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
Numeral system
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Numeral systems
2.
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
Eastern Arabic numerals
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Numeral systems
Eastern Arabic numerals
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Arabic style Eastern Arabic numerals on a clock in the Cairo Metro
Eastern Arabic numerals
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Ottoman clocks tended to use Eastern Arabic numerals styled to look like Roman
3.
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
Khmer numerals
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The number 605 in Khmer numerals, from the Sambor inscriptions in 683 AD. The earliest known material use of zero as a decimal figure.
Khmer numerals
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Numeral systems
4.
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
Lao alphabet
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Numeral systems
Lao alphabet
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Lao
5.
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
Thai numerals
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Numeral systems
6.
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
Korean numerals
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Numeral systems
7.
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
Vietnamese numerals
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Numeral systems
8.
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
Counting rods
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Numeral systems
Counting rods
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Yang Hui (Pascal's) triangle, as depicted by Zhu Shijie in 1303, using rod numerals.
Counting rods
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rod numeral place value from Yongle Encyclopedia: 71,824
Counting rods
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Japanese counting board with grids
9.
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
Armenian numerals
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Numeral systems
10.
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
Ge'ez script
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Geʻez script used to advertise injera (እንጀራ) to the Ethiopian diaspora in the USA.
Ge'ez script
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Sign in Amharic using the Geʻez script at the Ethiopian millennium celebration
Ge'ez script
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Genesis 29.11–16 in Ge’ez
Ge'ez script
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Numeral systems
11.
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
Georgian numerals
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Numeral systems
Georgian numerals
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An inscription at the Motsameta monastery, dating the expansion of the convent to ჩყმვ (1846).
12.
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
Greek numerals
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Numeral systems
Greek numerals
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A Constantinopolitan map of the British Isles from Ptolemy 's Geography (c. 1300), using Greek numerals for its graticule: 52–63°N of the equator and 6–33°E from Ptolemy's Prime Meridian at the Fortunate Isles.
13.
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
Roman numerals
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Entrance to section LII (52) of the Colosseum, with numerals still visible
Roman numerals
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Numeral systems
Roman numerals
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A typical clock face with Roman numerals in Bad Salzdetfurth, Germany
Roman numerals
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An inscription on Admiralty Arch, London. The number is 1910, for which MCMX would be more usual
14.
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
Brahmi numerals
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Numeral systems
Brahmi numerals
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This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please improve this article by introducing more precise citations. (July 2012)
15.
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
Egyptian numerals
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Numeral systems
16.
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
Inuit numerals
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Numeral systems
17.
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
Radix
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Numeral systems
18.
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
Binary number
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Numeral systems
Binary number
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Arithmetic values represented by parts of the Eye of Horus
Binary number
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Gottfried Leibniz
Binary number
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George Boole
19.
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
Quaternary numeral system
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Numeral systems
20.
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
Quinary
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Numeral systems
21.
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
Vigesimal
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Numeral systems
Vigesimal
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The Maya numerals are a base-20 system.
22.
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
Non-standard positional numeral systems
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Numeral systems
23.
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
Bijective numeration
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Numeral systems
24.
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
Signed-digit representation
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Numeral systems
25.
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
Balanced ternary
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Numeral systems
26.
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
Factorial number system
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Numeral systems
Factorial number system
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Permutohedron graph showing permutations and their inversion vectors (compare version with factorial numbers) The arrows indicate the bitwise less or equal relation.
27.
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
Quater-imaginary base
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Numeral systems
28.
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
Non-integer representation
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Numeral systems
29.
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#
Mixed radix
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Numeral systems
30.
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
List of numeral systems
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Numeral systems
List of numeral systems
31.
