1.
Numeral system
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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. It can be seen as the context that allows the symbols 11 to be interpreted as the symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases. The number the numeral represents is called its value, 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 decimal representation of whole numbers gives every nonzero whole number a unique representation as a finite sequence of digits. Etc. all of which have the same meaning except for some scientific, such systems are, however, not the topic of this article. 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 notation in the 5th century. The numeral system and the concept, developed by the Hindus in India, slowly spread to other surrounding countries due to their commercial. 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 world due to their trade links with them. The Western world modified them and called them the Arabic numerals, 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 used in India. The simplest numeral system is the numeral system, in which every natural number is represented by a corresponding number of symbols. If the symbol / is chosen, for example, then the 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 used in data compression. The unary notation can be abbreviated by introducing different symbols for new values. The ancient Egyptian numeral system was of type, and the Roman numeral system was a modification of this idea
2.
Arabic numerals
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In this numeral system, a sequence of digits such as 975 is read as a single number, using the position of the digit in the sequence to interpret its value. The symbol for zero is the key to the effectiveness of the system, the system was adopted by Arab mathematicians in Baghdad and passed on to the Arabs farther west. There is some evidence to suggest that the numerals in their current form developed from Arabic letters in the Maghreb, the current form of the numerals developed in North Africa, distinct in form from the Indian and eastern Arabic numerals. The use of Arabic numerals spread around the world through European trade, books, the term Arabic numerals is ambiguous. It most commonly refers to the widely used in Europe. Arabic numerals is also the name for the entire family of related numerals of Arabic. It may also be intended to mean the numerals used by Arabs and it would be more appropriate to refer to the Arabic numeral system, where the value of a digit in a number depends on its position. The decimal Hindu–Arabic numeral system was developed in India by AD700, the development was gradual, spanning several centuries, but the decisive step was probably provided by Brahmaguptas formulation of zero as a number in AD628. The system was revolutionary by including zero in positional notation, thereby limiting the number of digits to ten. It is considered an important milestone in the development of mathematics, one may distinguish between this positional system, which is identical throughout the family, and the precise glyphs used to write the numerals, which varied regionally. The glyphs most commonly used in conjunction with the Latin script since early modern times are 0123456789. The first universally accepted inscription containing the use of the 0 glyph in India is first recorded in the 9th century, in an inscription at Gwalior in Central India dated to 870. Numerous Indian documents on copper plates exist, with the symbol for zero in them, dated back as far as the 6th century AD. Inscriptions in Indonesia and Cambodia dating to AD683 have also been found and their work was principally responsible for the diffusion of the Indian system of numeration in the Middle East and the West. In the 10th century, Middle-Eastern mathematicians extended the decimal system to include fractions. The decimal point notation was introduced by Sind ibn Ali, who wrote the earliest treatise on Arabic numerals. Ghubar numerals themselves are probably of Roman origin, some popular myths have argued that the original forms of these symbols indicated their numeric value through the number of angles they contained, but no evidence exists of any such origin. In 825 Al-Khwārizmī wrote a treatise in Arabic, On the Calculation with Hindu Numerals, Algoritmi, the translators rendition of the authors name, gave rise to the word algorithm
3.
Eastern Arabic numerals
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These numbers are known as أرقام هندية in Arabic. They are sometimes also called Indic numerals in English, however, that is sometimes discouraged as it can lead to confusion with Indian numerals, used in Brahmic scripts of India. Each numeral in the Persian variant has a different Unicode point even if it looks identical to the Eastern Arabic numeral counterpart, however the variants used with Urdu, Sindhi and other South Asian languages are not encoded separately from the Persian variants. See U+0660 through U+0669 and U+06F0 through U+06F9, written numerals are arranged with their lowest-value digit to the right, with higher value positions added to the left. That is identical to the arrangement used by Western texts using Hindu-Arabic numerals even though Arabic script is read from right to left. There is no conflict unless numerical layout is necessary, as is the case for arithmetic problems and lists of numbers, Eastern Arabic numerals remain strongly predominant vis-à-vis Western Arabic numerals in many countries to the East of the Arab world, particularly in Iran and Afghanistan. In Pakistan, Western Arabic numerals are more used as a considerable majority of the population is anglophone. Eastern numerals still continue to see use in Urdu publications and newspapers, in North Africa, only Western Arabic numerals are now commonly used. In medieval times, these used a slightly different set
4.
Indian numerals
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Indian numerals are the symbols representing numbers in India. These numerals are used in the context of the decimal Hindu–Arabic numeral system. Below is a list of the Indian numerals in their modern Devanagari form, the corresponding Hindu-Arabic equivalents, their Hindi and Sanskrit pronunciation, since Sanskrit is an Indo-European language, it is obvious that the words for numerals closely resemble those of Greek and Latin. The word Shunya for zero was translated into Arabic as صفر sifr, meaning nothing which became the zero in many European languages from Medieval Latin. The five Indian languages that have adapted the Devanagari script to their use also naturally employ the numeral symbols above, of course, for numerals in Tamil language see Tamil numerals. For numerals in Telugu language see Telugu numerals, Tamil and Malayalam scripts also have distinct forms for 10,100,1000 numbers, ௰, ௱, ௲and ൰, ൱, ൲ respectively in tamil and scripts. A decimal place system has been traced back to ca.500 in India, before that epoch, the Brahmi numeral system was in use, that system did not encompass the concept of the place-value of numbers. Instead, Brahmi numerals included additional symbols for the tens, as well as symbols for hundred. The Indian place-system numerals spread to neighboring Persia, where they were picked up by the conquering Arabs, in 662, Severus Sebokht - a Nestorian bishop living in Syria wrote, I will omit all discussion of the science of the Indians. Of their subtle discoveries in astronomy — discoveries that are more ingenious than those of the Greeks, I wish only to say that this computation is done by means of nine signs. But it is in Khmer numerals of modern Cambodia where the first extant material evidence of zero as a numerical figure, as it was from the Arabs that the Europeans learned this system, the Europeans called them Arabic numerals, the Arabs refer to their numerals as Indian numerals. In academic circles they are called the Hindu–Arabic or Indo–Arabic numerals, but what was the net achievement in the field of reckoning, the earliest art practiced by man. An inflexible numeration so crude as to progress well nigh impossible. Man used these devices for thousands of years without contributing an important idea to the system. Even when compared with the growth of ideas during the Dark Ages. When viewed in light, the achievements of the unknown Hindu. Sanskrit Siddham Numbers Karl Menninger, Number Words and Number Symbols - A Cultural History of Numbers ISBN 0-486-27096-3 David Eugene Smith and Louis Charles Karpinski, The Hindu-Arabic Numerals
5.
Sinhala numerals
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Sinhalese belongs to the Indo-European language family with its roots deeply associated with Indo-Aryan sub family to which the languages such as Persian and Hindi belong. It is also surmised that Sinhala had evolved from an ancient variant of Apabramsa which is known as ‘Elu’, when tracing history of Elu, it was preceded by Hela or Pali Sihala. The Sinhala script had evolved from Southern Brahmi script from which almost all the Southern Indic Scripts such as Telugu, later Sinhala was influenced by Grantha writing of Southern India. Since 1250 AD, the Sinhala script had remained the same with few changes, although some scholars are of the view that the Brahmi Script arrived with the Buddhism, Mahavamsa speaks of written language even right after the arrival of Vijaya. Archeologists had found pottery fragments in Anuradhapura Sri Lanka with older Brahmi script inscriptions, the earliest Brahmi Script found in India had been dated to 6th Century BC in Tamil Nadu though most of Brahmi writing found in India had been attributed to emperor Ashoka in the 3rd century BC. Sinhala letters are round-shaped and are written left to right. The evolution of the script to the present shapes may have taken place due to writing on Ola leaves, unlike chiseling on a rock, writing on palm leaves has to be more round-shaped to avoid the stylus ripping the Palm leaf while writing on it. When drawing vertical or horizontal lines on Ola leaf, the leaves would have been ripped. Instead a stylistic stop which was known as ‘Kundaliya’ is used, period and commas were later introduced into Sinhala script after the introduction of paper due to the influence of Western languages. In modern Sinhala, Arabic numerals, which were introduced by Portuguese, Dutch and English, is used for writing numbers and it is accepted that Arabic numerals had evolved from Brahmi numerals. This article will touch upon Brahmi numerals, which were found in Sri Lanka. It had been found five different types of numerations were used in the Sinhala language at the time of the invasion of the Kandyan kingdom by the British. Out of the five types of numerations, two sets of numerations were in use in the century mainly for astrological calculations and to express traditional year. The five types or sets of numerals or numerations are listed below, according to Mr. Gunesekera, these numerals were used for ordinary calculations and to express simple numbers. These numerals had separate Symbols for 10,40,50,100,1000 and these numerals were also regarded as Lith Lakunu or ephemeris numbers by W. A. De Silva in his “Catalogue of Palm leaf manuscripts in the library of Colombo Museum”. This set of numerals was known as Sinhala illakkam or Sinhala archaic numerals, Arabic Figures are now universally used. For the benefit of the student, the old numerals are given in the plate opposite,11 clauses had been numbered in Arabic numerals in the English part of the agreement and in parallel Sinhala clauses were numbered in Sinhala archaic numerals. Numbers of lith illakkam look Sinhala letters and vowel modifiers, the number six is known as ‘akma’ in the Lith Illakkam
6.
Tamil numerals
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Traditionally Vattezhuttu characters were used, but now Arabic numerals have become commonplace. Old Tamil possesses a special character for zero and it is read as andru. But yet Modern Tamil renounces the use of its native character, Modern Tamil words for zero include சுழியம் or பூச்சியம். Tamil has a prefix for each number from 1 to 9. For instance, the word for fifty, ஐம்பது is a combination of ஐ, the prefix for nine changes with respect to the succeeding base 10. தொ+ the unvoiced consonant of the succeeding base 10 forms the prefix for nine, for instance,90 is தொ+ண், hence, தொண்ணூறு). These are typically void in the Tamil language except for some Hindu and Christian religious references, example அட்ட இலட்சுமிகள் in a Hindu context, unlike other Indian languages, Tamil has distinct digits for 10,100, and 1000. It also has characters for other number-based aspects of day-to-day life. − − − − − − − − − − − − There are two systems that can be used in the Tamil language, the Tamil system which is as follows. The following are the numbers of the Ancient Tamil Country. Sanskrit based multiples like lakhs are also followed just like other Indian languages and you can transcribe any fraction, by affixing -இல் after the denominator followed by the numerator. For instance, 1/41 can be said as நாற்பத்து ஒன்றில் ஒன்று, the suffixing of the -இல் requires you to change the last consonant of the number to its இ form. For example, மூன்று+இல் becomes மூன்றில், note the உ has been omitted, common fractions have names already allocated to them, hence, these names are often used rather than the above method. Other fractions are, Anu was considered as lowest fraction by ancient Tamils as size of smallest physical object, later, this term went to Sanskrit to refer directly atom. Decimal point is called புள்ளி in Tamil, for example,1.1 would be read as ஒன்று புள்ளி ஒன்று. Percentage is known as விழுக்காடு in Tamil or சதவீதம் and these words are simply added after a number to form percentages. For instance, four percent is நான்கு சதவீதம் or நான்கு விழுக்காடு, percentage symbol is also recognised and used. Ordinal numbers are formed by adding the suffix -ஆம் after the number, as always, when blending two words into one, an unvoiced form of the consonant as the one that the second starts with, is placed in between to blend
7.
Balinese numerals
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The Balinese language has an elaborate decimal numeral system. The numerals 1–10 have basic, combining, and independent forms, the combining forms are used to form higher numbers. In some cases there is more than one word for a numeral, reflecting the Balinese register system, final orthographic -a is a schwa. * A less productive combining form of a-1 is sa- and it, ulung-, and sangang- are from Javanese. Dasa 10 is from Sankrit désa, like English, Balinese has compound forms for the teens and tens, however, it also has a series of compound tweens, 21–29. The teens are based on a root *-welas, the tweens on -likur, hyphens are not used in the orthography, but have been added to the table below to clarify their derivation. The high-register combining forms kalih-2 and tigang-3 are used with -likur, -dasa, and higher numerals, the teens are from Javanese, where the -olas forms are regular, apart from pele-kutus 18, which is suppletive. Sa-laé25, and se-ket 50 are also suppletive, and cognate with Javanese səlawé25, there are additional numerals pasasur ~ sasur 35 and se-timahan ~ se-timan 45, and a compound telung-benang for 75. The unit combining forms are combined with atus 100, atak 200, amas 400, tali 1000, laksa 10,000, keti 100,000, in addition, there is karobelah 150, lebak 175, and sepa for 1600. At least karobelah has a cognate in Javanese, ro-bəlah, where ro- is the form for two
8.
Burmese numerals
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Burmese numerals are a set of numerals traditionally used in the Burmese language, although the Arabic numerals are also used. Burmese numerals follow the Hindu-Arabic numeral system used in the rest of the world. 1 Burmese for zero comes from Sanskrit śūnya.2 Can be abbreviated to IPA, in list contexts, spoken Burmese has innate pronunciation rules that govern numbers when they are combined with another word, be it a numerical place or a measure word. Other suffixes such as ထောင်, သောင်း, သိန်း, and သန်း all shift to, for six and eight, no pronunciation shift occurs. These pronunciation shifts are exclusively confined to spoken Burmese and are not spelt any differently,1 Shifts to voiced consonant following three, four, five, and nine. Ten to nineteen are almost always expressed without including တစ်, another pronunciation rule shifts numerical place name from the low tone to the creaky tone. Number places from 10 up to 107 has increment of 101, beyond those Number places, larger number places have increment of 107. 1014 up to 10140 has increment of 107, numbers in the hundreds place, shift from ရာ to ရာ့, except for numbers divisible by 100. Numbers in the place, shift from ထောင် to ထောင့်. Hence, a number like 301 is pronounced, while 300 is pronounced, the digits of a number are expressed in order of decreasing digits place. When a number is used as an adjective, the word order is. However, for numbers, the word order is flipped to. The exception to rule is the number 10, which follows the standard word order. Ordinal numbers, from first to tenth, are Burmese pronunciations of their Pali equivalents and they are prefixed to the noun. Beyond that, cardinal numbers can be raised to the ordinal by suffixing the particle မြောက် to the number in the order, number + measure word + မြောက်. Colloquially, decimal numbers are formed by saying ဒသမ where the separator is located. For example,10.1 is ဆယ် ဒသမ တစ်, half is expressed primarily by တစ်ဝက်, although ထက်ဝက်, အခွဲ and အခြမ်း are also used. Quarter is expressed with အစိတ် or တစ်စိတ်, other fractional numbers are verbally expressed as follows, denominator + ပုံ + numerator + ပုံ
9.
Dzongkha numerals
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Dzongkha, the national language of Bhutan, has two numeral systems, one vigesimal, and a modern decimal system. The vigesimal system remains in robust use, ten is an auxiliary base, the teens are formed with ten and the numerals 1–9. *When it appears on its own, ten is usually said cu-tʰãm a full ten, in combinations it is simply cu. Factors of 20 are formed from kʰe, intermediate factors of ten are formed with pɟʱe-da half to,400 ɲiɕu is the next unit, ɲiɕu ciː400, ɲiɕu ɲi 800, etc. Higher powers are 8000 kʰecʰe and jãːcʰe 160,000, the decimal system is the same as the vigesimal system up to 19. Then decades, however, are formed as unit–ten, as in Chinese,20 is reported to be ɲiɕu, the vigesimal numeral 400, this may be lexical interference for the expected *ɲi-cu. Mazaudon & Lacito,2002, Les principes de construction du nombre dans les langues tibeto-birmanes, in François, ed
10.
Gujarati numerals
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Gujarati numerals is the numeral system of the Gujarati script of South Asia, which is a derivative of Devanagari numerals. It is the numeral system of Gujarat, India. It is also recognized in India and as a minor script in Pakistan. The following table shows Gujarati numbers and the Gujarati word for each of them in various scripts, Gujarati script Gurmukhi numerals Devanagari alphabet
11.
Javanese numerals
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The Javanese language has a decimal numeral system with distinct words for the tweens from 21 to 29, called likuran. The basic numerals 1–10 have independent and combining forms, the latter derived via a suffix -ng, the combining forms are used to form the tens, hundreds, thousands, and millions. The numerals 1–5 and 10 have distinct high-register and low register forms, the halus forms are listed below in italics. Like English, Javanese has compound forms for the teens, however, it also has a series of compound tweens, the teens are based on a root -las, the tweens on -likur, and the tens are formed by the combining forms. Hyphens are not used in the orthography, but have added to the table below to clarify their derivation. Final orthographic -a tends to in many dialects, as does any preceding a, parallel to the tens are the hundreds, the thousands, and the millions, except that the compounds of five and six are formed with limang- and nem-. The names of the Old Javanese numerals were derived from their names in the Sanskrit language, balinese numerals, a related but yet more complex numeral system
12.
