is about modern humans. ... This article does not cite any references or sources. ... Many numeral systems with base 10 use a superimposed larger base of 100, 1000, 10000 or 1000000. ...

### Twelve

Base-12 systems (duodecimal or dozenal) have been popular because multiplication and division are easier than in base-10, with addition and subtracting being just as easy. 12 is a useful base because it has many factors. It is the smallest multiple of one through four and of six. We still have a special word for "dozen" and just like there is a word for 102, hundred, there is also a word for 122, gross. Base-12 could have originated from the number of knuckles in the four fingers of a hand excluding the thumb, which is used as a pointer in counting. The duodecimal (also known as base-12 or dozenal) system is a numeral system using twelve as its base. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ...

There are 24 hours per day, usually counted till 12 until noon (p.m.) and once again until midnight (a.m.), often further divided per 6 hours in counting (for instance in Thailand) or as switches between using terms like 'night', 'morning', 'afternoon', and 'evening', whereas other languages use such terms with durations of 3 to 9 hours often according to switches at some of the 3 hour interval marks. The 12-hour clock is a timekeeping convention in which the 24 hours of the day are divided into two periods called ante meridiem (AM, Latin for before noon) and post meridiem (PM, Latin for after noon). Each period consists of 12 hours numbered 12, 1, 2, 3, 4, 5... The 12-hour clock is a timekeeping convention in which the 24 hours of the day are divided into two periods called ante meridiem (AM, Latin for before noon) and post meridiem (PM, Latin for after noon). Each period consists of 12 hours numbered 12, 1, 2, 3, 4, 5...

Multiples of 12 have been in common use as English units of resolution in the analog and digital printing world, where 1 point equals 1/72 of an inch and 12 points equal 1 pica, and printer resolutions like 360, 600, 720, 1200 or 1440 dpi (dots per inch) are common. These are combinations of base-12 and base-10 factors: (3×12)×10, 12×(5×10), (6×12)×10, 12×(10×10) and (12×12)×10. Point, in typography, may also refer to a dot grapheme (e. ... A pica (pronounced PIKE-ah, SAMPA /paIk@/) is a unit of measure traditionally used in document layout. ...

### Twenty

Possible remnants of a base-20 system also exist in French, as seen in the names of the numbers from 60 through 99. For example, sixty-five is soixante-cinq (literally, "sixty [and] five"), while seventy-five is soixante-quinze (literally, "sixty [and] fifteen"). Furthermore, for any number between 80 and 99, the "tens-column" number is expressed as a multiple of twenty (somewhat similar to the archaic English manner of speaking of "scores"). For example, eighty-two is quatre-vingt-deux (literally, four twenty[s] [and] two), while ninety-two is quatre-vingt-douze (literally, four twenty[s] [and] twelve). 20 (twenty) is the natural number following 19 and preceding 21. ...

Danish numerals display a similar base-20 structure. Danish (dansk) is one of the North Germanic languages (also called Scandinavi languages), a sub-group of the Germanic branch of the Indo-European languages. ...

### Dual base (five and twenty)

Many ancient counting systems use 5 as a primary base, almost surely coming from the number of fingers on a person's hand. Often these systems are supplemented with a secondary base, sometimes ten, sometimes twenty. In some African languages the word for 5 is the same as "hand" or "fist" (Dyola language of Guinea-Bissau, Banda language of Central Africa). Counting continues by adding 1, 2, 3, or 4 to combinations of 5, until the secondary base is reached. In the case of twenty, this word often means "man complete". This system is referred to as quinquavigesimal. It is found in many languages of the Sudan region. Map showing the distribution of African language families and some major African languages. ... Banda is a brass-based form of traditional Mexican music. ...