Hindu-Arabic numeral system
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The Hindu–Arabic numeral system a positional decimal numeral system, is the most common system for the symbolic representation of numbers in the world. It was invented between the 1st and 4th centuries by Indian mathematicians, the system was adopted by Arabic mathematicians by the 9th century. It later spread to medieval Europe by the High Middle Ages, the system is based upon ten different glyphs. The symbols used to represent the system are in principle independent of the system itself, the glyphs in actual use are descended from Brahmi numerals and have split into various typographical variants since the Middle Ages. This numerical system is used worldwide today. The Hindu-Arabic numerals were invented by mathematicians in India, perso-Arabic mathematicians called them Hindu numerals. Later they came to be called Arabic numerals in Europe, because they were introduced to the West by Arab merchants, the Hindu–Arabic system is designed for positional notation in a decimal system. In a more developed form, positional notation also uses a decimal marker, in modern usage, this latter symbol is usually a vinculum. In this more developed form, the system can symbolize any rational number using only 13 symbols. Although generally found in text written with the Arabic abjad, numbers written with these numerals also place the most-significant digit to the left, the requisite changes in reading direction are found in text that mixes left-to-right writing systems with right-to-left systems. Various symbol sets are used to represent numbers in the Hindu–Arabic numeral system, the Arabic–Indic or Eastern Arabic numerals used with Arabic script, developed primarily in what is now Iraq. A variant of the Eastern Arabic numerals is used in Persian, the Hindu numerals in use with scripts of the Brahmic family in India and Southeast Asia. Each of the roughly dozen major scripts of India has its own numeral glyphs, as in many numbering systems, the numbers 1,2, and 3 represent simple tally marks,1 being a single line,2 being two lines and 3 being three lines. After three, numbers tend to more complex symbols. Theorists believe that this is because it becomes difficult to count objects past three. The Brahmi numerals at the basis of the system predate the Common Era and they replaced the earlier Kharosthi numerals used since the 4th century BC. Brahmi and Kharosthi numerals were used alongside one another in the Maurya Empire period, buddhist inscriptions from around 300 BC use the symbols that became 1,4 and 6. One century later, their use of the symbols that became 2,4,6,7 and 9 was recorded
Hindu-Arabic numeral system
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Numeral systems
Hindu-Arabic numeral system
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Arabic and Western Arabic numerals on a road sign in Abu Dhabi
Hindu-Arabic numeral system
Hindu-Arabic numeral system
32.
Decimal
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This article aims to be an accessible introduction. For the mathematical definition, see Decimal representation, the decimal numeral system has ten as its base, which, in decimal, is written 10, as is the base in every positional numeral system. It is the base most widely used by modern civilizations. Decimal fractions have terminating decimal representations and other fractions have repeating decimal representations, Decimal notation is the writing of numbers in a base-ten numeral system. Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, Roman numerals have symbols for the decimal powers and secondary symbols for half these values. Brahmi numerals have symbols for the nine numbers 1–9, the nine decades 10–90, plus a symbol for 100, Chinese numerals have symbols for 1–9, and additional symbols for powers of ten, which in modern usage reach 1072. Positional decimal systems include a zero and use symbols for the ten values to represent any number, positional notation uses positions for each power of ten, units, tens, hundreds, thousands, etc. The position of each digit within a number denotes the multiplier multiplied with that position has a value ten times that of the position to its right. There were at least two independent sources of positional decimal systems in ancient civilization, the Chinese counting rod system. Ten is the number which is the count of fingers and thumbs on both hands, the English word digit as well as its translation in many languages is also the anatomical term for fingers and toes. In English, decimal means tenth, decimate means reduce by a tenth, however, the symbols used in different areas are not identical, for instance, Western Arabic numerals differ from the forms used by other Arab cultures. A decimal fraction is a fraction the denominator of which is a power of ten. g, Decimal fractions 8/10, 1489/100, 24/100000, and 58900/10000 are expressed in decimal notation as 0.8,14.89,0.00024,5.8900 respectively. In English-speaking, some Latin American and many Asian countries, a period or raised period is used as the separator, in many other countries, particularly in Europe. The integer part, or integral part of a number is the part to the left of the decimal separator. The part from the separator to the right is the fractional part. It is usual for a number that consists only of a fractional part to have a leading zero in its notation. Any rational number with a denominator whose only prime factors are 2 and/or 5 may be expressed as a decimal fraction and has a finite decimal expansion. 1/2 =0.5 1/20 =0.05 1/5 =0.2 1/50 =0.02 1/4 =0.25 1/40 =0.025 1/25 =0.04 1/8 =0.125 1/125 =0.008 1/10 =0
Decimal
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The world's earliest decimal multiplication table was made from bamboo slips, dating from 305 BC, during the Warring States period in China.