Khmer numerals
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Khmer numerals are the numerals used in the Khmer language. They have been in use since at least the early 7th century, with the earliest known use being on a stele dated to AD604 found in Prasat Bayang, Cambodia, having been derived from the Hindu numerals, modern Khmer numerals also represent a decimal positional notation system. It is the script with the first extant material evidence of zero as a figure, dating its use back to the seventh century. However, Old Khmer, or Angkorian Khmer, also possessed separate symbols for the numbers 10,20 and this inconsistency with its decimal system suggests that spoken Angkorian Khmer used a vigesimal system. For example,6 is formed from 5 plus 1, with the exception of the number 0, which stems from Sanskrit, the etymology of the Khmer numbers from 1 to 5 is of proto-Mon–Khmer origin. For details of the various alternative romanization systems, see Romanization of Khmer, some authors may alternatively mark as the pronunciation for the word two, and either or for the word three. In neighbouring Thailand the number three is thought to bring good luck, however, in Cambodia, taking a picture with three people in it is considered bad luck, as it is believed that the person situated in the middle will die an early death. As mentioned above, the numbers from 6 to 9 may be constructed by adding any number between 1 and 4 to the base number 5, so that 7 is literally constructed as 5 plus 2. Beyond that, Khmer uses a base, so that 14 is constructed as 10 plus 4, rather than 2 times 5 plus 4. In constructions from 6 to 9 that use 5 as a base, /pram/ may alternatively be pronounced, giving and this is especially true in dialects which elide /r/, but not necessarily restricted to them, as the pattern also follows Khmers minor syllable pattern. The numbers from thirty to ninety in Khmer bear many resemblances to both the modern Thai and Cantonese numbers, informally, a speaker may choose to omit the final and the number is still understood. For example, it is possible to say instead of the full, Language Comparisons, Words in parenthesis indicate literary pronunciations, while words preceded with an asterisk mark are non-productive. The standard Khmer numbers starting from one hundred are as follows, Although មួយកោដិ is most commonly used to mean ten million, in some areas this is also colloquially used to refer to one billion. In order to avoid confusion, sometimes ដប់លាន is used to mean ten million, along with មួយរយលាន for one hundred million, different Cambodian dialects may also employ different base number constructions to form greater numbers above one thousand. As a result of prolonged literary influence from both the Sanskrit and Pali languages, Khmer may occasionally use borrowed words for counting. One reason for the decline of numbers is that a Khmer nationalism movement. The Khmer Rouge also attempted to cleanse the language by removing all words which were considered politically incorrect, Khmer ordinal numbers are formed by placing the word ទី in front of a cardinal number. This is similar to the use of ที่ thi in Thai and it is generally assumed that the Angkorian and pre-Angkorian numbers also represented a dual base system, with both base 5 and base 20 in use
13.
Lao alphabet
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Lao script, or Akson Lao, is the primary script used to write the Lao language and other minority languages in Laos. It was also used to write the Isan language, but was replaced by the Thai script and it has 27 consonants,7 consonantal ligatures,33 vowels, and 4 tone marks. Akson Lao is a system to the Thai script, with which it shares many similarities. However, Lao has fewer characters and is formed in a curvilinear fashion than Thai. Lao is traditionally written from left to right, Lao is considered an abugida, in which certain implied vowels are unwritten. However, due to spelling reforms by the communist Lao Peoples Revolutionary Party, despite this, most Lao outside of Laos, and many inside Laos, continue to write according to former spelling standards, so vernacular Lao functions as a pure abugida. For example, the old spelling of ສເລີມ to hold a ceremony, vowels can be written above, below, in front of, or behind consonants, with some vowel combinations written before, over and after. Spaces for separating words and punctuation were traditionally not used, but a space is used, the letters have no majuscule or minuscule differentiation. The Lao script was standardized in the Mekong River valley after the various Tai principalities of the region were merged under Lan Xang in the 14th century. This script, sometimes known as Tai Noi, has changed little since its inception and continued use in the Lao-speaking regions of modern-day Laos, conversely, the Thai alphabet continued to evolve, but the scripts still share similarities. This script was derived locally from the Khmer script of Angkor with additional influence from Mon, traditionally, only secular literature were written with the Lao alphabet. Religious literature was written in Tua Tham, a Mon-based script that is still used for the Tai Khün, Tai Lue. Mystical, magical, and some literature was written in a modified version of the Khmer alphabet. Essentially Thai and Lao are almost typographic variants of other just as in the Javanese and Balinese scripts. The Lao and Thai alphabets share the same roots, but Lao has fewer characters and is written in a curvilinear fashion than Thai. However this is apparent today due to the communist party simplifying the spelling to be phonetic. There is speculation that the Lao and Thai script both derive from a common script due to the similarities between the scripts. When examining older forms of Thai scripts, many letters are almost identical to the Lao alphabet, some minority languages use separate writing systems, The Hmong have adopted the Roman Alphabet
14.
Thai numerals
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The Thai language lacks grammatical number. A count is expressed in the form of an uninflected noun followed by a number. In Thai, counting is kannap, the classifier, laksananam Variations to this pattern do occur, a partial list of Thai words that also classify nouns can be found in Wiktionary category, Thai classifiers. Thai sūn is written as oval 0 when using Arabic numerals, but a small circle ๐ when using traditional numerals and it is from Sanskrit śūnya, as are the alternate names for numbers one to four given below, but not the counting 1. Thai names for N +1 and the regular digits 2 through 9 as shown in the table, below, resemble those in Chinese varieties as spoken in Southern China, Thai and Lao words for numerals are almost identical, however, the numerical digits vary somewhat in shape. Shown below is a comparison between three languages using Cantonese and Minnan characters and pronunciations, the Thai transliteration uses the Royal Thai General System of Transcription. Sanskrit lakh designates the place value of a digit, which are named for the powers of ten, the place is lak nuai, tens place, lak sip, hundreds place, lak roi. The number one following any multiple of sip becomes et, the number ten is the same as Minnan 十. Numbers from twenty to twenty nine begin with yi sip, names of the lak sip for 30 to 90, and for the lak of 100,1000,10,000,100,000 and million, are almost identical to those of the like Khmer numerals. For the numbers twenty-one through twenty-nine, the part signifying twenty, yi sip, see the alternate numbers section below. The hundreds are formed by combining roi with the tens and ones values, for example, two hundred and thirty-two is song roi sam sip song. The words roi, phan, muen, and saen should occur with a preceding numeral, nueng never precedes sip, so song roi nueng sip is incorrect. Native speakers will sometimes use roi nueng with different tones on nueng to distinguish one hundred from one hundred, however, such distinction is often not made, and ambiguity may follow. To resolve this problem, if the number 101 is intended, numbers above a million are constructed by prefixing lan with a multiplier. For example, ten million is sip lan, and a trillion is lan lan, colloquially, decimal numbers are formed by saying chut where the decimal separator is located. For example,1.01 is nueng chut sun nueng, fractional numbers are formed by placing nai between the numerator and denominator or using x suan y to clearly indicate. For example, ⅓ is nueng nai sam or nueng suan sam, the word set can be omitted. The word khrueng is used for half and it precedes the measure word if used alone, but it follows the measure word when used with another number
15.
Chinese numerals
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Chinese numerals are words and characters used to denote numbers in Chinese. Today speakers of Chinese use three written numeral systems, the system of Arabic numerals used worldwide, and two indigenous systems, the more familiar indigenous system is based on Chinese characters that correspond to numerals in the spoken language. These are shared with languages of the Chinese cultural sphere such as Japanese, Korean. The other indigenous system is the Suzhou numerals, or huama, a positional system and these were once used by Chinese mathematicians, and later in Chinese markets, such as those in Hong Kong before the 1990s, but have been gradually supplanted by Arabic numerals. The Chinese character numeral system consists of the Chinese characters used by the Chinese written language to write spoken numerals, similar to spelling-out numbers in English, it is not an independent system per se. Since it reflects spoken language, it not use the positional system as in Arabic numerals. There are characters representing the numbers zero through nine, and other characters representing larger numbers such as tens, hundreds, thousands, there are two sets of characters for Chinese numerals, one for everyday writing and one for use in commercial or financial contexts known as dàxiě. A forger could easily change the everyday characters 三十 to 五千 just by adding a few strokes and that would not be possible when writing using the financial characters 參拾 and 伍仟. They are also referred to as bankers numerals, anti-fraud numerals, for the same reason, rod numerals were never used in commercial records. T denotes Traditional Chinese characters, S denotes Simplified Chinese characters, in the PLA, some numbers will have altered names when used for clearer radio communications. They are,0, renamed 洞 lit, hole 1, renamed 幺 lit. small 2, renamed 两 lit. Double 7, renamed 拐 lit. cane, kidnap, turn 9, hook For numbers larger than 10,000, similarly to the long and short scales in the West, there have been four systems in ancient and modern usage. The original one, with names for all powers of ten up to the 14th, is ascribed to the Yellow Emperor in the 6th century book by Zhen Luan. To avoid problems arising from the ambiguity, the PRC government never uses this character in official documents, the ROC government in Taiwan uses 兆 to mean 1012 in official documents. Numerals beyond 載 zài come from Buddhist texts in Sanskrit, but are found in ancient texts. Some of the words are still being used today. The following are characters used to denote small order of magnitude in Chinese historically, with the introduction of SI units, some of them have been incorporated as SI prefixes, while the rest have fallen into disuse. In the Peoples Republic of China, the translations for the SI prefixes in 1981 were different from those used today, the Republic of China defined 百萬 as the translation for mega
16.
Suzhou numerals
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The Suzhou numerals, also known as Suzhou mazi or huama, is a numeral system used in China before the introduction of Arabic numerals. The Suzhou numeral system is the only surviving variation of the rod numeral system, the rod numeral system is a positional numeral system used by the Chinese in mathematics. Suzhou numerals are a variation of the Southern Song rod numerals, Suzhou numerals were used as shorthand in number-intensive areas of commerce such as accounting and bookkeeping. At the same time, standard Chinese numerals were used in formal writing, Suzhou numerals were once popular in Chinese marketplaces, such as those in Hong Kong along with local transportation before the 1990s, but they have gradually been supplanted by Arabic numerals. This is similar to what had happened in Europe with Roman numerals used in ancient and medieval Europe for mathematics, nowadays, the Suzhou numeral system is only used for displaying prices in Chinese markets or on traditional handwritten invoices. In the Suzhou numeral system, special symbols are used for digits instead of the Chinese characters, the digits of the Suzhou numerals are defined between U+3021 and U+3029 in Unicode. An additional three code points starting from U+3038 were added later, the numbers one, two, and three are all represented by vertical bars. This can cause confusion when they next to each other. Standard Chinese ideographs are often used in this situation to avoid ambiguity, for example,21 is written as 〢一 instead of 〢〡 which can be confused with 3. The first character of such sequences is usually represented by the Suzhou numeral, the full numerical notations are written in two lines to indicate numerical value, order of magnitude, and unit of measurement. Following the rod system, the digits of the Suzhou numerals are always written horizontally from left to right. The first line contains the values, in this example. The second line consists of Chinese characters that represents the order of magnitude, in this case 十元 which stands for ten yuan. When put together, it is read as 40.22 yuan. Zero is represented by the character for zero, leading and trailing zeros are unnecessary in this system. This is very similar to the scientific notation for floating point numbers where the significant digits are represented in the mantissa. Also, the unit of measurement, with the first digit indicator, is aligned to the middle of the numbers row. In the Unicode standard version 3.0, these characters are incorrectly named Hangzhou style numerals, in the episode The Blind Banker of the 2010 BBC television series Sherlock, Sherlock Holmes erroneously refers to the number system as Hangzhou instead of the correct Suzhou
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Japanese numerals
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The system of Japanese numerals is the system of number names used in the Japanese language. The Japanese numerals in writing are based on the Chinese numerals. Two sets of pronunciations for the numerals exist in Japanese, one is based on Sino-Japanese readings of the Chinese characters, there are two ways of writing the numbers in Japanese, in Hindu-Arabic numerals or in Chinese numerals. The Hindu-Arabic numerals are often used in horizontal writing. Numerals with multiple On readings use the Go-on and Kan-on variants respectively, * The special reading 〇 maru is also found. It may be used when reading individual digits of a number one after another. A popular example is the famous 109 store in Shibuya, Tokyo which is read as ichi-maru-kyū and this usage of maru for numerical 0 is similar to reading numeral 0 in English as oh. However, as a number, it is written as 0 or rei. Additionally, two and five are pronounced with a vowel in phone numbers Starting at 万, numbers begin with 一 if no digit would otherwise precede. That is,100 is just 百 hyaku, and 1000 is just 千 sen and this differs from Chinese as numbers begin with 一 if no digit would otherwise precede starting at 百. And, if 千 sen directly precedes the name of powers of myriad, 一 ichi is normally attached before 千 sen and that is,10,000,000 is normally read as 一千万 issenman. But if 千 sen does not directly precede the name of powers of myriad or if numbers are lower than 2,000 and that is,15,000,000 is read as 千五百万 sengohyakuman or 一千五百万 issengohyakuman, and 1,500 as 千五百 sengohyaku or 一千五百 issengohyaku. The numbers 4 and 9 are considered unlucky in Japanese,4, pronounced shi, is a homophone for death,9, the number 13 is sometimes considered unlucky, though this is a carryover from Western tradition. On the contrary, numbers 7 and sometimes 8 are considered lucky in Japanese, in modern Japanese, cardinal numbers are given the on readings except 4 and 7, which are called yon and nana respectively. Alternate readings are used in names, day-of-month names. For instance, the decimal fraction 4.79 is always read yon-ten nana kyū, though April, July, and September are called shi-gatsu, shichi-gatsu, the on readings are also used when shouting out headcounts. Intermediate numbers are made by combining elements, Tens from 20 to 90 are -jū as in 二十 to 九十. Hundreds from 200 to 900 are -hyaku, thousands from 2000 to 9000 are -sen
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Korean numerals
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The Korean language has two regularly used sets of numerals, a native Korean system and Sino-Korean system. For both native and Sino- Korean numerals, the teens are represented by a combination of tens, for instance,15 would be sib-o, but not usually il-sib-o in the Sino-Korean system, and yeol-daseot in native Korean. Twenty through ninety are likewise represented in this manner in the Sino-Korean system, while Native Korean has its own unique set of words. The grouping of large numbers in Korean follow the Chinese tradition of myriads rather than thousands, the Sino-Korean system is nearly entirely based on the Chinese numerals. The distinction between the two systems is very important. Everything that can be counted will use one of the two systems, but seldom both, Sino-Korean words are sometimes used to mark ordinal usage, yeol beon means ten times while sip beon means number ten. When denoting the age of a person, one will usually use sal for the native Korean numerals, for example, seumul-daseot sal and i-sib-o se both mean twenty-five-year-old. See also East Asian age reckoning, the Sino-Korean numerals are used to denote the minute of time. For example, sam-sib-o bun means __,35 or thirty-five minutes, the native Korean numerals are used for the hours in the 12-hour system and for the hours 0,00 to 12,00 in the 24-hour system. The hours 13,00 to 24,00 in the 24-hour system are denoted using both the native Korean numerals and the Sino-Korean numerals. For example, se si means 03,00 or 3,00 a. m. /p. m. for counting above 100, Sino-Korean words are used, sometimes in combination,101 can be baek-hana or baeg-il. The usual liaison and consonant-tensing rules apply, so for example, 예순여섯 yesun-yeoseot is pronounced like, beon, ho, cha, and hoe are always used with Sino-Korean or Arabic ordinal numerals. For example, Yihoseon is Line Number Two in a subway system. 906호 is Apt #906 in a mailing address,906 without ho is not used in spoken Korean to imply apartment number or office suite number. The special prefix je is usually used in combination with suffixes to designate a specific event in sequential things such as the Olympics, in commerce or the financial sector, some hanja for each Sino-Korean numbers are replaced by alternative ones to prevent ambiguity or retouching. For verbally communicating number sequences such as numbers, ID numbers, etc. especially over the phone. For the same reason, military transmissions are known to use mixed native Korean and Sino-Korean numerals, note 1, ^ Korean assimilation rules apply as if the underlying form were 십륙 |sip. ryuk|, giving sim-nyuk instead of the expected sib-yuk. Note 2, ^ ^ ^ ^ ^ These names are considered archaic, note 3, ^ ^ ^ ^ ^ ^ ^ The numbers higher than 1020 are not usually used
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Vietnamese numerals
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Historically Vietnamese has two sets of numbers, one is etymologically native Vietnamese, the other uses Sino-Vietnamese vocabulary. In the modern language the native Vietnamese vocabulary is used for both everyday counting and mathematical purposes, the Sino-Vietnamese vocabulary is used only in fixed expressions or in Sino-Vietnamese words. This is somewhat analogous to the way in which Latin and Greek numerals are used in modern English, Sino-Vietnamese words are also used for units of ten thousand or above, where native vocabulary was lacking. Among the languages of the Chinese cultural sphere, Japanese and Korean both use two systems, one native and one Chinese-based. The Chinese-based vocabulary is the one in common use, in Vietnamese, on the other hand, the Chinese-based system is not in everyday use. Numbers from 1 to 1000 are expressed using native Vietnamese vocabulary, in the modern Vietnamese writing system, numbers are written in the romanized script quốc ngữ or Arabic numerals. Prior to the 20th century Vietnam officially used Classical Chinese as a written language, for non-official purposes Vietnamese also had a writing system known as Hán-Nôm. Under this system, Sino-Vietnamese numbers were written in Hán tự, basic features of the Vietnamese numbering system include the following, Unlike other sinoxenic numbering systems, Vietnamese separates place values in thousands rather than myriads. The Sino-Vietnamese numbers are not in frequent use in modern Vietnamese, number values for these words follow usage in Ancient China, with each numeral increasing tenfold in digit value, 億 being the number for 105, 兆 for 106, et cetera. As a result, the value of triệu differs from modern Chinese 兆, outside of fixed Sino-Vietnamese expressions, Sino-Vietnamese words are usually used in combination with native Vietnamese words. For instance, mười triệu combines native mười and Sino-Vietnamese triệu, the following table is an overview of the basic Vietnamese numeric figures, provided in both Native and Sino-Viet forms. For each number, the form that is commonly used is highlighted. Where there are differences between the Hanoi and Saigon dialects of Vietnamese, readings between each are differentiated below within the notes, when the number 1 appears after 20 in the unit digit, the pronunciation changes to mốt. When the number 4 appears after 20 in the digit, it is more common to use Sino-Viet tư／四. When the number 5 appears after 10 in the unit digit, when mười appears after 20, the pronunciation changes to mươi. Vietnamese ordinal numbers are preceded by the prefix thứ, which is a Sino-Viet word which corresponds to 次. For the ordinal numbers of one and four, the Sino-Viet readings nhất／一 and tư／四 are more commonly used, in all other cases, the native Vietnamese number is used. Chinese numerals Japanese numerals Korean numerals