### Base names

` 1 - unary 2 - binary 3 - ternary / trinary 4 - quaternary 5 - quinary / quinternary 6 - senary / heximal / hexary 7 - septenary / septuary 8 - octal / octonary / octonal / octimal 9 - nonary / novary / noval 10 - decimal / denary 11 - undecimal / undenary / unodecimal 12 - dozenal / duodecimal / duodenary 13 - tridecimal / tredecimal / triodecimal 14 - tetradecimal / quadrodecimal / quattuordecimal 15 - pentadecimal / quindecimal 16 - hexadecimal / sexadecimal / sedecimal 17 - septendecimal / heptadecimal 18 - octodecimal / decennoctal 19 - nonadecimal / novodecimal / decennoval 20 - vigesimal / bigesimal / bidecimal 21 - unovigesimal / unobigesimal 22 - duovigesimal 23 - triovigesimal 24 - quadrovigesimal / quadriovigesimal 26 - hexavigesimal / sexavigesimal 27 - heptovigesimal 28 - octovigesimal 29 - novovigesimal 30 - trigesimal / triogesimal 31 - unotrigesimal (...repeat naming pattern...) 36 - hexatridecimal / sexatrigesimal (...repeat naming pattern...) 40 - quadragesimal / quadrigesimal 41 - unoquadragesimal (...repeat naming pattern...) 50 - quinquagesimal / pentagesimal 51 - unoquinquagesimal (...repeat naming pattern...) 60 - sexagesimal (...repeat naming pattern...) 64 - quadrosexagesimal (...repeat naming pattern...) 70 - septagesimal / heptagesimal 80 - octagesimal / octogesimal 90 - nonagesimal / novagesimal 100 - centimal / centesimal (...repeat naming pattern...) 110 - decacentimal 111 - unodecacentimal (...repeat naming pattern...) 200 - bicentimal / bicentesimal (...repeat naming pattern...) 210 - decabicentimal 211 - unodecabicentimal (...repeat naming pattern...) 300 - tercentimal / tricentesimal 400 - quattrocentimal / quadricentesimal 500 - quincentimal / pentacentesimal 600 - hexacentimal / hexacentesimal 700 - heptacentimal / heptacentesimal 800 - octacentimal / octocentimal / octacentesimal / octocentesimal 900 - novacentimal / novacentesimal 1000 - millesimal 2000 - bimillesimal (...repeat naming pattern...) 10000 - decamillesimal `

The unary numeral system is the simplest numeral system to represent natural numbers: in order to represent a number N, an arbitrarily chosen symbol is repeated N times. ... The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ... Ternary or trinary is the base-3 numeral system. ... Quaternary is the base four numeral system. ... Quinary (base-5) is a numeral system with five as the base. ... A senary numeral system is a base-six numeral system. ... A senary numeral system is a base-six numeral system. ... The septenary numeral system is the base seven number system, and uses the digits 0-6. ... The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. ... Nonary is a base 9 numeral system, typically using the digits 0-8, but not the digit 9. ... For other uses, see Decimal (disambiguation). ... Decimal, or denary, notation is the most common way of writing the base 10 numeral system, which uses various symbols for ten distinct quantities (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, called digits) together with the decimal point and the sign symbols + (plus) and &#8722; (minus... The undecimal (base-11) positional notation system is based on the number eleven, rather than ten as in decimal or eight in octal and so on. ... Duodecimal is a base 12 numeral system. ... The words Tredecimal and Tridecimal (i. ... The words Tredecimal and Tridecimal (i. ... The tetradecimal (base-14) positional notation system is based on the number fourteen. ... The pentadecimal (base-15) positional notation system is based on the number fifteen. ... In mathematics and computer science, hexadecimal, base-16, or simply hex, is a numeral system with a radix, or base, of 16, usually written using the symbols 0â€“9 and Aâ€“F, or aâ€“f. ... The vigesimal or base-20 numeral system is based on twenty (in the same way in which the ordinary decimal numeral system is based on ten). ... As there are 24 hours in a day a numbering system based upon 24, and as the base 12 is convenient here some examples of the base 24 (quadrovigesimal) system. ... A Hexavigesimal numeral system has a base of twenty-six. ... Base 30 or trigesimal is a positional numeral system using 30 as the radix. ... Base 36 is a positional numeral system using 36 as the radix. ... The sexagesimal (base-sixty) is a numeral system with sixty as the base. ... Base 64 or quadrosexagesimal is a positional notation using a base of 64. ...

## Positional systems in detail

Also see Positional notation. A positional notation or place-value notation system is a numeral system in which each position is related to the next by a constant multiplier, a common ratio, called the base or radix of that numeral system. ...

In a positional base-b numeral system (with b a positive natural number known as the radix), b basic symbols (or digits) corresponding to the first b natural numbers including zero are used. To generate the rest of the numerals, the position of the symbol in the figure is used. The symbol in the last position has its own value, and as it moves to the left its value is multiplied by b. In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... The radix (Latin for root), also called base, is the number of various unique symbols (or digits or numerals) a positional numeral system uses to represent numbers. ...