Decimal
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Numeral systems
Decimal
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Ten fingers on two hands, the possible starting point of the decimal counting.
Decimal
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Diagram of the world's earliest decimal multiplication table (c. 305 BC) from the Warring States period
33.
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
Number
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The number 605 in Khmer numerals, from an inscription from 683 AD. An early use of zero as a decimal figure.
Number
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Subsets of the complex numbers.
34.
Hindu-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
Hindu-Arabic numerals
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Numeral systems
Hindu-Arabic numerals
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Arabic numerals sans-serif
Hindu-Arabic numerals
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The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.
Hindu-Arabic numerals
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Modern-day Arab telephone keypad with two forms of Arabic numerals: Western Arabic/European numerals on the left and Eastern Arabic numerals on the right
35.
Latin
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Latin is a classical language belonging to the Italic branch of the Indo-European languages. The Latin alphabet is derived from the Etruscan and Greek alphabets, Latin was originally spoken in Latium, in the Italian Peninsula. Through the power of the Roman Republic, it became the dominant language, Vulgar Latin developed into the Romance languages, such as Italian, Portuguese, Spanish, French, and Romanian. Latin, Italian and French have contributed many words to the English language, Latin and Ancient Greek roots are used in theology, biology, and medicine. By the late Roman Republic, Old Latin had been standardised into Classical Latin, Vulgar Latin was the colloquial form spoken during the same time and attested in inscriptions and the works of comic playwrights like Plautus and Terence. Late Latin is the language from the 3rd century. Later, Early Modern Latin and Modern Latin evolved, Latin was used as the language of international communication, scholarship, and science until well into the 18th century, when it began to be supplanted by vernaculars. Ecclesiastical Latin remains the language of the Holy See and the Roman Rite of the Catholic Church. Today, many students, scholars and members of the Catholic clergy speak Latin fluently and it is taught in primary, secondary and postsecondary educational institutions around the world. The language has been passed down through various forms, some inscriptions have been published in an internationally agreed, monumental, multivolume series, the Corpus Inscriptionum Latinarum. Authors and publishers vary, but the format is about the same, volumes detailing inscriptions with a critical apparatus stating the provenance, the reading and interpretation of these inscriptions is the subject matter of the field of epigraphy. The works of several hundred ancient authors who wrote in Latin have survived in whole or in part and they are in part the subject matter of the field of classics. The Cat in the Hat, and a book of fairy tales, additional resources include phrasebooks and resources for rendering everyday phrases and concepts into Latin, such as Meissners Latin Phrasebook. The Latin influence in English has been significant at all stages of its insular development. From the 16th to the 18th centuries, English writers cobbled together huge numbers of new words from Latin and Greek words, dubbed inkhorn terms, as if they had spilled from a pot of ink. Many of these words were used once by the author and then forgotten, many of the most common polysyllabic English words are of Latin origin through the medium of Old French. Romance words make respectively 59%, 20% and 14% of English, German and those figures can rise dramatically when only non-compound and non-derived words are included. Accordingly, Romance words make roughly 35% of the vocabulary of Dutch, Roman engineering had the same effect on scientific terminology as a whole
Latin
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Latin inscription, in the Colosseum
Latin
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Julius Caesar 's Commentarii de Bello Gallico is one of the most famous classical Latin texts of the Golden Age of Latin. The unvarnished, journalistic style of this patrician general has long been taught as a model of the urbane Latin officially spoken and written in the floruit of the Roman republic.
Latin
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A multi-volume Latin dictionary in the University Library of Graz
Latin
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Latin and Ancient Greek Language - Culture - Linguistics at Duke University in 2014.
36.