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Counting rods
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Counting rods are small bars, typically 3–14 cm long, that were used by mathematicians for calculation in ancient China, Japan, Korea, and Vietnam. They are placed horizontally or vertically to represent any integer or rational number. The written forms based on them are called rod numerals and they are a true positional numeral system with digits for 1–9 and a blank for 0, from the Warring states period to the 16th century. Counting rods were used by ancient Chinese for more two thousand years. In 1954, forty-odd counting rods of the Warring States period were found in Zuǒjiāgōngshān Chu Grave No.15 in Changsha, in 1973, archeologists unearthed a number of wood scripts from a Han dynasty tomb in Hubei. On one of the scripts was written, “当利二月定算”. This is one of the earliest examples of using counting rod numerals in writing, in 1976, a bundle of Western Han counting rods made of bones was unearthed from Qianyang County in Shaanxi. The use of counting rods must predate it, Laozi said a good calculator doesnt use counting rods, the Book of Han recorded, they calculate with bamboo, diameter one fen, length six cun, arranged into a hexagonal bundle of two hundred seventy one pieces. At first calculating rods were round in section, but by the time of the Sui dynasty triangular rods were used to represent positive numbers. After the abacus flourished, counting rods were abandoned except in Japan, counting rods represent digits by the number of rods, and the perpendicular rod represents five. To avoid confusion, vertical and horizontal forms are alternately used, generally, vertical rod numbers are used for the position for the units, hundreds, ten thousands, etc. while horizontal rod numbers are used for the tens, thousands, hundred thousands etc. It is written in Sunzi Suanjing that one is vertical, ten is horizontal, red rods represent positive numbers and black rods represent negative numbers. Ancient Chinese clearly understood negative numbers and zero, though they had no symbol for the latter, later, a go stone was sometimes used to represent zero. This alternation of vertical and horizontal rod numeral form is important to understanding written transcription of rod numerals on manuscripts correctly. In the same manuscript,405 was transcribed as, with a space in between for obvious reasons, and could in no way be interpreted as 45. In other words, transcribed rod numerals may not be positional, the value of a number depends on its physical position on the counting board. A9 at the rightmost position on the stands for 9. Moving the batch of rods representing 9 to the one position gives 9 or 90
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Abjad numerals
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The Abjad numerals are a decimal numeral system in which the 28 letters of the Arabic alphabet are assigned numerical values. They have been used in the Arabic-speaking world since before the century when Arabic numerals were adopted. In modern Arabic, the word abjadīyah means alphabet in general, in the Abjad system, the first letter of the Arabic alphabet, alif, is used to represent 1, the second letter, bāʾ, is used to represent 2, etc. Individual letters also represent 10s and 100s, yāʾ for 10, kāf for 20, qāf for 100, the word abjad itself derives from the first four letters in the Phoenician alphabet, Aramaic alphabet, Hebrew alphabet and other scripts for Semitic languages. These older alphabets contained only 22 letters, stopping at taw, the Arabic Abjad system continues at this point with letters not found in other alphabets, ṯāʾ=500, etc. The Abjad order of the Arabic alphabet has two different variants. Loss of samekh was compensated for by the split of shin ש into two independent Arabic letters, ش and ﺱ, which moved up to take the place of samekh. The most common Abjad sequence, read right to left, is, This is commonly vocalized as follows. Before the introduction of the Hindu–Arabic numeral system, the numbers were used for all mathematical purposes. In modern Arabic, they are used for numbering outlines, items in lists. In English, points of information are sometimes referred to as A, B, and C, the abjad numbers are also used to assign numerical values to Arabic words for purposes of numerology. The common Islamic phrase بسم الله الرحمن الرحيم bismillāh al-Raḥmān al-Raḥīm has a value of 786. The name Allāh الله by itself has the value 66, a few of the numerical values are different in the alternative Abjad order. For four Persian letters these values are used, The Abjad numerals are equivalent to the earlier Hebrew numerals up to 400, the Hebrew numeral system is known as Gematria and is used in Kabbalistic texts and numerology. Like the Abjad order, it is used in times for numbering outlines and points of information. The Greek numerals differ in a number of ways from the Abjad ones, the Greek language system of letters-as-numbers is called isopsephy
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Armenian numerals
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The system of Armenian numerals is a historic numeral system created using the majuscules of the Armenian alphabet. There was no notation for zero in the old system, the principles behind this system are the same as for the Ancient Greek numerals and Hebrew numerals. In modern Armenia, the familiar Arabic numerals are used, Armenian numerals are used more or less like Roman numerals in modern English, e. g. Գարեգին Բ. means Garegin II and Գ. Since not all browsers can render Unicode Armenian letters, the transliteration is given. The final two letters of the Armenian alphabet, o and fe were added to the Armenian alphabet only after Arabic numerals were already in use, thus, they do not have a numerical value assigned to them. Numbers in the Armenian numeral system are obtained by simple addition, although the order of the numerals is irrelevant since only addition is performed, the convention is to write them in decreasing order of value. This is done by drawing a line over them, indicating their value is to be multiplied by 10000, Ա =10000 Ջ =9000000 ՌՃԽԳՌՄԾԵ =11431255
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Cyrillic numerals
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Cyrillic numerals are a numeral system derived from the Cyrillic script, developed in the First Bulgarian Empire in the late 10th century. It was used in the First Bulgarian Empire and by South, the system was used in Russia as late as the early 18th century, when Peter the Great replaced it with Arabic numerals as part of his civil script reform initiative. By 1725, Russian Imperial coins had transitioned to Arabic numerals, the Cyrillic numerals may still be found in books written in the Church Slavonic language. The system is an alphabetic system, equivalent to the Ionian numeral system. The order is based on the original Greek alphabet rather than the standard Cyrillic alphabetical order, a separate letter is assigned to each unit, each multiple of ten, and each multiple of one hundred. To distinguish numbers from text, a titlo is drawn over the numbers. Examples, –1706 –7118 To evaluate a Cyrillic number, the values of all the figures are added up, for example, ѰЗ is 700 +7, making 707. If the number is greater than 999, the sign is used to multiply the numbers value, for example, ҂Ѕ is 6000, while ҂Л҂В is parsed as 30,000 +2000. To produce larger numbers, a sign is used to encircle the number being multiplied. Glagolitic numerals are similar to Cyrillic numerals except that values are assigned according to the native alphabetic order of the Glagolitic alphabet. Glyphs for the ones, tens, and hundreds values are combined to form more precise numbers, for example, ⰗⰑⰂ is 500 +80 +3 or 583. As with Cyrillic numerals, the numbers 11 through 19 are typically written with the ones digit before the glyph for 10, for example ⰅⰊ is 6 +10, early Cyrillic alphabet Glagolitic alphabet Relationship of Cyrillic and Glagolitic scripts Greek numerals Combining Cyrillic Millions
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Ge'ez script
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Geez is a script used as an abugida for several languages of Ethiopia and Eritrea. It originated as an abjad and was first used to write Geez, now the language of the Ethiopian Orthodox Tewahedo Church. In Amharic and Tigrinya, the script is often called fidäl, the Geez script has been adapted to write other, mostly Semitic, languages, particularly Amharic in Ethiopia, and Tigrinya in both Eritrea and Ethiopia. It is also used for Sebatbeit, Meen, and most other languages of Ethiopia, in Eritrea it is used for Tigre, and it has traditionally been used for Blin, a Cushitic language. Tigre, spoken in western and northern Eritrea, is considered to resemble Geez more than do the other derivative languages, some other languages in the Horn of Africa, such as Oromo, used to be written using Geez, but have migrated to Latin-based orthographies. For the representation of sounds, this uses a system that is common among linguists who work on Ethiopian Semitic languages. This differs somewhat from the conventions of the International Phonetic Alphabet, see the articles on the individual languages for information on the pronunciation. The earliest inscriptions of Semitic languages in Eritrea and Ethiopia date to the 9th century BC in Epigraphic South Arabian, after the 7th and 6th centuries BC, however, variants of the script arose, evolving in the direction of the Geez abugida. This evolution can be seen most clearly in evidence from inscriptions in Tigray region in northern Ethiopia, at least one of Wazebas coins from the late 3rd or early 4th century contains a vocalized letter, some 30 or so years before Ezana. It has been argued that the marking pattern of the script reflects a South Asian system. On the other hand, emphatic P̣ait ጰ, a Geez innovation, is a modification of Ṣädai ጸ, while Pesa ፐ is based on Tawe ተ. Thus, there are 24 correspondences of Geez and the South Arabian alphabet, Many of the names are cognate with those of Phoenician. Two alphabets were used to write the Geez language, an abjad and later an abugida. The abjad, used until c.330 AD, had 26 consonantal letters, h, l, ḥ, m, ś, r, s, ḳ, b, t, ḫ, n, ʾ, k, w, ʿ, z, y, d, g, ṭ, p̣, ṣ, ṣ́, f, p Vowels were not indicated. Modern Geez is written left to right. The Geez abugida developed under the influence of Christian scripture by adding obligatory vocalic diacritics to the consonantal letters. The diacritics for the vowels, u, i, a, e, ə, o, were fused with the consonants in a recognizable but slightly irregular way, the original form of the consonant was used when the vowel was ä, the so-called inherent vowel. The resulting forms are shown below in their traditional order, for some consonants, there is an eighth form for the diphthong -wa or -oa, and a ninth for -yä
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Georgian numerals
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The Georgian numerals are the system of number names used in Georgian, a language spoken in the country of Georgia. The Georgian numerals from 30 to 99 are constructed using a system, similar to the scheme used in Basque, French for numbers 80 through 99. An older method for writing numerals exists in which most of letters of the Georgian alphabet are assigned a numeric value. The Georgian cardinal numerals up to ten are primitives, as are the words for 20 and 100, other cardinal numbers are formed from these primitives via a mixture of decimal and vigesimal structural principles. The following chart shows the forms of the primitive numbers. Except for rva and tskhra, these words are all consonant-final stems, numbers from 11 to 19 are formed from 1 through 9, respectively, by prefixing t and adding meti. In some cases, the prefixed t coalesces with the consonant of the root word to form a single consonant. Numbers between 20 and 99 use a vigesimal system. g, the hundreds are formed by linking 2,3. 10 directly to the word for 100,1000 is expressed as atasi, and multiples of 1000 are expressed using atasi — so, for example,2000 is ori atasi. The final i is dropped when a number is added to a multiple of 100. The Georgian numeral system is a system of representing numbers using letters of the Georgian alphabet, numerical values in this system are obtained by simple addition of the component numerals, which are written greatest-to-least from left to right. *Both letters ჳ and უ are equal to 400 in numerical value and these letters have no numerical value
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Greek numerals
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Greek numerals are a system of writing numbers using the letters of the Greek alphabet. These alphabetic numerals are known as Ionic or Ionian numerals, Milesian numerals. In modern Greece, they are used for ordinal numbers. For ordinary cardinal numbers, however, Greece uses Arabic numerals, attic numerals, which were later adopted as the basis for Roman numerals, were the first alphabetic set. They were acrophonic, derived from the first letters of the names of the numbers represented and they ran =1, =5, =10, =100, =1000, and =10000. 50,500,5000, and 50000 were represented by the letter with minuscule powers of ten written in the top right corner, the same system was used outside of Attica, but the symbols varied with the local alphabets, in Boeotia, was 1000. The present system probably developed around Miletus in Ionia, 19th-century classicists placed its development in the 3rd century BC, the occasion of its first widespread use. The present system uses the 24 letters adopted by Euclid as well as three Phoenician and Ionic ones that were not carried over, digamma, koppa, and sampi. The position of characters within the numbering system imply that the first two were still in use while the third was not. Greek numerals are decimal, based on powers of 10, the units from 1 to 9 are assigned to the first nine letters of the old Ionic alphabet from alpha to theta. Each multiple of one hundred from 100 to 900 was then assigned its own separate letter as well and this alphabetic system operates on the additive principle in which the numeric values of the letters are added together to obtain the total. For example,241 was represented as, in ancient and medieval manuscripts, these numerals were eventually distinguished from letters using overbars, α, β, γ, etc. In medieval manuscripts of the Book of Revelation, the number of the Beast 666 is written as χξϛ, although the Greek alphabet began with only majuscule forms, surviving papyrus manuscripts from Egypt show that uncial and cursive minuscule forms began early. These new letter forms sometimes replaced the ones, especially in the case of the obscure numerals. The old Q-shaped koppa began to be broken up and simplified, the numeral for 6 changed several times. During antiquity, the letter form of digamma came to be avoided in favor of a special numerical one. By the Byzantine era, the letter was known as episemon and this eventually merged with the sigma-tau ligature stigma. In modern Greek, a number of changes have been made
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Hebrew numerals
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The system of Hebrew numerals is a quasi-decimal alphabetic numeral system using the letters of the Hebrew alphabet. The system was adapted from that of the Greek numerals in the late 2nd century BC, the current numeral system is also known as the Hebrew alphabetic numerals to contrast with earlier systems of writing numerals used in classical antiquity. The Greek system was adopted in Hellenistic Judaism and had been in use in Greece since about the 5th century BC, in this system, there is no notation for zero, and the numeric values for individual letters are added together. Each unit is assigned a letter, each tens a separate letter. The later hundreds are represented by the sum of two or three letters representing the first four hundreds, to represent numbers from 1,000 to 999,999, the same letters are reused to serve as thousands, tens of thousands, and hundreds of thousands. In Israel today, the system of Arabic numerals is used in almost all cases. The Hebrew numerals are used only in cases, such as when using the Hebrew calendar, or numbering a list. Numbers in Hebrew from zero to one million, Hebrew alphabet are used to a limited extent to represent numbers, widely used on calendars. In other situations Arabic numerals are used, cardinal and ordinal numbers must agree in gender with the noun they are describing. If there is no such noun, the form is used. For ordinal numbers greater than ten the cardinal is used and numbers above the value 20 have no gender, note, For ordinal numbers greater than 10, cardinal numbers are used instead. Note, For numbers greater than 20, gender does not apply, cardinal and ordinal numbers must agree in gender with the noun they are describing. If there is no such noun, the form is used. Ordinal numbers must also agree in number and definite status like other adjectives, the cardinal number precedes the noun, except for the number one which succeeds it. The number two is special - shnayim and shtayim become shney and shtey when followed by the noun they count, for ordinal numbers greater than ten the cardinal is used. The Hebrew numeric system operates on the principle in which the numeric values of the letters are added together to form the total. For example,177 is represented as קעז which corresponds to 100 +70 +7 =177, mathematically, this type of system requires 27 letters. In practice the last letter, tav is used in combination with itself and/or other letters from kof onwards, to numbers from 500
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Roman numerals
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The numeric system represented by Roman numerals originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet, Roman numerals, as used today, are based on seven symbols, The use of Roman numerals continued long after the decline of the Roman Empire. The numbers 1 to 10 are usually expressed in Roman numerals as follows, I, II, III, IV, V, VI, VII, VIII, IX, Numbers are formed by combining symbols and adding the values, so II is two and XIII is thirteen. Symbols are placed left to right in order of value. Named after the year of its release,2014 as MMXIV, the year of the games of the XXII Olympic Winter Games The standard forms described above reflect typical modern usage rather than a universally accepted convention. Usage in ancient Rome varied greatly and remained inconsistent in medieval, Roman inscriptions, especially in official contexts, seem to show a preference for additive forms such as IIII and VIIII instead of subtractive forms such as IV and IX. Both methods appear in documents from the Roman era, even within the same document, double subtractives also occur, such as XIIX or even IIXX instead of XVIII. Sometimes V and L are not used, with such as IIIIII. Such variation and inconsistency continued through the period and into modern times. Clock faces that use Roman numerals normally show IIII for four o’clock but IX for nine o’clock, however, this is far from universal, for example, the clock on the Palace of Westminster in London uses IV. Similarly, at the beginning of the 20th century, different representations of 900 appeared in several inscribed dates. For instance,1910 is shown on Admiralty Arch, London, as MDCCCCX rather than MCMX, although Roman numerals came to be written with letters of the Roman alphabet, they were originally independent symbols. The Etruscans, for example, used
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Attic numerals