For example, in the decimal system (base 10), the numeral 4327 means (4×103) + (3×102) + (2×101) + (7×100), noting that 100 = 1. For other uses, see Decimal (disambiguation). ...

In general, if b is the base, we write a number in the numeral system of base b by expressing it in the form anbn + an − 1bn − 1 + an − 2bn − 2 + ... + a0b0 and writing the enumerated digits anan − 1an − 2 ... a0 in descending order. The digits are natural numbers between 0 and b − 1, inclusive.

If a text (such as this one) discusses multiple bases, and if ambiguity exists, the base (itself represented in base 10) is added in subscript to the right of the number, like this: numberbase. Unless specified by context, numbers without subscript are considered to be decimal.

By using a dot to divide the digits into two groups, one can also write fractions in the positional system. For example, the base-2 numeral 10.11 denotes 1×21 + 0×20 + 1×2−1 + 1×2−2 = 2.75.

In general, numbers in the base b system are of the form:

$(a_na_{n-1}cdots a_1a_0.c_1 c_2 c_3cdots)_b = sum_{k=0}^n a_kb^k + sum_{k=1}^infty c_kb^{-k}$

The numbers bk and bk are the weights of the corresponding digits. The position k is the logarithm of the corresponding weight w, that is k = logbw = logbbk. The highest used position is close to the order of magnitude of the number. A weight function is a mathematical device used when performing a sum, integral, or average in order to give some elements more of a weight than others. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. ...

The number of tally marks required in the unary numeral system for describing the weight would have been w. In the positional system the number of digits required to describe it is only k + 1 = logbw + 1, for $k ge 0$. E.g. to describe the weight 1000 then 4 digits are needed since log101000 + 1 = 3 + 1. The number of digits required to describe the position is logbk + 1 = logblogbw + 1 (in positions 1, 10, 100... only for simplicity in the decimal example). A sign warning hikers on the trail to Hanakapiai Beach. ... The unary numeral system is the simplest numeral system to represent natural numbers: in order to represent a number N, an arbitrarily chosen symbol is repeated N times. ...

 Position Weight Digit Decimal example weight Decimal example digit 3 2 1 0 -1 -2 ... b3 b2 b1 b0 b − 1 b − 2 ... a3 a2 a1 a0 c1 c2 ... 1000 100 10 1 0.1 0.01 ... 4 3 2 7 0 0 ...

Note that a number has a terminating or repeating expansion if and only if it is rational; this does not depend on the base. A number that terminates in one base may repeat in another (thus 0.310 = 0.0100110011001...2). An irrational number stays unperiodic (infinite amount of unrepeating digits) in all integral bases. Thus, for example in base 2, π = 3.1415926...10 can be written down as the unperiodic 11.001001000011111...2. IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ...

If b = p is a prime number, one can define base-p numerals whose expansion to the left never stops; these are called the p-adic numbers. In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... In mathematics, the p-adic number systems were first described by Kurt Hensel in 1897. ...

A simple algorithm for converting integers between positive-integer radices is repeated division by the target radix; the remainders give the "digits" starting at the least significant. E.g., 1020304 base 10 into base 7: In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will terminate in a defined end-state. ...

` 1020304 / 7 = 145757 r 5 145757 / 7 = 20822 r 3 20822 / 7 = 2974 r 4 2974 / 7 = 424 r 6 424 / 7 = 60 r 4 60 / 7 = 8 r 4 8 / 7 = 1 r 1 1 / 7 = 0 r 1 => 11446435 `

E.g., 10110111 base 2 into base 5:

` 10110111 / 101 = 100100 r 11 (3) 100100 / 101 = 111 r 1 (1) 111 / 101 = 1 r 10 (2) 1 / 101 = 0 r 1 (1) => 1213 `

To convert a "decimal" fraction, do repeated multiplication, taking the protruding integer parts as the "digits". Unfortunately a terminating fraction in one base may not terminate in another. E.g., 0.1A4C base 16 into base 9:

` 0.1A4C × 9 = 0.ECAC 0.ECAC × 9 = 8.520C 0.520C × 9 = 2.E26C 0.E26C × 9 = 7.F5CC 0.F5CC × 9 = 8.A42C 0.A42C × 9 = 5.C58C => 0.082785... `

## Generalized variable-length integers

More general is using a notation (here written little-endian) like a0a1a2 for a0 + a1b1 + a2b1b2, etc. In computing, endianness is the byte (and sometimes bit) ordering in memory used to represent some kind of data. ...