Gray code
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The reflected binary code, also known as Gray code after Frank Gray, is a binary numeral system where two successive values differ in only one bit. The reflected binary code was designed to prevent spurious output from electromechanical switches. Today, Gray codes are used to facilitate error correction in digital communications such as digital terrestrial television. Bell Labs researcher Frank Gray introduced the term reflected binary code in his 1947 patent application and he derived the name from the fact that it may be built up from the conventional binary code by a sort of reflection process. The code was named after Gray by others who used it. Two different 1953 patent applications use Gray code as a name for the reflected binary code, one of those also lists minimum error code. A1954 patent application refers to the Bell Telephone Gray code, many devices indicate position by closing and opening switches. In the transition between the two shown above, all three switches change state. In the brief period while all are changing, the switches will read some spurious position, even without keybounce, the transition might look like 011 —001 —101 —100. When the switches appear to be in position 001, the observer cannot tell if that is the real position 001, if the output feeds into a sequential system, possibly via combinational logic, then the sequential system may store a false value. This is called the property of a Gray code. In the standard Gray coding the least significant bit follows a pattern of 2 on,2 off, the next digit a pattern of 4 on,4 off. These codes are known as single-distance codes, reflecting the Hamming distance of 1 between adjacent codes. Reflected binary codes were applied to mathematical puzzles before they became known to engineers, martin Gardner wrote a popular account of the Gray code in his August 1972 Mathematical Games column in Scientific American. The French engineer Émile Baudot used Gray codes in telegraphy in 1878 and he received the French Legion of Honor medal for his work. The Gray code is attributed, incorrectly, to Elisha Gray. The method and apparatus were patented in 1953 and the name of Gray stuck to the codes. The PCM tube apparatus that Gray patented was made by Raymond W. Sears of Bell Labs, working with Gray and William M. Goodall, Gray codes are used in position encoders, in preference to straightforward binary encoding
Gray code
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A Gray code absolute rotary encoder with 13 tracks. At the top can be seen the housing, interrupter disk, and light source; at the bottom can be seen the sensing element and support components.
Gray code
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Gray's patent introduces the term "reflected binary code"
37.
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
Exponentiation
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Graphs of y = b x for various bases b: base 10 (green), base e (red), base 2 (blue), and base 1 / 2 (cyan). Each curve passes through the point (0, 1) because any nonzero number raised to the power of 0 is 1. At x = 1, the value of y equals the base because any number raised to the power of 1 is the number itself.
38.
Statistics
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Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data. In applying statistics to, e. g. a scientific, industrial, or social problem, populations can be diverse topics such as all people living in a country or every atom composing a crystal. Statistics deals with all aspects of data including the planning of data collection in terms of the design of surveys, statistician Sir Arthur Lyon Bowley defines statistics as Numerical statements of facts in any department of inquiry placed in relation to each other. When census data cannot be collected, statisticians collect data by developing specific experiment designs, representative sampling assures that inferences and conclusions can safely extend from the sample to the population as a whole. In contrast, an observational study does not involve experimental manipulation, inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena. A standard statistical procedure involves the test of the relationship between two data sets, or a data set and a synthetic data drawn from idealized model. A hypothesis is proposed for the relationship between the two data sets, and this is compared as an alternative to an idealized null hypothesis of no relationship between two data sets. Rejecting or disproving the hypothesis is done using statistical tests that quantify the sense in which the null can be proven false. Working from a hypothesis, two basic forms of error are recognized, Type I errors and Type II errors. Multiple problems have come to be associated with this framework, ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis, measurement processes that generate statistical data are also subject to error. Many of these errors are classified as random or systematic, the presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems. Statistics continues to be an area of research, for example on the problem of how to analyze Big data. Statistics is a body of science that pertains to the collection, analysis, interpretation or explanation. Some consider statistics to be a mathematical science rather than a branch of mathematics. While many scientific investigations make use of data, statistics is concerned with the use of data in the context of uncertainty, mathematical techniques used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure-theoretic probability theory. In applying statistics to a problem, it is practice to start with a population or process to be studied. Populations can be diverse topics such as all living in a country or every atom composing a crystal. Ideally, statisticians compile data about the entire population and this may be organized by governmental statistical institutes
Statistics
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Scatter plots are used in descriptive statistics to show the observed relationships between different variables.