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Attic numerals were used by the 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. They are also known as acrophonic numerals because the symbols derive from the first letters of the words that the symbols represent, five, ten, hundred, thousand and ten thousand. The use of Η for 100 reflects the date of this numbering system. It wasnt until Aristophanes of Byzantium introduced the various accent markings during the Hellenistic period that the spiritus asper began to represent /h/, thus the word for a hundred would originally have been written ΗΕΚΑΤΟΝ, as compared to the now more familiar spelling ἑκατόν. In modern Greek, the /h/ phoneme has disappeared altogether, unlike the more familiar Modern Roman numeral system, the Attic system contains only additive forms. Thus, the number 4 is written ΙΙΙΙ, not ΙΠ, the numerals representing 50,500, and 5,000 were composites of pi and a tiny version of the applicable power of ten. For example, is five times one thousand, specific numeral symbols were used to represent one drachma, to represent talents and staters, to represent ten mnas and to represent one half and one quarter. Attic numerals in Unicode Etruscan numerals
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Babylonian numerals
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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. The Babylonians, who were famous for their observations and calculations. Neither of the predecessors was a positional system and this system first appeared around 2000 BC, its structure reflects the decimal lexical numerals of Semitic languages rather than Sumerian lexical numbers. However, the use of a special Sumerian sign for 60 attests to a relation with the Sumerian system. The Babylonian system is credited as being the first known positional numeral system and this was an extremely important development, because non-place-value systems require unique symbols to represent each power of a base, which can make calculations more difficult. Only two symbols were used to notate the 59 non-zero digits and these symbols and their values were combined to form a digit in a sign-value notation quite similar to that of Roman numerals, for example, the combination represented the digit for 23. A space was left to indicate a place value, similar to the modern-day zero. Babylonians later devised a sign to represent this empty place and they lacked a symbol to serve the function of radix point, so the place of the units had to be inferred from context, could have represented 23 or 23×60 or 23×60×60 or 23/60, etc. A common theory is that 60, a highly composite number, was chosen due to its prime factorization, 2×2×3×5, which makes it divisible by 1,2,3,4,5,6,10,12,15,20. Integers and fractions were represented identically — a radix point was not written, the Babylonians did not technically have a digit for, nor a concept of, the number zero. Although they understood the idea of nothingness, it was not seen as a number—merely the lack of a number, what the Babylonians had instead was a space to mark the nonexistence of a digit in a certain place value. Babylon Babylonia History of zero Numeral system Menninger, Karl W. Number Words and Number Symbols, Number, From Ancient Civilisations to the Computer. CESCNC - a handy and easy-to use numeral converter
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Brahmi numerals
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The Brahmi numerals are a numeral system attested from the 3rd century BCE. They are the direct ancestors of the modern Indian and Hindu–Arabic numerals. However, they were distinct from these later systems, as they were not used as a positional system with a zero. Rather, there were separate numerals for each of the tens, there were also symbols for 100 and 1000 which were combined in ligatures with the units to signify 200,300,2000,3000, etc. In the oldest inscriptions,4 is a +, reminiscent of the X of neighboring Kharoṣṭhī, however, the other unit numerals appear to be arbitrary symbols in even the oldest inscriptions. Likewise, the units for the tens are not obviously related to other or to the units. With a similar writing instrument, the forms of such groups of strokes could easily be broadly similar as well. Another possibility is that the numerals were acrophonic, like the Attic numerals, and based on the Kharoṣṭhī alphabet. For instance, chatur 4 early on took a ¥ shape much like the Kharosthi letter ch, panca 5 looks remarkably like Kharosthi p, and so on through shat 6, sapta 7, however, there are problems of timing and lack of records. The full set of numerals is not attested until the 1st-2nd century CE,400 years after Ashoka, both suggestions, that the numerals derive from tallies or that theyre alphabetic, are purely speculative at this point, with little evidence to decide between them. Brahmi script Georges Ifrah, The Universal History of Numbers, From Prehistory to the Invention of the Computer, translated by David Bellos, Sophie Wood, pub. Karl Menninger, Number Words and Number Symbols - A Cultural History of Numbers ISBN 0-486-27096-3 David Eugene Smith and Louis Charles Karpinski, The Hindu-Arabic Numerals
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Egyptian numerals
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The system of ancient Egyptian numerals was used in Ancient Egypt around 3000 BC until the early first millennium AD. It was a system of numeration based on the scale of ten, often rounded off to the power, written in hieroglyphs. The hieratic form of numerals stressed an exact finite series notation, the Ancient Egyptian system used bases of ten. The following hieroglyphics were used to denote powers of ten, Multiples of these values were expressed by repeating the symbol as many times as needed, for instance, a stone carving from Karnak shows the number 4622 as Egyptian hieroglyphs could be written in both directions. The symbol nfr, meaning beautiful, was used to indicate the base level in drawings of tombs and pyramids. Rational numbers could also be expressed, but only as sums of fractions, i. e. sums of reciprocals of positive integers, except for 2⁄3. The hieroglyph indicating a fraction looked like a mouth, which meant part, Fractions were written with this fractional solidus, i. e. the numerator 1, and the positive denominator below. As with most modern day languages, the ancient Egyptian language could also write out numerals as words phonetically, just like one can write thirty instead of 30 in English. The word, for instance, was written as while the numeral was This was, however, uncommon for most numbers other than one, instances of numerals written in hieratic can be found as far back as the Early Dynastic Period. The Old Kingdom Abusir Papyri are an important corpus of texts that utilize hieratic numerals. A large number like 9999 could thus be written only four signs—combining the signs for 9000,900,90. Boyer saw the new hieratic numerals as ciphered, mapping one number onto one Egyptian letter for the first time in human history, greeks adopted the new system, mapping their counting numbers onto two of their alphabets, the Doric and Ionian. In the oldest hieratic texts the individual numerals were written in a ciphered relationship to the Egyptian alphabet. But during the Old Kingdom a series of standardized writings had developed for sign-groups containing more than one numeral, however, repetition of the same numeral for each place-value was not allowed in the hieratic script. As the hieratic writing system developed over time, these sign-groups were further simplified for quick writing, two famous mathematical papyri using hieratic script are the Moscow Mathematical Papyrus and the Rhind Mathematical Papyrus. The majuscule letter A in some reconstructed forms means that the quality of that remains uncertain, Ancient Egypt Egyptian language Egyptian mathematics Allen. Middle Egyptian, An Introduction to the Language and Culture of Hieroglyphs, Egyptian Grammar, Being an Introduction to the Study of Hieroglyphs. Hieratische Paläographie, Die aegyptische Buchschrift in ihrer Entwicklung von der Fünften Dynastie bis zur römischen Kaiserzeit, Introduction Egyptian numerals Numbers and dates http, //egyptianmath. blogspot. com
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Etruscan numerals
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The Etruscan numerals were used by the ancient Etruscans. The system was adapted from the Greek Attic numerals and formed the inspiration for the later Roman numerals via the Old Italic script, there is very little surviving evidence of these numerals. Examples are known of the symbols for numbers, but it is unknown which symbol represents which number. Thanks to the written out on the Tuscania dice, there is agreement that zal, ci, huθ. The assignment depends on whether the numbers on opposite faces of Etruscan dice add up to seven, some dice found did not show this proposed pattern. An aspect of the Etruscan numeral system is that some numbers, so 17 is not written *semφ-śar as users of the Hindu-Arabic numerals might reason. One instead finds ci-em zaθrum, literally three from twenty, the numbers 17,18 and 19 are all written in this way. The general agreement among Etruscologists nowadays is the following, Archaeological evidence strongly supports the correspondence 4/huth and 6/sa. In the same necropolis, in the Tomb of the Anina, which contains six burial places, an inscription reads, sa suthi cherichunce, however, other scholars disagree with this attribution. In this connection, in October 2011, Artioli and colleagues presented evidence from 93 Etruscan dice allowing the firm attribution of the numeral 6 to the graphical value huth and 4 to sa. In 2006, S. A. Yatsemirsky presented evidence that zar = śar meant ‘12’ while halχ meant ‘10’, according to his interpretation, the attested form huθzar was used for ‘sixteen’, not ‘fourteen’, assuming huθ meant four. Much debate has been carried out about a possible Indo-European origin of the Etruscan cardinals, in the words of Larissa Bonfante, What these numerals show, beyond any shadow of a doubt, is the non-Indo-European nature of the Etruscan language
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Inuit numerals
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Inuit, like other Eskimo languages, uses a vigesimal counting system. Inuit counting has sub-bases at 5,10, and 15, arabic numerals, consisting of 10 distinct digits are not adequate to represent a base-20 system. The numeral system has helped to revive counting in Inuit, which had been falling into disuse among Inuit speakers due to the prevalence of the system in schools. The picture below shows the numerals 1–19 and then 0, twenty is written with a one and a zero, forty with a two and a zero, and four hundred with a one and two zeros. The corresponding spoken forms are, In Greenlandic Inuit language
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Maya numerals
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The Maya numeral system is a vigesimal positional notation used in the Maya civilization to represent numbers. The numerals are made up of three symbols, zero, one and five, for example, thirteen is written as three dots in a horizontal row above two horizontal lines stacked above each other. Numbers after 19 were written vertically in powers of twenty, for example, thirty-three would be written as one dot above three dots, which are in turn atop two lines. The first dot represents one twenty or 1×20, which is added to three dots and two bars, or thirteen, upon reaching 202 or 400, another row is started. The number 429 would be written as one dot above one dot above four dots, the powers of twenty are numerals, just as the Hindu-Arabic numeral system uses powers of tens. Other than the bar and dot notation, Maya numerals can be illustrated by face type glyphs or pictures, the face glyph for a number represents the deity associated with the number. These face number glyphs were used, and are mostly seen on some of the most elaborate monumental carving. Addition and subtraction, Adding and subtracting numbers below 20 using Maya numerals is very simple, addition is performed by combining the numeric symbols at each level, If five or more dots result from the combination, five dots are removed and replaced by a bar. If four or more bars result, four bars are removed, similarly with subtraction, remove the elements of the subtrahend symbol from the minuend symbol, If there are not enough dots in a minuend position, a bar is replaced by five dots. If there are not enough bars, a dot is removed from the next higher minuend symbol in the column, the Maya/Mesoamerican Long Count calendar required the use of zero as a place-holder within its vigesimal positional numeral system. A shell glyph – – was used as a symbol for these Long Count dates. However, since the eight earliest Long Count dates appear outside the Maya homeland, it is assumed that the use of zero predated the Maya, indeed, many of the earliest Long Count dates were found within the Olmec heartland. However, the Olmec civilization had come to an end by the 4th century BC, in the Long Count portion of the Maya calendar, a variation on the strictly vigesimal numbering is used. The Long Count changes in the place value, it is not 20×20 =400, as would otherwise be expected. This is supposed to be because 360 is roughly the number of days in a year, subsequent place values return to base-twenty. In fact, every known example of large numbers uses this modified vigesimal system and it is reasonable to assume, but not proven by any evidence, that the normal system in use was a pure base-20 system. Maya Mathematics - online converter from decimal numeration to Maya numeral notation, anthropomorphic Maya numbers - online story of number representations
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Quipu
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Quipus, sometimes known as khipus or talking knots, were recording devices historically used in a number of cultures and particularly in the region of Andean South America. A quipu usually consisted of colored, spun, and plied thread or strings made from cotton or camelid fiber. For the Inca, the system aided in collecting data and keeping records, ranging from monitoring tax obligations, properly collecting census records, calendrical information, the cords contained numeric and other values encoded by knots in a base ten positional system. A quipu could have only a few or up to 2,000 cords, the configuration of the quipus have also been compared to string mops. Archaeological evidence has shown a use of finely carved wood as a supplemental. A relatively small number have survived, objects that can be identified unambiguously as quipus first appear in the archaeological record in the first millennium AD. As the region was subsumed under the invading Spanish Empire, the use of the quipu faded from use, however, in several villages, quipu continued to be important items for the local community, albeit for ritual rather than recording use. It is unclear as to where and how many intact quipus still exist, as many have been stored away in mausoleums, quipu is the Spanish spelling and the most common spelling in English. Khipu is the word for knot in Cusco Quechua, the kh is an aspirated k, in most Quechua varieties, the term is kipu. The word khipu, meaning knot or to knot, comes from the Quechua language word, quipu,1704, most information recorded on the quipus consists of numbers in a decimal system. In the early years of the Spanish conquest of Peru, Spanish officials often relied on the quipus to settle disputes over local tribute payments or goods production, Spanish chroniclers also concluded that quipus were used primarily as mnemonic devices to communicate and record numerical information. Quipucamayocs could be summoned to court, where their bookkeeping was recognised as valid documentation of past payments, some of the knots, as well as other features, such as color, are thought to represent non-numeric information, which has not been deciphered. It is generally thought that the system did not include phonetic symbols analogous to letters of the alphabet, however Gary Urton has suggested that the quipus used a binary system which could record phonological or logographic data. To date, no link has yet been found between a quipu and Quechua, the language of the Peruvian Andes. This suggests that quipus are not a writing system and have no phonetic referent. If this conjecture is correct, quipus are the known example of a complex language recorded in a 3-D system. Marcia and Robert Ascher, after having analyzed several hundred quipus, have shown that most information on quipus is numeric, and these numbers can be read. Each cluster of knots is a digit, and there are three types of knots, simple overhand knots, long knots, consisting of an overhand knot with one or more additional turns
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Prehistoric numerals
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Counting in prehistory was first assisted by using body parts, primarily the fingers. This is reflected in the etymology of certain names, such as in the names of ten and hundred in the Proto-Indo-European numerals. Early systems of counting using tally marks appear in the Upper Paleolithic, the first more complex systems develop in the Ancient Near East together with the development of early writing out of proto-writing systems. Numerals originally developed from the use of tally marks as a counting aid, counting aids like tally marks become more sophisticated in the Near Eastern Neolithic, developing into various types of proto-writing. The Cuneiform script develops out of proto-writing associated with keeping track of goods during the Chalcolithic, the Moksha people, whose existence dates to about the beginning of the 1st millennium BC, had a numeral system. The numerals were tally marks carved on wood, drawn on clay or birch bark, in some places they were preserved until the beginning of 20th century mostly among small traders, bee-keepers, and village elders. These numerals still can be found on old shepherd and tax-gatherer staffs, apiaries, evans, Writing in Prehistoric Greece, Journal of the Anthropological Institute of Great Britain and Ireland
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Positional notation
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Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the symbol for the different orders of magnitude. This greatly simplified arithmetic, leading to the spread of the notation across the world. With the use of a point, the notation can be extended to include fractions. The Hindu–Arabic numeral system, base-10, is the most commonly used system in the world today for most calculations, today, the base-10 system, which is likely motivated by counting with the ten fingers, is ubiquitous. Other bases have been used in the past however, and some continue to be used today, for example, the Babylonian numeral system, credited as the first positional numeral system, was base-60, but it lacked a real 0 value. Zero was indicated by a space between sexagesimal numerals, by 300 BC, a punctuation symbol was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish, the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges, the Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number, thus numbers like 2 and 120,3 and 180,4 and 240, looked the same because the larger numbers lacked a final sexagesimal placeholder. Counting rods and most abacuses have been used to represent numbers in a numeral system. This approach required no memorization of tables and could produce practical results quickly, for four centuries there was strong disagreement between those who believed in adopting the positional system in writing numbers and those who wanted to stay with the additive-system-plus-abacus. Although electronic calculators have largely replaced the abacus, the continues to be used in Japan. After the French Revolution, the new French government promoted the extension of the decimal system, some of those pro-decimal efforts—such as decimal time and the decimal calendar—were unsuccessful. Other French pro-decimal efforts—currency decimalisation and the metrication of weights and measures—spread widely out of France to almost the whole world. According to Joseph Needham and Lam Lay Yong, decimal fractions were first developed and used by the Chinese in the 1st century BC, the written Chinese decimal fractions were non-positional. However, counting rod fractions were positional, the Jewish mathematician Immanuel Bonfils used decimal fractions around 1350, anticipating Simon Stevin, but did not develop any notation to represent them. A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century. A key argument against the system was its susceptibility to easy fraud by simply putting a number at the beginning or end of a quantity, thereby changing 100 into 5100
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Radix
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In mathematical numeral systems, the radix or base is the number of unique digits, including zero, used to represent numbers in a positional numeral system. For example, for the system the radix is ten. For example,10 represents the one hundred, while 2 represents the number four. Radix is a Latin word for root, root can be considered a synonym for base in the arithmetical sense. In the system with radix 13, for example, a string of such as 398 denotes the number 3 ×132 +9 ×131 +8 ×130. More generally, in a system with radix b, a string of digits d1 … dn denotes the number d1bn−1 + d2bn−2 + … + dnb0, commonly used numeral systems include, For a larger list, see List of numeral systems. The octal and hexadecimal systems are used in computing because of their ease as shorthand for binary. Every hexadecimal digit corresponds to a sequence of four binary digits, a similar relationship holds between every octal digit and every possible sequence of three binary digits, since eight is the cube of two. However, other systems are possible, e. g. golden ratio base. Base Radix economy Non-standard positional numeral systems Base Convert, a floating-point base calculator MathWorld entry on base