This is used in punycode, one aspect of which is the representation of a sequence of non-negative integers of arbitrary size in the form of a sequence without delimiters, of "digits" from a collection of 36: a-z and 0-9, representing 0-25 and 26-35 respectively. A digit lower than a threshold value marks that it is the most-significant digit, hence the end of the number. The threshold value depends on the position in the number. For example, if the threshold value for the first digit is b (i.e. 1) then a (i.e. 0) marks the end of the number (it has just one digit), so in numbers of more than one digit the range is only b-9 (1-35), therefore the weight b1 is 35 instead of 36. Suppose the threshold values for the second and third digit are c (2), then the third digit has a weight 34 × 35 = 1190 and we have the following sequence: This article or section may be confusing for some readers, and should be edited to be clearer. ...

a (0), ba (1), ca (2), .., 9a (35), bb (36), cb (37), .., 9b (70), bca (71), .., 99a (1260), bcb (1261), etc.

Note that unlike a regular base-35 numeral system, we have numbers like 9b where 9 and b each represent 35; yet the representation is unique because ac and aca are not allowed.

The flexibility in choosing threshold values allows optimization depending on the frequency of occurrence of numbers of various sizes.

The case with all threshold values equal to 1 corresponds to bijective numeration, where the zeros correspond to separators of numbers with digits which are nonzero. Bijective numeration is any numeral system that establishes a bijection between the set of non-negative integers and the set of finite strings over a finite set of digits. ...

### Properties of numerical systems with integer bases

Numeral systems with base A, where A is a positive integer, possess the following properties:

If A is even and A/2 is odd, all integral powers greater than zero of the number (A/2)+1 will contain (A/2)+1 as their last digit
If both A and A/2 are even, then all integral powers greater than or equal to zero of the number (A/2)+1 will alternate between having (A/2)+1 and 1 as their last digit. (For odd powers it will be (A/2)+1, for even powers it will be 1)

Proof of the first property:

Define ${A over 2} + 1 = x$ Then x is even, and all xp for p greater than 0 must be even. The property is equivalent to

$! x^p equiv x (mbox{mod} a)$

We first check the case for p=1

$! x equiv x (mbox{mod} A)$

x is less than A, so the result is trivial. We then check for p=2:

$! x^2 = xx$
$! x^2 = x(x-1) + x$

Since $x-1 = ({A over 2} + 1) - 1 = {A over 2}$, then for all even N:

$! {NA over 2} = N(x-1) equiv 0 (mbox{mod} A) (1)$

Because x is even, then x(x − 1) is congruent to zero modulo a. Therefore:

$! x^2 equiv x (mbox{mod} A)$

Using induction, assuming that the property holds for p-1:

$! x^p = {x^{p-1}}x = {x^{p-1}}(x-1) + x^{p-1}$

Since the case holds for p-1, then ${x^{p-1}} equiv x (mbox{mod} A)$. Since

$! {x^{p-1}}(x-1)$

is a case of Equation 1, then ${x^{p-1}}(x-1) equiv 0 (mbox{mod} A)$. This leaves, for all p greater than 0,

$! x^p equiv x (mbox{mod} a)$

Q.E.D. Look up QED in Wiktionary, the free dictionary. ...

Proof of the second property:

Define ${A over 2} + 1 = x$ Then x is odd, and all xp for p greater than or equal to 0 must be odd. The property is equivalent to

$! x^p equiv 1 (mbox{mod} A); mbox{if} p equiv 0 (mbox{mod} 2)$
$! x^p equiv x (mbox{mod} A); mbox{if} p equiv 1 (mbox{mod} 2)$

Since $x-1 = ({A over 2} + 1) - 1 = {A over 2}$, then for all odd E:

$! {EA over 2} = E(x-1) equiv {A over 2} (mbox{mod} A) (2)$

The case is first checked for p=0:

$! x^0 = 1$
$! 1 equiv 1 (mbox{mod} A)$

This result is trivial

Next, for p=1:

$! x^1 = x$
$! x equiv x (mbox{mod} A)$

This result is also trivial

Next, for p=2:

$! x^2 = xx = x(x-1) + x$

Because x is odd, then x(x-1) is a case of Equation 2,

$x(x-1) + x equiv {{A over 2} + x} (mbox{mod} A)$
$! {A over 2} + x = {A over 2} + {A over 2} + 1 = A+1$
$! A+1 equiv 1 (mbox{mod} A), (mbox{so} x(x-1) + x = x^2 equiv 1 (mbox{mod} A)$