Statistics
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More probability density is found as one gets closer to the expected (mean) value in a normal distribution. Statistics used in standardized testing assessment are shown. The scales include standard deviations, cumulative percentages, percentile equivalents, Z-scores, T-scores, standard nines, and percentages in standard nines.
Statistics
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Gerolamo Cardano, the earliest pioneer on the mathematics of probability.
Statistics
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Karl Pearson, a founder of mathematical statistics.
39.
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
Rational number
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A diagram showing a representation of the equivalent classes of pairs of integers
40.
Prime factor
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In number theory, the prime factors of a positive integer are the prime numbers that divide that integer exactly. The prime factorization of an integer is a list of the integers prime factors, together with their multiplicities. The fundamental theorem of arithmetic says that every integer has a single unique prime factorization. To shorten prime factorizations, factors are expressed in powers. For example,360 =2 ×2 ×2 ×3 ×3 ×5 =23 ×32 ×5, in which the factors 2,3 and 5 have multiplicities of 3,2 and 1, respectively. For a prime factor p of n, the multiplicity of p is the largest exponent a for which pa divides n exactly. For a positive n, the number of prime factors of n. Perfect square numbers can be recognized by the fact all of their prime factors have even multiplicities. For example, the number 144 has the prime factors 144 =2 ×2 ×2 ×2 ×3 ×3 =24 ×32. These can be rearranged to make the more visible,144 =2 ×2 ×2 ×2 ×3 ×3 = × =2 =2. Because every prime factor appears a number of times, the original number can be expressed as the square of some smaller number. In the same way, perfect cube numbers will have prime factors whose multiplicities are multiples of three, and so on, positive integers with no prime factors in common are said to be coprime. Two integers a and b can also be defined as if their greatest common divisor gcd =1. Euclids algorithm can be used to determine whether two integers are coprime without knowing their prime factors, the runs in a time that is polynomial in the number of digits involved. The integer 1 is coprime to every integer, including itself. This is because it has no prime factors, it is the empty product and this implies that gcd =1 for any b ≥1. The function, ω, represents the number of prime factors of n, while the function, Ω. If n = ∏ i =1 ω p i α i, for example,24 =23 ×31, so ω =2 and Ω =3 +1 =4
Prime factor
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This image demonstrates how to find the prime factorization of 864. A shorthand way of writing the resulting prime factors is 2 5 × 3 3
41.
Long division
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In arithmetic, long division is a standard division algorithm suitable for dividing multidigit numbers that is simple enough to perform by hand. It breaks down a division problem into a series of easier steps, as in all division problems, one number, called the dividend, is divided by another, called the divisor, producing a result called the quotient. It enables computations involving arbitrarily large numbers to be performed by following a series of simple steps, the abbreviated form of long division is called short division, which is almost always used instead of long division when the divisor has only one digit. Chunking is a less-efficient form of long division which may be easier to understand, while related algorithms have existed since the 12th century AD, the specific algorithm in modern use was introduced by Henry Briggs c.1600 AD. In English-speaking countries, long division does not use the division slash ⟨∕⟩ or obelus ⟨÷⟩ signs, the divisor is separated from the dividend by a right parenthesis ⟨)⟩ or vertical bar ⟨|⟩, the dividend is separated from the quotient by a vinculum. The combination of two symbols is sometimes known as a long division symbol or division bracket. It developed in the 18th century from an earlier single-line notation separating the dividend from the quotient by a left parenthesis, the process is begun by dividing the left-most digit of the dividend by the divisor. The quotient becomes the first digit of the result, and the remainder is calculated and this remainder carries forward when the process is repeated on the following digit of the dividend. When all digits have been processed and no remainder is left, an example is shown below, representing the division of 500 by 4. The largest number that the divisor 4 can be multiplied by without exceeding 5 is 1, next the 4 under the 5 is subtracted from the 5 to get the remainder,1, which is placed under the 4 under the 5. This remainder 1 is necessarily smaller than the divisor 4, next the first as-yet