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Binary number
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The base-2 system is a positional notation with a radix of 2. Because of its implementation in digital electronic circuitry using logic gates. Each digit is referred to as a bit, the modern binary number system was devised by Gottfried Leibniz in 1679 and appears in his article Explication de lArithmétique Binaire. Systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, Leibniz was specifically inspired by the Chinese I Ching. The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions, the method used for ancient Egyptian multiplication is also closely related to binary numbers. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus, the I Ching dates from the 9th century BC in China. The binary notation in the I Ching is used to interpret its quaternary divination technique and it is based on taoistic duality of yin and yang. Eight trigrams and a set of 64 hexagrams, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China. The Song Dynasty scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, the Indian scholar Pingala developed a binary system for describing prosody. He used binary numbers in the form of short and long syllables, Pingalas Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter. The binary representations in Pingalas system increases towards the right, the residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Africa, sets of binary combinations similar to the I Ching have also been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy. The base-2 system utilized in geomancy had long been applied in sub-Saharan Africa. Leibnizs system uses 0 and 1, like the modern binary numeral system, Leibniz was first introduced to the I Ching through his contact with the French Jesuit Joachim Bouvet, who visited China in 1685 as a missionary. Leibniz saw the I Ching hexagrams as an affirmation of the universality of his own beliefs as a Christian. Binary numerals were central to Leibnizs theology and he believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing. Is not easy to impart to the pagans, is the ex nihilo through Gods almighty power. In 1854, British mathematician George Boole published a paper detailing an algebraic system of logic that would become known as Boolean algebra
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Ternary numeral system
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The ternary numeral system has three as its base. Analogous to a bit, a digit is a trit. One trit is equivalent to bits of information. Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary, for example, decimal 365 corresponds to binary 101101101 and to ternary 111112. However, they are far less compact than the corresponding representations in bases such as decimal – see below for a compact way to codify ternary using nonary. The value of a number with n bits that are all 1 is 2n −1. Then N = M, N = /, and N = bd −1, for a three-digit ternary number, N =33 −1 =26 =2 ×32 +2 ×31 +2 ×30 =18 +6 +2. Nonary or septemvigesimal can be used for representation of ternary. A base-three system is used in Islam to keep track of counting Tasbih to 99 or to 100 on a hand for counting prayers. In certain analog logic, the state of the circuit is often expressed ternary and this is most commonly seen in Transistor–transistor logic using 7406 open collector logic. The output is said to either be low, high, or open, in this configuration the output of the circuit is actually not connected to any voltage reference at all. Where the signal is usually grounded to a reference, or at a certain voltage level. Thus, the voltage level is sometimes unpredictable. A rare ternary point is used to denote fractional parts of an inning in baseball, since each inning consists of three outs, each out is considered one third of an inning and is denoted as.1. For example, if a player pitched all of the 4th, 5th and 6th innings, plus 2 outs of the 7th inning, his Innings pitched column for that game would be listed as 3.2, meaning 3⅔. In this usage, only the part of the number is written in ternary form. Ternary numbers can be used to convey self-similar structures like the Sierpinski triangle or the Cantor set conveniently, additionally, it turns out that the ternary representation is useful for defining the Cantor set and related point sets, because of the way the Cantor set is constructed. The Cantor set consists of the points from 0 to 1 that have an expression that does not contain any instance of the digit 1
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Quaternary numeral system
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Quaternary is the base-4 numeral system. It uses the digits 0,1,2 and 3 to represent any real number. Four is the largest number within the range and one of two numbers that is both a square and a highly composite number, making quaternary a convenient choice for a base at this scale. Despite being twice as large, its economy is equal to that of binary. However, it no better in the localization of prime numbers. See decimal and binary for a discussion of these properties, as with the octal and hexadecimal numeral systems, quaternary has a special relation to the binary numeral system. Each radix 4,8 and 16 is a power of 2, so the conversion to and from binary is implemented by matching each digit with 2,3 or 4 binary digits, for example, in base 4,302104 =11001001002. Although octal and hexadecimal are widely used in computing and computer programming in the discussion and analysis of binary arithmetic and logic, by analogy with byte and nybble, a quaternary digit is sometimes called a crumb. There is a surviving list of Ventureño language number words up to 32 written down by a Spanish priest ca, the Kharosthi numerals have a partial base 4 counting system from 1 to decimal 10. Quaternary numbers are used in the representation of 2D Hilbert curves, here a real number between 0 and 1 is converted into the quaternary system. Every single digit now indicates in which of the respective 4 sub-quadrants the number will be projected, parallels can be drawn between quaternary numerals and the way genetic code is represented by DNA. The four DNA nucleotides in order, abbreviated A, C, G and T, can be taken to represent the quaternary digits in numerical order 0,1,2. With this encoding, the complementary digit pairs 0↔3, and 1↔2 match the complementation of the pairs, A↔T and C↔G. For example, the nucleotide sequence GATTACA can be represented by the quaternary number 2033010, quaternary line codes have been used for transmission, from the invention of the telegraph to the 2B1Q code used in modern ISDN circuits
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Quinary
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Quinary is a numeral system with five as the base. A possible origination of a system is that there are five fingers on either hand. The base five is stated from 0–4, in the quinary place system, five numerals, from 0 to 4, are used to represent any real number. According to this method, five is written as 10, twenty-five is written as 100, today, the main usage of base 5 is as a biquinary system, which is decimal using five as a sub-base. Another example of a system, is sexagesimal, base 60. Each quinary digit has log25 bits of information, many languages use quinary number systems, including Gumatj, Nunggubuyu, Kuurn Kopan Noot, Luiseño and Saraveca. Gumatj is a true 5–25 language, in which 25 is the group of 5. The Gumatj numerals are shown below, In the video game Riven and subsequent games of the Myst franchise, a decimal system with 2 and 5 as a sub-bases is called biquinary, and is found in Wolof and Khmer. Roman numerals are a biquinary system, the numbers 1,5,10, and 50 are written as I, V, X, and L respectively. Eight is VIII and seventy is LXX, most versions of the abacus use a biquinary system to simulate a decimal system for ease of calculation. Urnfield culture numerals and some tally mark systems are also biquinary, units of currencies are commonly partially or wholly biquinary. A vigesimal system with 4 and 5 as a sub-bases is found in Nahuatl, pentimal system Quibinary Yan Tan Tethera References, Quinary Base Conversion, includes fractional part, from Math Is Fun Media related to Quinary numeral system at Wikimedia Commons
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Senary
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The senary numeral system has six as its base. It has been adopted independently by a number of cultures. Like decimal, it is a semiprime, though being the product of the two consecutive numbers that are both prime it has a high degree of mathematical properties for its size. As six is a highly composite number, many of the arguments made in favor of the duodecimal system also apply to this base-6. Senary may be considered interesting in the study of numbers, since all primes other than 2 and 3. That is, for every number p greater than 3, one has the modular arithmetic relations that either p ≡1 or 5. This property maximizes the probability that the result of an integer multiplication will end in zero, E. g. if three fingers are extended on the left hand and four on the right, 34senary is represented. This is equivalent to 3 ×6 +4 which is 22decimal, flipping the sixes hand around to its backside may help to further disambiguate which hand represents the sixes and which represents the units. While most developed cultures count by fingers up to 5 in very similar ways, beyond 5 non-Western cultures deviate from Western methods, such as with Chinese number gestures. More abstract finger counting systems, such as chisanbop or finger binary, allow counting to 99,1,023, or even higher depending on the method. The English monk and historian Bede, in the first chapter of De temporum ratione, titled Tractatus de computo, vel loquela per gestum digitorum, the Ndom language of Papua New Guinea is reported to have senary numerals. Mer means 6, mer an thef means 6 ×2 =12, nif means 36, another example from Papua New Guinea are the Morehead-Maro languages. In these languages, counting is connected to ritualized yam-counting and these languages count from a base six, employing words for the powers of six, running up to 66 for some of the languages. One example is Kómnzo with the numerals, nimbo, féta, tarumba, ntamno, wärämäkä. Some Niger-Congo languages have been reported to use a number system, usually in addition to another. For some purposes, base 6 might be too small a base for convenience. The choice of 36 as a radix is convenient in that the digits can be represented using the Arabic numerals 0–9 and the Latin letters A–Z, this choice is the basis of the base36 encoding scheme. Base36 encoding scheme Binary Ternary Duodecimal Sexagesimal Shacks Base Six Dialectic Digital base 6 clock Analog Clock Designer capable of rendering a base 6 clock Senary base conversion
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Octal
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The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Octal numerals can be made from binary numerals by grouping binary digits into groups of three. For example, the representation for decimal 74 is 1001010. Two zeroes can be added at the left,1001010, corresponding the octal digits 112, in the decimal system each decimal place is a power of ten. For example,7410 =7 ×101 +4 ×100 In the octal system each place is a power of eight. The Yuki language in California and the Pamean languages in Mexico have octal systems because the speakers count using the spaces between their fingers rather than the fingers themselves and it has been suggested that the reconstructed Proto-Indo-European word for nine might be related to the PIE word for new. Based on this, some have speculated that proto-Indo-Europeans used a number system. In 1716 King Charles XII of Sweden asked Emanuel Swedenborg to elaborate a number based on 64 instead of 10. Swedenborg however argued that for people with less intelligence than the king such a big base would be too difficult, in 1718 Swedenborg wrote a manuscript, En ny rekenkonst som om vexlas wid Thalet 8 i stelle then wanliga wid Thalet 10. The numbers 1-7 are there denoted by the l, s, n, m, t, f, u. Thus 8 = lo,16 = so,24 = no,64 = loo,512 = looo etc, numbers with consecutive consonants are pronounced with vowel sounds between in accordance with a special rule. Writing under the pseudonym Hirossa Ap-Iccim in The Gentlemans Magazine, July 1745, Hugh Jones proposed a system for British coins, weights. In 1801, James Anderson criticized the French for basing the Metric system on decimal arithmetic and he suggested base 8 for which he coined the term octal. In the mid 19th century, Alfred B. Taylor concluded that Our octonary radix is, therefore, so, for example, the number 65 would be spoken in octonary as under-un. Taylor also republished some of Swedenborgs work on octonary as an appendix to the above-cited publications, in the 2009 film Avatar, the language of the extraterrestrial Navi race employs an octal numeral system, probably due to the fact that they have four fingers on each hand. In the TV series Stargate SG-1, the Ancients, a race of beings responsible for the invention of the Stargates, in the tabletop game series Warhammer 40,000, the Tau race use an octal number system. Octal became widely used in computing systems such as the PDP-8, ICL1900. Octal was an abbreviation of binary for these machines because their word size is divisible by three
46.
Duodecimal
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The duodecimal system is a positional notation numeral system using twelve as its base. In this system, the number ten may be written by a rotated 2 and this notation was introduced by Sir Isaac Pitman. These digit forms are available as Unicode characters on computerized systems since June 2015 as ↊ and ↋, other notations use A, T, or X for ten and B or E for eleven. The number twelve is written as 10 in duodecimal, whereas the digit string 12 means 1 dozen and 2 units. Similarly, in duodecimal 100 means 1 gross,1000 means 1 great gross, the number twelve, a superior highly composite number, is the smallest number with four non-trivial factors, and the smallest to include as factors all four numbers within the subitizing range. As a result, duodecimal has been described as the number system. Of its factors,2 and 3 are prime, which means the reciprocals of all 3-smooth numbers have a representation in duodecimal. In particular, the five most elementary fractions all have a terminating representation in duodecimal. This all makes it a convenient number system for computing fractions than most other number systems in common use, such as the decimal, vigesimal, binary. Although the trigesimal and sexagesimal systems do even better in respect, this is at the cost of unwieldy multiplication tables. In this section, numerals are based on decimal places, for example,10 means ten,12 means twelve. Languages using duodecimal number systems are uncommon, germanic languages have special words for 11 and 12, such as eleven and twelve in English. However, they are considered to come from Proto-Germanic *ainlif and *twalif, historically, units of time in many civilizations are duodecimal. There are twelve signs of the zodiac, twelve months in a year, traditional Chinese calendars, clocks, and compasses are based on the twelve Earthly Branches. There are 12 inches in a foot,12 troy ounces in a troy pound,12 old British pence in a shilling,24 hours in a day. The Romans used a system based on 12, including the uncia which became both the English words ounce and inch. The importance of 12 has been attributed to the number of cycles in a year. It is possible to count to 12 with the acting as a pointer
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Hexadecimal
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In mathematics and computing, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, Hexadecimal numerals are widely used by computer system designers and programmers. As each hexadecimal digit represents four binary digits, it allows a more human-friendly representation of binary-coded values, one hexadecimal digit represents a nibble, which is half of an octet or byte. For example, a byte can have values ranging from 00000000 to 11111111 in binary form. In a non-programming context, a subscript is typically used to give the radix, several notations are used to support hexadecimal representation of constants in programming languages, usually involving a prefix or suffix. The prefix 0x is used in C and related languages, where this value might be denoted as 0x2AF3, in contexts where the base is not clear, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously, a numerical subscript can give the base explicitly,15910 is decimal 159,15916 is hexadecimal 159, which is equal to 34510. Some authors prefer a text subscript, such as 159decimal and 159hex, or 159d and 159h. example. com/name%20with%20spaces where %20 is the space character, thus ’, represents the right single quotation mark, Unicode code point number 2019 in hex,8217. In the Unicode standard, a value is represented with U+ followed by the hex value. Color references in HTML, CSS and X Window can be expressed with six hexadecimal digits prefixed with #, white, CSS allows 3-hexdigit abbreviations with one hexdigit per component, #FA3 abbreviates #FFAA33. *nix shells, AT&T assembly language and likewise the C programming language, to output an integer as hexadecimal with the printf function family, the format conversion code %X or %x is used. In Intel-derived assembly languages and Modula-2, hexadecimal is denoted with a suffixed H or h, some assembly languages use the notation HABCD. Ada and VHDL enclose hexadecimal numerals in based numeric quotes, 16#5A3#, for bit vector constants VHDL uses the notation x5A3. Verilog represents hexadecimal constants in the form 8hFF, where 8 is the number of bits in the value, the Smalltalk language uses the prefix 16r, 16r5A3 PostScript and the Bourne shell and its derivatives denote hex with prefix 16#, 16#5A3. For PostScript, binary data can be expressed as unprefixed consecutive hexadecimal pairs, in early systems when a Macintosh crashed, one or two lines of hexadecimal code would be displayed under the Sad Mac to tell the user what went wrong. Common Lisp uses the prefixes #x and #16r, setting the variables *read-base* and *print-base* to 16 can also used to switch the reader and printer of a Common Lisp system to Hexadecimal number representation for reading and printing numbers. Thus Hexadecimal numbers can be represented without the #x or #16r prefix code, MSX BASIC, QuickBASIC, FreeBASIC and Visual Basic prefix hexadecimal numbers with &H, &H5A3 BBC BASIC and Locomotive BASIC use & for hex. TI-89 and 92 series uses a 0h prefix, 0h5A3 ALGOL68 uses the prefix 16r to denote hexadecimal numbers, binary, quaternary and octal numbers can be specified similarly
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Vigesimal
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The vigesimal or base 20 numeral system is based on twenty. In a vigesimal system, twenty individual numerals are used. One modern method of finding the extra needed symbols is to write ten as the letter A20, to write nineteen as J20, and this is similar to the common computer-science practice of writing hexadecimal numerals over 9 with the letters A–F. Another method skips over the letter I, in order to avoid confusion between I20 as eighteen and one, so that the number eighteen is written as J20, the number twenty is written as 1020. According to this notation,2020 means forty in decimal = + D020 means two hundred and sixty in decimal = +10020 means four hundred in decimal = + +, in the rest of this article below, numbers are expressed in decimal notation, unless specified otherwise. For example,10 means ten,20 means twenty, in decimal, dividing by three twice only gives one digit periods because 9 is the number below ten. 21, however, the adjacent to 20 that is divisible by 3, is not divisible by 9. Ninths in vigesimal have six-digit periods, the prime factorization of twenty is 22 ×5, so it is not a perfect power. However, its part,5, is congruent to 1. Thus, according to Artins conjecture on primitive roots, vigesimal has infinitely many cyclic primes, but the fraction of primes that are cyclic is not necessarily ~37. 395%. An UnrealScript program that computes the lengths of recurring periods of various fractions in a set of bases found that, of the first 15,456 primes. In many European languages,20 is used as a base, vigesimal systems are common in Africa, for example in Yoruba. Ogún,20, is the basic numeric block, ogójì,40, =20 multiplied by 2. Ogota,60, =20 multiplied by 3, ogorin,80, =20 multiplied by 4. Ogorun,100, =20 multiplied by 5, twenty was a base in the Maya and Aztec number systems. The Maya used the names for the powers of twenty, kal, bak, pic, calab, kinchil. See also Maya numerals and Maya calendar, Mayan languages, Yucatec, the Aztec called them, cempoalli, centzontli, cenxiquipilli, cempoalxiquipilli, centzonxiquipilli and cempoaltzonxiquipilli. Note that the ce prefix at the beginning means one and is replaced with the number to get the names of other multiples of the power
49.