Next, for p=3:

$! x^3 = {x^2}x = {x^2}(x-1) + x^2$

Because x2 is odd, x2(x − 1) + x2 is a case of Equation 2,

$! {x^2}(x-1) + x^2 equiv {{A over 2} + x^2} (mbox{mod} A)$

Since $x^2 equiv 1 (mbox{mod} A)$,

$! {x^2}(x-1) + x^2 equiv {{A over 2} + 1} (mbox{mod} A)$

${{A over 2} + 1} = x$, so $x^3 equiv x (mbox{mod} A)$.

Using induction, assuming that the property holds for p-1:

$! x^p equiv {x^{p-1}}(x-1) + x^{p-1}$

If p is odd:

$! x^{p-1} equiv 1 (mbox{mod} A)$

Since xp − 1(x − 1) is a case of Equation (2), ${x^{p-1}}(x-1) + x^{p-1} equiv {{A over 2} + 1} (mbox{mod} A)$, so

$x^p equiv x (mbox{mod} A)$

If p is even:

$! x^{p-1} equiv x (mbox{mod} A)$

Since xp − 1(x − 1) is a case of Equation (2), ${x^{p-1}}(x-1) + x^{p-1} equiv {{A over 2} + x} (mbox{mod} A)$.

${A over 2} + x = {A over 2} + {A over 2} + 1 = A+1$

$A+1 equiv 1 (mbox{mod} A)$, so

$x^p equiv 1 (mbox{mod} A)$

Q.E.D. Look up QED in Wiktionary, the free dictionary. ...

The term computer numbering formats refers to the schemes implemented in digital computer and calculator hardware and software to represent numbers. ... Subtractive notation is an early form of positional notation used with Roman numerals as a shorthand to replace four or five characters in a numeral representing a number with usually just two characters. ... A recurring or repeating decimal is a number which when expressed as a decimal has a set of final digits which repeat an infinite number of times. ... Inca Quipu. ... Babylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record. ... Golden ratio base refers to the use of the golden ratio, the irrational number â‰ˆ1. ... In linguistics, grammatical number is a morphological category characterized by the expression of quantity through inflection or agreement. ...

## References

• Georges Ifrah. The Universal History of Numbers : From Prehistory to the Invention of the Computer, Wiley, 1999. ISBN 0-471-37568-3.
• D. Knuth. The Art of Computer Programming. Volume 2, 3rd Ed. Addison-Wesley. pp.194–213, "Positional Number Systems".
• J.P. Mallory and D.Q. Adams, Encyclopedia of Indo-European Culture, Fitzroy Dearborn Publishers, London and Chicago, 1997.
• Hans J. Nissen, P. Damerow, R. Englund, Archaic Bookkeeping, University of Chicago Press, 1993, ISBN 0-226-58659-6.
• Denise Schmandt-Besserat, How Writing Came About, University of Texas Press, 1992, ISBN 0-292-77704-3.
• Claudia Zaslavsky, Africa Counts: Number and Pattern in African Cultures, Lawrence Hill Books, 1999, ISBN 1-55652-350-5.

Donald Ervin Knuth ( or Ka-NOOTH[1], Chinese: [2]) (b. ... Pearson can mean Pearson PLC the media conglomerate. ...

Results from FactBites:

 Numeral system (2673 words) Numeral systems are sometimes called number systems, but that name is misleading: different systems of numbers, such as the system of real numbers, the system of complex numbers, the system of p-adic numbers, etc., are not the topic of this article. The simplest numeral system is the unary numeral system, in which every natural number is represented by a corresponding number of symbols. The numeral system of English is of this type ("three hundred [and] four"), as are those of virtually all other spoken languages, regardless of what written systems they have adopted.
 Numeral system - Wikipedia, the free encyclopedia (3482 words) Numeral systems are sometimes called number systems, but that name is misleading: different systems of numbers, such as the system of real numbers, the system of complex numbers, the system of p-adic numbers, etc., are not the topic of this article. The simplest numeral system is the unary numeral system, in which every natural number is represented by a corresponding number of symbols. The numeral system of English is of this type ("three hundred [and] four"), as are those of virtually all other spoken languages, regardless of what written systems they have adopted.
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