Non-standard positional numeral systems
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Each numeral represents one of the values 0,1,2, etc. up to b −1, but the value also depends on the position of the digit in a number. The value of a string like pqrs in base b is given by the polynomial form p × b 3 + q × b 2 + r × b + s. The numbers written in superscript represent the powers of the base used, and a minus sign −, all real numbers can be represented. This article summarizes facts on some non-standard positional numeral systems, in most cases, the polynomial form in the description of standard systems still applies. Some historical numeral systems may be described as non-standard positional numeral systems, however, most of the non-standard systems listed below have never been intended for general use, but are deviced by mathematicians or engineers for special academic or technical use. A bijective numeral system with base b uses b different numerals to represent all non-negative integers, however, the numerals have values 1,2,3, etc. up to and including b, whereas zero is represented by an empty digit string. For example, it is possible to have decimal without a zero, unary is the bijective numeral system with base b =1. In unary, one numeral is used to represent all positive integers, the value of the digit string pqrs given by the polynomial form can be simplified into p + q + r + s since bn =1 for all n. Non-standard features of this include, The value of a digit does not depend on its position. Thus, one can argue that unary is not a positional system at all. Introducing a radix point in this system will not enable representation of non-integer values, the single numeral represents the value 1, not the value 0 = b −1. The value 0 cannot be represented, in some systems, while the base is a positive integer, negative digits are allowed. Non-adjacent form is a system where the base is b =2. In the balanced system, the base is b =3. The reflected binary code, also known as the Gray code, is related to binary numbers. A few positional systems have been suggested in which the base b is not a positive integer, negative-base systems include negabinary, negaternary and negadecimal, in base −b the number of different numerals used is b. All integers, positive and negative, can be represented without a sign, in purely imaginary base bi the b2 numbers from 0 to b2 −1 are used as digits. It can be generalized to other bases, Complex-base system
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Bijective numeration
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Bijective numeration is any numeral system in which every non-negative integer can be represented in exactly one way using a finite string of digits. The name derives from this bijection between the set of integers and the set of finite strings using a finite set of symbols. Most ordinary numeral systems, such as the decimal system, are not bijective because more than one string of digits can represent the same positive integer. In particular, adding leading zeroes does not change the value represented, even though only the first is usual, the fact that the others are possible means that decimal is not bijective. However, unary, with one digit, is bijective. A bijective base-k numeration is a positional notation. It uses a string of digits from the set to encode each positive integer, the base-k bijective numeration system uses the digit-set to uniquely represent every non-negative integer, as follows, The integer zero is represented by the empty string. The integer represented by the nonempty digit-string anan−1, a1a0 is an kn + an−1 kn−1 +. The digit-string representing the integer m >0 is anan−1, for a given base k ≥1, there are exactly kn bijective base-k numerals of length n ≥0. Thus, using 0 to denote the empty string, the base 1,2,3,8,10,12, 119A = 1×103 + 1×102 + 9×101 + 10×1 =1200. The bijective base-10 system is a base ten positional system that does not use a digit to represent zero. It instead has a digit to represent ten, such as A, as with conventional decimal, each digit position represents a power of ten, so for example 123 is one hundred, plus two tens, plus three units. All positive integers that are represented solely with non-zero digits in conventional decimal have the same representation in decimal without a zero. Addition and multiplication in decimal without a zero are essentially the same as with conventional decimal, except that carries occur when a position exceeds ten, rather than when it exceeds nine. So to calculate 643 +759, there are twelve units, ten tens, thirteen hundreds, in the bijective base-26 system one may use the Latin alphabet letters A to Z to represent the 26 digit values one to twenty-six. With this choice of notation, the sequence begins A, B, C. Each digit position represents a power of twenty-six, so for example, many spreadsheets including Microsoft Excel use this system to assign labels to the columns of a spreadsheet, starting A, B, C. For instance, in Excel 2013, there can be up to 16384 columns, a variant of this system is used to name variable stars
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Unary numeral system
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The unary numeral system is the bijective base-1 numeral system. It is the simplest numeral system to represent natural numbers, in order to represent a number N, for examples, the numbers 1,2,3,4,5. Would be represented in this system as 1,11,111,1111,11111 and these numbers should be distinguished from repunits, which are also written as sequences of ones but have their usual decimal numerical interpretation. This system is used in tallying, for example, using the tally mark |, the number 3 is represented as |||. In East Asian cultures, the three is represented as “三”, a character that is drawn with three strokes. Addition and subtraction are particularly simple in the system, as they involve little more than string concatenation. The Hamming weight or population count operation that counts the number of bits in a sequence of binary values may also be interpreted as a conversion from unary to binary numbers. However, multiplication is more cumbersome and has often used as a test case for the design of Turing machines. Compared to standard positional numeral systems, the system is inconvenient. It occurs in some decision problem descriptions in theoretical computer science, therefore, while the run-time and space requirement in unary looks better as function of the input size, it does not represent a more efficient solution. In computational complexity theory, unary numbering is used to distinguish strongly NP-complete problems from problems that are NP-complete, for such a problem, there exist hard instances for which all parameter values are at most polynomially large. Unary is used as part of data compression algorithms such as Golomb coding. It also forms the basis for the Peano axioms for formalizing arithmetic within mathematical logic, a form of unary notation called Church encoding is used to represent numbers within lambda calculus. Sloanes A000042, Unary representation of natural numbers, the On-Line Encyclopedia of Integer Sequences
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Signed-digit representation
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In mathematical notation for numbers, signed-digit representation is a positional system with signed digits, the representation may not be unique. Signed-digit representation can be used to accomplish fast addition of integers because it can eliminate chains of dependent carries, in the binary numeral system, a special case signed-digit representation is the non-adjacent form, which can offer speed benefits with minimal space overhead. Challenges in calculation stimulated early authors Colson and Cauchy to use signed-digit representation, the further step of replacing negated digits with new ones was suggested by Selling and Cajori. In balanced form, the digits are drawn from a range −k to − k, for balanced forms, odd base numbers are advantageous. With an odd number, truncation and rounding become the same operation. A notable example is balanced ternary, where the base is b =3, balanced ternary uses the minimum number of digits in a balanced form. Balanced decimal uses digits from −5 to +4, balanced base nine, with digits from −4 to +4 provides the advantages of an odd-base balanced form with a similar number of digits, and is easy to convert to and from balanced ternary. Other notable examples include Booth encoding and non-adjacent form, both of which use a base of b =2, and both of which use numerals with the values −1,0, and +1, note that signed-digit representation is not necessarily unique. The oral and written forms of numbers in the Punjabi language use a form of a numeral one written as una or un. This negative one is used to form 19,29, …,89 from the root for 20,30, similarly, the Sesotho language utilizes negative numerals to form 8s and 9s. 8 robeli meaning break two i. e. two fingers down 9 robong meaning break one i. e. one finger down In 1928, Florian Cajori noted the theme of signed digits, starting with Colson. In his book History of Mathematical Notations, Cajori titled the section Negative numerals, eduard Selling advocated inverting the digits 1,2,3,4, and 5 to indicate the negative sign. He also suggested snie, jes, jerd, reff, most of the other early sources used a bar over a digit to indicate a negative sign for a it. For completeness, Colson uses examples and describes addition, multiplication and division using a table of multiples of the divisor and he explains the convenience of approximation by truncation in multiplication. Colson also devised an instrument that calculated using signed digits, Negative base Redundant binary representation J. P. Balantine A Digit for Negative One, American Mathematical Monthly 32,302. Augustin-Louis Cauchy Sur les moyens deviter les erreurs dans les calculs numerique, also found in Oevres completes Ser. Lui Han, Dongdong Chen, Seok-Bum Ko, Khan A. Wahid Non-speculative Decimal Signed Digit Adder from Department of Electrical and Computer Engineering, rudolf Mehmke Numerisches Rechen, §4 Beschränkung in den verwendeten Ziffern, Kleins encyclopedia, I-2, p.944
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Balanced ternary
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Balanced ternary is a non-standard positional numeral system, useful for comparison logic. While it is a number system, in the standard ternary system. The digits in the balanced ternary system have values −1,0, different sources use different glyphs used to represent the three digits in balanced ternary. In this article, T represents −1, while 0 and 1 represent themselves, other conventions include using − and + to represent −1 and 1 respectively, or using Greek letter theta, which resembles a minus sign in a circle, to represent −1. In Setun printings, −1 is represented as overturned 1,1, the notation has a number of computational advantages over regular binary. Particularly, the plus–minus consistency cuts down the rate in multi-digit multiplication. Balanced ternary also has a number of advantages over traditional ternary. Particularly, the multiplication table has no carries in balanced ternary. A possible use of balanced ternary is to represent if a list of values in a list is less than, equal to or greater than the corresponding value in a second list. Balanced ternary can also represent all integers without using a separate minus sign, in the balanced ternary system the value of a digit n places left of the radix point is the product of the digit and 3n. This is useful when converting between decimal and balanced ternary, in the following the strings denoting balanced ternary carry the suffix, bal3. For instance, −2/3dec = −1 + 1/3 = −1×30 + 1×3−1 = T. 1bal3, an integer is divisible by three if and only if the digit in the units place is zero. We may check the parity of a balanced ternary integer by checking the parity of the sum of all trits and this sum has the same parity as the integer itself. Balanced ternary can also be extended to fractional numbers similar to how decimal numbers are written to the right of the radix point, in decimal or binary, integer values and terminating fractions have multiple representations. For example,110 =0.1 =0.10 =0.09, and,12 =0. 1bin =0. 10bin =0. 01bin. Some balanced ternary fractions have multiple representations too, for example,16 =0. 1Tbal3 =0. 01bal3. Certainly, in the decimal and binary, we may omit the rightmost trailing infinite 0s after the radix point, but, in balanced ternary, we cant omit the rightmost trailing infinite –1s after the radix point in order to gain a representations of integer or terminating fraction. Donald Knuth has pointed out that truncation and rounding are the operation in balanced ternary — they produce exactly the same result
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Factorial number system
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In combinatorics, the factorial number system, also called factoradic, is a mixed radix numeral system adapted to numbering permutations. It is also called factorial base, although factorials do not function as base, by converting a number less than n. General mixed radix systems were studied by Georg Cantor, the term factorial number system is used by Knuth, while the French equivalent numération factorielle was first used in 1888. The term factoradic, which is a portmanteau of factorial and mixed radix, appears to be of more recent date. The factorial number system is a mixed radix numeral system, the i-th digit from the right has base i, which means that the digit must be less than i. From this it follows that the rightmost digit is always 0, the second can be 0 or 1, the third 0,1 or 2, the factorial number system is sometimes defined with the 0. Place omitted because it is always zero, in this article, a factorial number representation will be flagged by a subscript. Stands for 354413021100, whose value is = 3×5, general properties of mixed radix number systems also apply to the factorial number system. Reading the remainders backward gives 341010, in principle, this system may be extended to represent fractional numbers, though rather than the natural extension of place values. Etc. which are undefined, the choice of radix values n =0,1,2,3,4. Again, the 0 and 1 places may be omitted as these are always zero, the corresponding place values are therefore 1/1, 1/1, 1/2, 1/6, 1/24. The following sortable table shows the 24 permutations of four elements with different inversion related vectors, the left and right inversion counts l and r are particularly eligible to be interpreted as factorial numbers. L gives the position in reverse colexicographic order, and the latter the position in lexicographic order. Sorting by a column that has the omissible 0 on the right makes the numbers in that column correspond to the index numbers in the immovable column on the left. The small columns are reflections of the next to them. The rightmost column shows the digit sums of the factorial numbers, for another example, the greatest number that could be represented with six digits would be 543210. Which equals 719 in decimal, 5×5, clearly the next factorial number representation after 543210. is 1000000. =72010, the value for the radix-7 digit
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Negative base
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A negative base may be used to construct a non-standard positional numeral system. The need to store the information normally contained by a sign often results in a negative-base number being one digit longer than its positive-base equivalent. Negative numerical bases were first considered by Vittorio Grünwald in his work Giornale di Matematiche di Battaglini, Grünwald gave algorithms for performing addition, subtraction, multiplication, division, root extraction, divisibility tests, and radix conversion. Negative bases were later rediscovered by A. J. Kempner in 1936 and Zdzisław Pawlak. Negabinary was implemented in the early Polish computer BINEG, built 1957–59, based on ideas by Z. Pawlak, implementations since then have been rare. The base −r expansion of a is given by the string dndn-1…d1d0. Negative-base systems may thus be compared to signed-digit representations, such as balanced ternary, some numbers have the same representation in base −r as in base r. For example, the numbers from 100 to 109 have the same representations in decimal and negadecimal, similarly,17 =24 +20 =4 +0 and is represented by 10001 in binary and 10001 in negabinary. The base −r expansion of a number can be found by repeated division by −r, recording the non-negative remainders of 0,1, …, r −1, and concatenating those remainders, starting with the last. Note that if a / b = c, remainder d, then bc + d = a, to arrive at the correct conversion, the value for c must be chosen such that d is non-negative and minimal. This is exemplified in the line of the following example wherein –5 ÷ –3 must be chosen to equal 2 remainder 1 instead of 1 remainder –2. Note that in most programming languages, the result of dividing a number by a negative number is rounded towards 0. In such a case we have a = c + d = c + d − r + r = +, because |d| < r, is the positive remainder. The conversion from integer to some negative base, Visual Basic implementation, The conversion to negabinary allows a remarkable shortcut, the bitwise XOR portion is originally due to Schroeppel. Adding negabinary numbers proceeds bitwise, starting from the least significant bits, while adding two negabinary numbers, every time a carry is generated an extra carry should be propagated to next bit. Unary negation, −x, can be computed as binary subtraction from zero,0 − x, shifting to the left multiplies by −2, shifting to the right divides by −2. To multiply, multiply like normal decimal or binary numbers, but using the rules for adding the carry. It is possible to compare negabinary numbers by slightly adjusting a normal unsigned binary comparator, when comparing the numbers A and B, invert each odd positioned bit of both numbers
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Quater-imaginary base
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The quater-imaginary numeral system was first proposed by Donald Knuth in 1960. It is a positional numeral system which uses the imaginary number 2i as its base. It is able to represent every complex number using only the digits 0,1,2. The real and imaginary parts of complex number are thus readily expressed in base −4 as … d 4 d 2 d 0. D −2 … and 2 ⋅ respectively, to convert a digit string from the quater-imaginary system to the decimal system, the standard formula for positional number systems can be used. Additionally, for a given string d in the form d w −1, d w −2, every complex number has a quater-imaginary representation. Most numbers have a unique quater-imaginary representation, but just as 1 has the two representations 1 =0.999. in decimal notation, so 1/5 has the two quater-imaginary representations 1. …2i =0. …2i. For example, the representation of 6i is calculated by multiplying 6i • 2i = –12, which is expressed as 3002i. Finding the quater-imaginary representation of an arbitrary real number can be done manually by solving a system of simultaneous equations. But there are methods for both, real and imaginary, integers, as shown in section Negative base#To Negaquaternary. As an example of a number we can try to find the quater-imaginary counterpart of the decimal number 7. Since it is hard to exactly how long the digit string will be for a given decimal number. In this case, a string of six digits can be chosen, when an initial guess at the size of the string eventually turns out to be insufficient, a larger string can be used. Now the value of the coefficients d0, d2 and d4, because d0 −4 d2 +16 d4 =7 and because—by the nature of the quater-imaginary system—the coefficients can only be 0,1,2 or 3 the value of the coefficients can be found. A possible configuration could be, d0 =3, d2 =3 and this configuration gives the resulting digit string for 710. 710 =0103032 i =103032 i, finding a quater-imaginary representation of a purely imaginary integer number ∈ iZ is analogous to the method described above for a real number. For example, to find the representation of 6i, it is possible to use the general formula, then all coefficients of the real part have to be zero and the complex part should make 6. However, for 6i it is seen by looking at the formula that if d1 =3 and all other coefficients are zero
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Non-integer representation
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A non-integer representation uses non-integer numbers as the radix, or bases, of a positional numbering system. For a non-integer radix β >1, the value of x = d n … d 2 d 1 d 0, the numbers di are non-negative integers less than β. This is also known as a β-expansion, an introduced by Rényi. Every real number has at least one β-expansion, there are applications of β-expansions in coding theory and models of quasicrystals. β-expansions are a generalization of decimal expansions, while infinite decimal expansions are not unique, all finite decimal expansions are unique. However, even finite β-expansions are not necessarily unique, for example φ +1 = φ2 for β = φ, the golden ratio. A canonical choice for the β-expansion of a real number can be determined by the following greedy algorithm, essentially due to Rényi. Let β >1 be the base and x a non-negative real number, denote by ⌊x⌋ the floor function of x, that is, the greatest integer less than or equal to x, and let = x − ⌊x⌋ be the fractional part of x. There exists a k such that βk ≤ x < βk+1. Set d k = ⌊ x / β k ⌋ and r k =, for k −1 ≥ j > −∞, put d j = ⌊ β r j +1 ⌋, r j =. In other words, the canonical β-expansion of x is defined by choosing the largest dk such that βkdk ≤ x, then choosing the largest dk−1 such that βkdk + βk−1dk−1 ≤ x, thus it chooses the lexicographically largest string representing x. With an integer base, this defines the usual radix expansion for the number x and this construction extends the usual algorithm to possibly non-integer values of β. See Golden ratio base, 11φ = 100φ, with base e the natural logarithm behaves like the common logarithm as ln =0, ln =1, ln =2 and ln =3. This means that every integer can be expressed in base √2 without the need of a decimal point, another use of the base is to show the silver ratio as its representation in base √2 is simply 11√2. In no positional number system can every number be expressed uniquely, for example, in base ten, the number 1 has two representations,1.000. and 0.999. Another problem is to classify the real numbers whose β-expansions are periodic, let β >1, and Q be the smallest field extension of the rationals containing β. Then any real number in [0, 1) having a periodic β-expansion must lie in Q, on the other hand, the converse need not be true. The converse does hold if β is a Pisot number, although necessary and sufficient conditions are not known
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Golden ratio base
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Golden ratio base is a non-integer positional numeral system that uses the golden ratio as its base. It is sometimes referred to as base-φ, golden mean base, phi-base, or, colloquially, any non-negative real number can be represented as a base-φ numeral using only the digits 0 and 1, and avoiding the digit sequence 11 – this is called a standard form. A base-φ numeral that includes the digit sequence 11 can always be rewritten in standard form, despite using an irrational number base, when using standard form, all non-negative integers have a unique representation as a terminating base-φ expansion. Other numbers have standard representations in base-φ, with rational numbers having recurring representations and these representations are unique, except that numbers with a terminating expansion also have a non-terminating expansion, as they do in base-10, for example,1 =0. 99999…. In the following example the notation 1 is used to represent −1. 211. 01φ is not a standard base-φ numeral, since it contains a 11 and a 2, which isnt a 0 or 1, and contains a 1 = −1, which isnt a 0 or 1 either. To standardize a numeral, we can use the following substitutions, 011φ = 100φ, 0200φ = 1001φ, 010φ = 101φ and we can apply the substitutions in any order we like, as the result is the same. Below, the applied to the number on the previous line are on the right. Any positive number with a non-standard terminating base-φ representation can be standardized in this manner. If we get to a point where all digits are 0 or 1, except for the first digit being negative and this can be converted to the negative of a base-φ representation by negating every digit, standardizing the result, and then marking it as negative. For example, use a sign, or some other significance to denote negative numbers. If the arithmetic is being performed on a computer, a message may be returned. We can either consider our integer to be the digit of a nonstandard base-φ numeral, therefore, we can compute + =, − = and × =. So, using integer values only, we can add, subtract and multiply numbers of the form, > if and only if 2 − > × √5. If one side is negative, the positive, the comparison is trivial. Otherwise, square sides, to get an integer comparison, reversing the comparison direction if both sides were negative. On squaring both sides, the √5 is replaced with the integer 5, so, using integer values only, we can also compare numbers of the form. To convert an integer x to a number, note that x =
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Mixed radix
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Mixed radix numeral systems are non-standard positional numeral systems in which the numerical base varies from position to position. Such numerical representation applies when a quantity is expressed using a sequence of units that are each a multiple of the smaller one. 32,5,7,45,15,500. ∞,7,24,60,60,1000 or as 32∞577244560.15605001000 In the tabular format, the digits are written above their base, and a semicolon indicates the radix point. In numeral format, each digit has its base attached as a subscript. The base for each digit is the number of corresponding units that make up the larger unit. As a consequence there is no base for the first digit, the most familiar example of mixed radix systems is in timekeeping and calendars. Western time radices include decimal centuries, decades and years as well as duodecimal months, trigesimal days, overlapped with base 52 weeks, one variant uses tridecimal months, quaternary weeks, and septenary days. Time is further divided by quadrivigesimal hours, sexagesimal minutes and seconds, a mixed radix numeral system can often benefit from a tabular summary. m. On Wednesday, and 070201202602460 would be 12,02,24 a. m. on Sunday, ad hoc notations for mixed radix numeral systems are commonplace. The Maya calendar consists of several overlapping cycles of different radices, a short count tzolkin overlaps vigesimal named days with tridecimal numbered days. A haab consists of vigesimal days, octodecimal months, and base-52 years forming a round, in addition, a long count of vigesimal days, octodecimal winal, then vigesimal tun, katun, baktun, etc. tracks historical dates. So, for example, in the UK, banknotes are printed for £50, £20, £10 and £5, mixed-radix numbers of the same base can be manipulated using a generalization of manual arithmetic algorithms. APL and J include operators to convert to and from mixed-radix systems, another proposal is the so-called factorial number system, For example, the biggest number that could be represented with six digits would be 543210 which equals 719 in decimal, 5×5. It might not be clear at first sight but the factorial based numbering system is unambiguous and complete. Every number can be represented in one and only one way because the sum of respective factorials multiplied by the index is always the next factorial minus one, −1 There is a natural mapping between the integers 0. N. −1 and permutations of n elements in lexicographic order, the above equation is a particular case of the following general rule for any radix base representation which expresses the fact that any radix base representation is unambiguous and complete. The Art of Computer Programming, Volume 2, Seminumerical Algorithms, Über einfache Zahlensysteme, Zeitschrift für Math. Mixed Radix Calculator — Mixed Radix Calculator in C#
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List of numeral systems
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This is a list of numeral systems, that is, writing systems for expressing numbers. Numeral systems are classified here as to whether they use positional notation, the common names are derived somewhat arbitrarily from a mix of Latin and Greek, in some cases including roots from both languages within a single name. In this Youtube video, Matt Parker jokingly invented a base-1082 system and this turns out to be 1925. Radix Radix economy Table of bases List of numbers in various languages Numeral prefix
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10 (number)
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10 is an even natural number following 9 and preceding 11. Ten is the base of the numeral system, by far the most common system of denoting numbers in both spoken and written language. The reason for the choice of ten is assumed to be that humans have ten fingers, a collection of ten items is called a decade. The ordinal adjective is decimal, the adjective is denary. Increasing a quantity by one order of magnitude is most widely understood to mean multiplying the quantity by ten, to reduce something by one tenth is to decimate. A theoretical highest number in topics that require a rating, by contrast having 0 or 1 as the lowest number, Ten is a composite number, its proper divisors being 1,2 and 5. Ten is the smallest noncototient, a number that cannot be expressed as the difference between any integer and the number of coprimes below it. Ten is the discrete semiprime and the second member of the discrete semiprime family. Ten has an aliquot sum σ of 8 and is accordingly the first discrete semiprime to be in deficit, all subsequent discrete semiprimes are in deficit. The aliquot sequence for 10 comprises five members with this number being the second member of the 7-aliquot tree. Ten is the smallest semiprime that is the sum of all the prime numbers from its lower factor through its higher factor Only three other small semiprimes share this attribute. It is the sum of only one number the discrete semiprime 14. Ten is the sum of the first three numbers, of the four first numbers, of the square of the two first odd numbers and also of the first four factorials. Ten is the eighth Perrin number, preceded in the sequence by 5,5,7, a polygon with ten sides is a decagon, and 10 is a decagonal number. Because 10 is the product of a power of 2 with nothing but distinct Fermat primes, Ten is also a triangular number, a centered triangular number, and a tetrahedral number. Ten is the number of n queens problem solutions for n =5, Ten is the smallest number whose status as a possible friendly number is unknown. As is the case for any base in its system, ten is the first two-digit number in decimal, any integer written in the decimal system can be multiplied by ten by adding a zero to the end. The Roman numeral for ten is X, it is thought that the V for five is derived from an open hand, incidentally, the Chinese word numeral for ten, is also a cross, 十
62.
Positional numeral system
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Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the symbol for the different orders of magnitude. This greatly simplified arithmetic, leading to the spread of the notation across the world. With the use of a point, the notation can be extended to include fractions. The Hindu–Arabic numeral system, base-10, is the most commonly used system in the world today for most calculations, today, the base-10 system, which is likely motivated by counting with the ten fingers, is ubiquitous. Other bases have been used in the past however, and some continue to be used today, for example, the Babylonian numeral system, credited as the first positional numeral system, was base-60, but it lacked a real 0 value. Zero was indicated by a space between sexagesimal numerals, by 300 BC, a punctuation symbol was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish, the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges, the Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number, thus numbers like 2 and 120,3 and 180,4 and 240, looked the same because the larger numbers lacked a final sexagesimal placeholder. Counting rods and most abacuses have been used to represent numbers in a numeral system. This approach required no memorization of tables and could produce practical results quickly, for four centuries there was strong disagreement between those who believed in adopting the positional system in writing numbers and those who wanted to stay with the additive-system-plus-abacus. Although electronic calculators have largely replaced the abacus, the continues to be used in Japan. After the French Revolution, the new French government promoted the extension of the decimal system, some of those pro-decimal efforts—such as decimal time and the decimal calendar—were unsuccessful. Other French pro-decimal efforts—currency decimalisation and the metrication of weights and measures—spread widely out of France to almost the whole world. According to Joseph Needham and Lam Lay Yong, decimal fractions were first developed and used by the Chinese in the 1st century BC, the written Chinese decimal fractions were non-positional. However, counting rod fractions were positional, the Jewish mathematician Immanuel Bonfils used decimal fractions around 1350, anticipating Simon Stevin, but did not develop any notation to represent them. A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century. A key argument against the system was its susceptibility to easy fraud by simply putting a number at the beginning or end of a quantity, thereby changing 100 into 5100
63.
Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
64.
Number
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A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1,2,3, a notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are used for labels, for ordering. In common usage, number may refer to a symbol, a word, calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation. Their study or usage is called arithmetic, the same term may also refer to number theory, the study of the properties of numbers. Besides their practical uses, numbers have cultural significance throughout the world, for example, in Western society the number 13 is regarded as unlucky, and a million may signify a lot. Though it is now regarded as pseudoscience, numerology, the belief in a significance of numbers, permeated ancient. Numerology heavily influenced the development of Greek mathematics, stimulating the investigation of problems in number theory which are still of interest today. During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers, among the first were the hypercomplex numbers, which consist of various extensions or modifications of the complex number system. Numbers should be distinguished from numerals, the used to represent numbers. Boyer showed that Egyptians created the first ciphered numeral system, Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets. The number five can be represented by digit 5 or by the Roman numeral Ⅴ, notations used to represent numbers are discussed in the article numeral systems. The Roman numerals require extra symbols for larger numbers, different types of numbers have many different uses. Numbers can be classified into sets, called number systems, such as the natural numbers, the same number can be written in many different ways. For different methods of expressing numbers with symbols, such as the Roman numerals, each of these number systems may be considered as a proper subset of the next one. This is expressed, symbolically, by writing N ⊂ Z ⊂ Q ⊂ R ⊂ C, the most familiar numbers are the natural numbers,1,2,3, and so on. Traditionally, the sequence of numbers started with 1 However, in the 19th century, set theorists. Today, different mathematicians use the term to both sets, including 0 or not
65.
Decimal mark
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A decimal mark is a symbol used to separate the integer part from the fractional part of a number written in decimal form. Different countries officially designate different symbols for the decimal mark, the choice of symbol for the decimal mark also affects the choice of symbol for the thousands separator used in digit grouping, so the latter is also treated in this article. In mathematics the decimal mark is a type of radix point, in the Middle Ages, before printing, a bar over the units digit was used to separate the integral part of a number from its fractional part, e. g.9995. His Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear, a similar notation remains in common use as an underbar to superscript digits, especially for monetary values without a decimal mark, e. g.9995. Later, a separatrix between the units and tenths position became the norm among Arab mathematicians, e. g. 99ˌ95, when this character was typeset, it was convenient to use the existing comma or full stop instead. The separatrix was also used in England as an L-shaped or vertical bar before the popularization of the period, gerbert of Aurillac marked triples of columns with an arc when using his Hindu–Arabic numeral-based abacus in the 10th century. Fibonacci followed this convention when writing numbers such as in his influential work Liber Abaci in the 13th century, in France, the full stop was already in use in printing to make Roman numerals more readable, so the comma was chosen. Many other countries, such as Italy, also chose to use the comma to mark the decimal units position and it has been made standard by the ISO for international blueprints. However, English-speaking countries took the comma to separate sequences of three digits, in some countries, a raised dot or dash may be used for grouping or decimal mark, this is particularly common in handwriting. In the United States, the stop or period was used as the standard decimal mark. g. However, as the mid dot was already in use in the mathematics world to indicate multiplication. In the event, the point was decided on by the Ministry of Technology in 1968, the three most spoken international auxiliary languages, Ido, Esperanto, and Interlingua, all use the comma as the decimal mark. Interlingua has used the comma as its decimal mark since the publication of the Interlingua Grammar in 1951, Esperanto also uses the comma as its official decimal mark, while thousands are separated by non-breaking spaces,12345678,9. Idos Kompleta Gramatiko Detaloza di la Linguo Internaciona Ido officially states that commas are used for the mark while full stops are used to separate thousands, millions. So the number 12,345,678.90123 for instance, the 1931 grammar of Volapük by Arie de Jong uses the comma as its decimal mark, and uses the middle dot as the thousands separator. In 1958, disputes between European and American delegates over the representation of the decimal mark nearly stalled the development of the ALGOL computer programming language. ALGOL ended up allowing different decimal marks, but most computer languages, the 22nd General Conference on Weights and Measures declared in 2003 that the symbol for the decimal marker shall be either the point on the line or the comma on the line. It further reaffirmed that numbers may be divided in groups of three in order to facilitate reading, neither dots nor commas are ever inserted in the spaces between groups
66.
Fraction (mathematics)
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A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction consists of an integer numerator displayed above a line, numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals. The numerator represents a number of parts, and the denominator. For example, in the fraction 3/4, the numerator,3, tells us that the fraction represents 3 equal parts, the picture to the right illustrates 34 or ¾ of a cake. Fractional numbers can also be written without using explicit numerators or denominators, by using decimals, percent signs, an integer such as the number 7 can be thought of as having an implicit denominator of one,7 equals 7/1. Other uses for fractions are to represent ratios and to represent division, thus the fraction ¾ is also used to represent the ratio 3,4 and the division 3 ÷4. The test for a number being a number is that it can be written in that form. In a fraction, the number of parts being described is the numerator. Informally, they may be distinguished by placement alone but in formal contexts they are separated by a fraction bar. The fraction bar may be horizontal, oblique, or diagonal and these marks are respectively known as the horizontal bar, the slash or stroke, the division slash, and the fraction slash. In typography, horizontal fractions are known as en or nut fractions and diagonal fractions as em fractions. The denominators of English fractions are expressed as ordinal numbers. When the denominator is 1, it may be expressed in terms of wholes but is commonly ignored. When the numerator is one, it may be omitted, a fraction may be expressed as a single composition, in which case it is hyphenated, or as a number of fractions with a numerator of one, in which case they are not. Fractions should always be hyphenated when used as adjectives, alternatively, a fraction may be described by reading it out as the numerator over the denominator, with the denominator expressed as a cardinal number. The term over is used even in the case of solidus fractions, Fractions with large denominators that are not powers of ten are often rendered in this fashion while those with denominators divisible by ten are typically read in the normal ordinal fashion. A simple fraction is a number written as a/b or a b
67.
Nonnegative integer
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In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers, the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1,10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory
68.
Real number
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In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by René Descartes, the real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the numbers, such as π. Real numbers can be thought of as points on a long line called the number line or real line. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, the real line can be thought of as a part of the complex plane, and complex numbers include real numbers. These descriptions of the numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions satisfy the definition and are thus equivalent. The statement that there is no subset of the reals with cardinality greater than ℵ0. Simple fractions were used by the Egyptians around 1000 BC, the Vedic Sulba Sutras in, c.600 BC, around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of number and magnitude into a general idea of real numbers. In the 16th century, Simon Stevin created the basis for modern decimal notation, in the 17th century, Descartes introduced the term real to describe roots of a polynomial, distinguishing them from imaginary ones. In the 18th and 19th centuries, there was work on irrational and transcendental numbers. Johann Heinrich Lambert gave the first flawed proof that π cannot be rational, Adrien-Marie Legendre completed the proof, Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, and Ferdinand von Lindemann, lindemanns proof was much simplified by Weierstrass, still further by David Hilbert, and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871, in 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, the real number system can be defined axiomatically up to an isomorphism, which is described hereafter. Another possibility is to start from some rigorous axiomatization of Euclidean geometry, from the structuralist point of view all these constructions are on equal footing
69.
Sequence (mathematics)
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In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members, the number of elements is called the length of the sequence. Unlike a set, order matters, and exactly the elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the numbers or the set of the first n natural numbers. The position of an element in a sequence is its rank or index and it depends on the context or of a specific convention, if the first element has index 0 or 1. For example, is a sequence of letters with the letter M first, also, the sequence, which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in these examples, or infinite, the empty sequence is included in most notions of sequence, but may be excluded depending on the context. A sequence can be thought of as a list of elements with a particular order, Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations, Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers. There are a number of ways to denote a sequence, some of which are useful for specific types of sequences. One way to specify a sequence is to list the elements, for example, the first four odd numbers form the sequence. This notation can be used for sequences as well. For instance, the sequence of positive odd integers can be written. Listing is most useful for sequences with a pattern that can be easily discerned from the first few elements. Other ways to denote a sequence are discussed after the examples, the prime numbers are the natural numbers bigger than 1, that have no divisors but 1 and themselves. Taking these in their natural order gives the sequence, the prime numbers are widely used in mathematics and specifically in number theory. The Fibonacci numbers are the integer sequence whose elements are the sum of the two elements. The first two elements are either 0 and 1 or 1 and 1 so that the sequence is, for a large list of examples of integer sequences, see On-Line Encyclopedia of Integer Sequences
70.
Rational number
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In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number. The set of all numbers, often referred to as the rationals, is usually denoted by a boldface Q, it was thus denoted in 1895 by Giuseppe Peano after quoziente. The decimal expansion of a rational number always either terminates after a number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number and these statements hold true not just for base 10, but also for any other integer base. A real number that is not rational is called irrational, irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating, since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. Rational numbers can be defined as equivalence classes of pairs of integers such that q ≠0, for the equivalence relation defined by ~ if. In abstract algebra, the numbers together with certain operations of addition and multiplication form the archetypical field of characteristic zero. As such, it is characterized as having no proper subfield or, alternatively, finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the numbers form a dense subset of the real numbers. The real numbers can be constructed from the numbers by completion, using Cauchy sequences, Dedekind cuts. The term rational in reference to the set Q refers to the fact that a number represents a ratio of two integers. In mathematics, rational is often used as a noun abbreviating rational number, the adjective rational sometimes means that the coefficients are rational numbers. However, a curve is not a curve defined over the rationals. Any integer n can be expressed as the rational number n/1, a b = c d if and only if a d = b c. Where both denominators are positive, a b < c d if and only if a d < b c. If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations, − a − b = a b, two fractions are added as follows, a b + c d = a d + b c b d
71.
Decimal digit
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A digit is a numeric symbol used in combinations to represent numbers in positional numeral systems. The name digit comes from the fact that the 10 digits of the hands correspond to the 10 symbols of the common base 10 numeral system, i. e. the decimal digits. In a given system, if the base is an integer. For example, the system has ten digits, whereas binary has two digits. In a basic system, a numeral is a sequence of digits. Each position in the sequence has a value, and each digit has a value. The value of the numeral is computed by multiplying each digit in the sequence by its place value, each digit in a number system represents an integer. For example, in decimal the digit 1 represents the one, and in the hexadecimal system. A positional number system must have a digit representing the integers from zero up to, but not including, thus in the positional decimal system, the numbers 0 to 9 can be expressed using their respective numerals 0 to 9 in the rightmost units position. The Hindu–Arabic numeral system uses a decimal separator, commonly a period in English, or a comma in other European languages, to denote the place or units place. Each successive place to the left of this has a value equal to the place value of the previous digit times the base. Similarly, each place to the right of the separator has a place value equal to the place value of the previous digit divided by the base. For example, in the numeral 10, the total value of the number is 1 ten,0 ones,3 tenths, and 4 hundredths. Note that the zero, which contributes no value to the number, the place value of any given digit in a numeral can be given by a simple calculation, which in itself is a compliment to the logic behind numeral systems. And to the right, the digit is multiplied by the base raised by a negative n, for example, in the number 10. 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 widely accepted. Instead of a zero, a dot was left in the numeral as a placeholder, the first widely acknowledged use of zero was in 876. The original numerals were very similar to the ones, even down to the glyphs used to represent digits
72.
Negative number
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In mathematics, a negative number is a real number that is less than zero. If positive represents movement to the right, negative represents movement to the left, if positive represents above sea level, then negative represents below level. If positive represents a deposit, negative represents a withdrawal and they are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset, if a quantity may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage, negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common idea of an opposite is reflected in arithmetic. For example, − −3 =3 because the opposite of an opposite is the original thing, negative numbers are usually written with a minus sign in front. For example, −3 represents a quantity with a magnitude of three, and is pronounced minus three or negative three. To help tell the difference between a subtraction operation and a number, occasionally the negative sign is placed slightly higher than the minus sign. Conversely, a number that is greater than zero is called positive, the positivity of a number may be emphasized by placing a plus sign before it, e. g. +3. In general, the negativity or positivity of a number is referred to as its sign, every real number other than zero is either positive or negative. The positive whole numbers are referred to as natural numbers, while the positive and negative numbers are referred to as integers. In bookkeeping, amounts owed are often represented by red numbers, or a number in parentheses, Liu Hui established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers, islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Western mathematicians accepted the idea of numbers by the 17th century. Prior to the concept of numbers, mathematicians such as Diophantus considered negative solutions to problems false. Negative numbers can be thought of as resulting from the subtraction of a number from a smaller. For example, negative three is the result of subtracting three from zero,0 −3 = −3, in general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers
73.
Minus sign
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The plus and minus signs are mathematical symbols used to represent the notions of positive and negative as well as the operations of addition and subtraction. Their use has extended to many other meanings, more or less analogous. Plus and minus are Latin terms meaning more and less, respectively, though the signs now seem as familiar as the alphabet or the Hindu-Arabic numerals, they are not of great antiquity. In Europe in the early 15th century the letters P and M were generally used, the symbols appeared for the first time in Luca Pacioli’s mathematics compendium, Summa de arithmetica, geometria, proportioni et proportionalità, first printed and published in Venice in 1494. The + is a simplification of the Latin et, the − may be derived from a tilde written over m when used to indicate subtraction, or it may come from a shorthand version of the letter m itself. In his 1489 treatise Johannes Widmann referred to the symbols − and + as minus and mer, was − ist, das ist minus, und das + ist das mer. They werent used for addition and subtraction here, but to indicate surplus and deficit, the plus sign is a binary operator that indicates addition, as in 2 +3 =5. It can also serve as an operator that leaves its operand unchanged. This notation may be used when it is desired to emphasize the positiveness of a number, the plus sign can also indicate many other operations, depending on the mathematical system under consideration. Many algebraic structures have some operation which is called, or is equivalent to and it is conventional to use the plus sign to only denote commutative operations. Subtraction is the inverse of addition, directly in front of a number and when it is not a subtraction operator it means a negative number. For instance −5 is negative 5, a unary operator that acts as an instruction to replace the operand by its additive inverse. For example, if x is 3, then −x is −3, similarly, − is equal to 2. The above is a case of this. All three uses can be referred to as minus in everyday speech, further, some textbooks in the United States encourage −x to be read as the opposite of x or the additive inverse of x to avoid giving the impression that −x is necessarily negative. However, in programming languages and Microsoft Excel in particular, unary operators bind strongest, so in those cases −5^2 is 25. Some elementary teachers use raised plus and minus signs before numbers to show they are positive or negative numbers. For example, subtracting −5 from 3 might be read as positive three take away negative 5 and be shown as 3 − −5 becomes 3 +5 =8, in grading systems, the plus sign indicates a grade one level higher and the minus sign a grade lower
74.
Europe
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Europe is a continent that comprises the westernmost part of Eurasia. Europe is bordered by the Arctic Ocean to the north, the Atlantic Ocean to the west, yet the non-oceanic borders of Europe—a concept dating back to classical antiquity—are arbitrary. Europe covers about 10,180,000 square kilometres, or 2% of the Earths surface, politically, Europe is divided into about fifty sovereign states of which the Russian Federation is the largest and most populous, spanning 39% of the continent and comprising 15% of its population. Europe had a population of about 740 million as of 2015. Further from the sea, seasonal differences are more noticeable than close to the coast, Europe, in particular ancient Greece, was the birthplace of Western civilization. The fall of the Western Roman Empire, during the period, marked the end of ancient history. Renaissance humanism, exploration, art, and science led to the modern era, from the Age of Discovery onwards, Europe played a predominant role in global affairs. Between the 16th and 20th centuries, European powers controlled at times the Americas, most of Africa, Oceania. The Industrial Revolution, which began in Great Britain at the end of the 18th century, gave rise to economic, cultural, and social change in Western Europe. During the Cold War, Europe was divided along the Iron Curtain between NATO in the west and the Warsaw Pact in the east, until the revolutions of 1989 and fall of the Berlin Wall. In 1955, the Council of Europe was formed following a speech by Sir Winston Churchill and it includes all states except for Belarus, Kazakhstan and Vatican City. Further European integration by some states led to the formation of the European Union, the EU originated in Western Europe but has been expanding eastward since the fall of the Soviet Union in 1991. The European Anthem is Ode to Joy and states celebrate peace, in classical Greek mythology, Europa is the name of either a Phoenician princess or of a queen of Crete. The name contains the elements εὐρύς, wide, broad and ὤψ eye, broad has been an epithet of Earth herself in the reconstructed Proto-Indo-European religion and the poetry devoted to it. For the second part also the divine attributes of grey-eyed Athena or ox-eyed Hera. The same naming motive according to cartographic convention appears in Greek Ανατολή, Martin Litchfield West stated that phonologically, the match between Europas name and any form of the Semitic word is very poor. Next to these there is also a Proto-Indo-European root *h1regʷos, meaning darkness. Most major world languages use words derived from Eurṓpē or Europa to refer to the continent, in some Turkic languages the originally Persian name Frangistan is used casually in referring to much of Europe, besides official names such as Avrupa or Evropa
75.
Nonnegative number
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In mathematics, the concept of sign originates from the property of every non-zero real number of being positive or negative. Zero itself is signless, although in some contexts it makes sense to consider a signed zero, along with its application to real numbers, change of sign is used throughout mathematics and physics to denote the additive inverse, even for quantities which are not real numbers. Also, the sign can indicate aspects of mathematical objects that resemble positivity and negativity. A real number is said to be if its value is greater than zero. The attribute of being positive or negative is called the sign of the number, zero itself is not considered to have a sign. Also, signs are not defined for complex numbers, although the argument generalizes it in some sense, in common numeral notation, the sign of a number is often denoted by placing a plus sign or a minus sign before the number. For example, +3 denotes positive three, and −3 denotes negative three, when no plus or minus sign is given, the default interpretation is that a number is positive. Because of this notation, as well as the definition of numbers through subtraction. In this context, it makes sense to write − = +3, any non-zero number can be changed to a positive one using the absolute value function. For example, the value of −3 and the absolute value of 3 are both equal to 3. In symbols, this would be written |−3| =3 and |3| =3, the number zero is neither positive nor negative, and therefore has no sign. In arithmetic, +0 and −0 both denote the same number 0, which is the inverse of itself. Note that this definition is culturally determined, in France and Belgium,0 is said to be both positive and negative. The positive resp. negative numbers without zero are said to be strictly positive resp, in some contexts, such as signed number representations in computing, it makes sense to consider signed versions of zero, with positive zero and negative zero being different numbers. One also sees +0 and −0 in calculus and mathematical analysis when evaluating one-sided limits and this notation refers to the behaviour of a function as the input variable approaches 0 from positive or negative values respectively, these behaviours are not necessarily the same. Because zero is positive nor negative, the following phrases are sometimes used to refer to the sign of an unknown number. A number is negative if it is less than zero, a number is non-negative if it is greater than or equal to zero. A number is non-positive if it is less than or equal to zero, thus a non-negative number is either positive or zero, while a non-positive number is either negative or zero
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Denominator
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A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction consists of an integer numerator displayed above a line, numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals. The numerator represents a number of parts, and the denominator. For example, in the fraction 3/4, the numerator,3, tells us that the fraction represents 3 equal parts, the picture to the right illustrates 34 or ¾ of a cake. Fractional numbers can also be written without using explicit numerators or denominators, by using decimals, percent signs, an integer such as the number 7 can be thought of as having an implicit denominator of one,7 equals 7/1. Other uses for fractions are to represent ratios and to represent division, thus the fraction ¾ is also used to represent the ratio 3,4 and the division 3 ÷4. The test for a number being a number is that it can be written in that form. In a fraction, the number of parts being described is the numerator. Informally, they may be distinguished by placement alone but in formal contexts they are separated by a fraction bar. The fraction bar may be horizontal, oblique, or diagonal and these marks are respectively known as the horizontal bar, the slash or stroke, the division slash, and the fraction slash. In typography, horizontal fractions are known as en or nut fractions and diagonal fractions as em fractions. The denominators of English fractions are expressed as ordinal numbers. When the denominator is 1, it may be expressed in terms of wholes but is commonly ignored. When the numerator is one, it may be omitted, a fraction may be expressed as a single composition, in which case it is hyphenated, or as a number of fractions with a numerator of one, in which case they are not. Fractions should always be hyphenated when used as adjectives, alternatively, a fraction may be described by reading it out as the numerator over the denominator, with the denominator expressed as a cardinal number. The term over is used even in the case of solidus fractions, Fractions with large denominators that are not powers of ten are often rendered in this fashion while those with denominators divisible by ten are typically read in the normal ordinal fashion. A simple fraction is a number written as a/b or a b
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Exponentiation
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Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent n. The exponent is usually shown as a superscript to the right of the base, Some common exponents have their own names, the exponent 2 is called the square of b or b squared, the exponent 3 is called the cube of b or b cubed. The exponent −1 of b, or 1 / b, is called the reciprocal of b, when n is a positive integer and b is not zero, b−n is naturally defined as 1/bn, preserving the property bn × bm = bn + m. The definition of exponentiation can be extended to any real or complex exponent. Exponentiation by integer exponents can also be defined for a variety of algebraic structures. The term power was used by the Greek mathematician Euclid for the square of a line, archimedes discovered and proved the law of exponents, 10a 10b = 10a+b, necessary to manipulate powers of 10. In the late 16th century, Jost Bürgi used Roman numerals for exponents, early in the 17th century, the first form of our modern exponential notation was introduced by Rene Descartes in his text titled La Géométrie, there, the notation is introduced in Book I. Nicolas Chuquet used a form of notation in the 15th century. The word exponent was coined in 1544 by Michael Stifel, samuel Jeake introduced the term indices in 1696. In the 16th century Robert Recorde used the square, cube, zenzizenzic, sursolid, zenzicube, second sursolid. Biquadrate has been used to refer to the power as well. Some mathematicians used exponents only for greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials, for example, as ax + bxx + cx3 + d, another historical synonym, involution, is now rare and should not be confused with its more common meaning. In 1748 Leonhard Euler wrote consider exponentials or powers in which the exponent itself is a variable and it is clear that quantities of this kind are not algebraic functions, since in those the exponents must be constant. With this introduction of transcendental functions, Euler laid the foundation for the introduction of natural logarithm as the inverse function for y = ex. The expression b2 = b ⋅ b is called the square of b because the area of a square with side-length b is b2, the expression b3 = b ⋅ b ⋅ b is called the cube of b because the volume of a cube with side-length b is b3. The exponent indicates how many copies of the base are multiplied together, for example,35 =3 ⋅3 ⋅3 ⋅3 ⋅3 =243. The base 3 appears 5 times in the multiplication, because the exponent is 5
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Integer part
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In mathematics and computer science, the floor and ceiling functions map a real number to the greatest preceding or the least succeeding integer, respectively. More precisely, floor = ⌊ x ⌋ is the greatest integer less than or equal to x, carl Friedrich Gauss introduced the square bracket notation for the floor function in his third proof of quadratic reciprocity. This remained the standard in mathematics until Kenneth E. Iverson introduced the names floor and ceiling, both notations are now used in mathematics, this article follows Iverson. e. The value of x rounded to an integer towards 0, the language APL uses ⌊x, other computer languages commonly use notations like entier, INT, or floor. In mathematics, it can also be written with boldface or double brackets, the ceiling function is usually denoted by ceil or ceiling in non-APL computer languages that have a notation for this function. The J Programming Language, a follow on to APL that is designed to use standard symbols, uses >. for ceiling. In mathematics, there is another notation with reversed boldface or double brackets ] ] x x[\. x[, the fractional part is the sawtooth function, denoted by for real x and defined by the formula = x − ⌊ x ⌋. HTML4.0 uses the names, &lfloor, &rfloor, &lceil. Unicode contains codepoints for these symbols at U+2308–U+230B, ⌈x⌉, ⌊x⌋, in the following formulas, x and y are real numbers, k, m, and n are integers, and Z is the set of integers. Floor and ceiling may be defined by the set equations ⌊ x ⌋ = max, ⌈ x ⌉ = min. Since there is exactly one integer in an interval of length one. Then ⌊ x ⌋ = m and ⌈ x ⌉ = n may also be taken as the definition of floor and these formulas can be used to simplify expressions involving floors and ceilings. In the language of order theory, the function is a residuated mapping. These formulas show how adding integers to the arguments affect the functions, negating the argument complements the fractional part, + = {0 if x ∈ Z1 if x ∉ Z. The floor, ceiling, and fractional part functions are idempotent, the result of nested floor or ceiling functions is the innermost function, ⌊ ⌈ x ⌉ ⌋ = ⌈ x ⌉, ⌈ ⌊ x ⌋ ⌉ = ⌊ x ⌋. If m and n are integers and n ≠0,0 ≤ ≤1 −1 | n |. If n is a positive integer ⌊ x + m n ⌋ = ⌊ ⌊ x ⌋ + m n ⌋, ⌈ x + m n ⌉ = ⌈ ⌈ x ⌉ + m n ⌉. For m =2 these imply n = ⌊ n 2 ⌋ + ⌈ n 2 ⌉