unused digit in the dividend, in this case the first digit 0 after the 5, is copied directly underneath itself and next to the remainder 1, to form the number 10. This remainder 2 is necessarily smaller than the divisor 4, the next digit of the dividend is copied directly below itself and next to the remainder 2, to form 20. Then the largest number by which the divisor 4 can be multiplied without exceeding 20 is ascertained, then this new quotient digit 5 is multiplied by the divisor 4 to get 20, which is written at the bottom below the existing 20. Then 20 is subtracted from 20, yielding 0, which is written below the 20 and we know we are done now because two things are true, there are no more digits to bring down from the dividend, and the last subtraction result was 0. If the last remainder when we ran out of dividend digits had been something other than 0, there would have been two possible courses of action. Or, we could extend the dividend by writing it as, say,500.000. and continue the process, in order to get a decimal answer, as in the following example. This example also illustrates that, at the beginning of the process, since the first digit 1 is less than the divisor 4, the first step is instead performed on the first two digits 12. Similarly, if the divisor were 13, one would perform the first step on 127 rather than 12 or 1, find the location of all decimal points in the dividend n and divisor m
Long division
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An example of long division performed without a calculator.
42.
Geometric series
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In mathematics, a geometric series is a series with a constant ratio between successive terms. For example, the series 12 +14 +18 +116 + ⋯ is geometric, Geometric series are among the simplest examples of infinite series with finite sums, although not all of them have this property. Historically, geometric series played an important role in the development of calculus. Geometric series are used throughout mathematics, and they have important applications in physics, engineering, biology, economics, computer science, queueing theory, the terms of a geometric series form a geometric progression, meaning that the ratio of successive terms in the series is constant. This relationship allows for the representation of a series using only two terms, r and a. The term r is the ratio, and a is the first term of the series. In the case above, where r is one half, the series has the sum one, if r is greater than one or less than minus one the terms of the series become larger and larger in magnitude. The sum of the terms also gets larger and larger, if r is equal to one, all of the terms of the series are the same. If r is one the terms take two values alternately. The sum of the oscillates between two values. This is a different type of divergence and again the series has no sum, see for example Grandis series,1 −1 +1 −1 + ···. The sum can be computed using the self-similarity of the series, consider the sum of the following geometric series, s =1 +23 +49 +827 + ⋯. This series has common ratio 2/3, if we multiply through by this common ratio, then the initial 1 becomes a 2/3, the 2/3 becomes a 4/9, and so on,23 s =23 +49 +827 +1681 + ⋯. This new series is the same as the original, except that the first term is missing, subtracting the new series s from the original series s cancels every term in the original but the first, s −23 s =1, so s =3. A similar technique can be used to evaluate any self-similar expression, as n goes to infinity, the absolute value of r must be less than one for the series to converge. When a =1, this can be simplified to 1 + r + r 2 + r 3 + ⋯ =11 − r, the formula also holds for complex r, with the corresponding restriction, the modulus of r is strictly less than one. Since = 1−rn+1 and rn+1 →0 for | r | <1, convergence of geometric series can also be demonstrated by rewriting the series as an equivalent telescoping series. Consider the function, g = r K1 − r
Geometric series
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Each of the purple squares has 1/4 of the area of the next larger square (1/2× 1/2 = 1/4, 1/4×1/4 = 1/16, etc.). The sum of the areas of the purple squares is one third of the area of the large square.
43.
Sign function
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In mathematics, the sign function or signum function is an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign function is represented as sgn. The signum function of a number x is defined as follows. Any real number can be expressed as the product of its value and its sign function. The numbers cancel and all we are left with is the sign of x, D | x | d x = sgn for x ≠0. The signum function is differentiable with derivative 0 everywhere except at 0, using this identity, it is easy to derive the distributional derivative, d sgn d x =2 d H d x =2 δ. The signum can also be using the Iverson bracket notation. The signum can also be using the floor and the absolute value functions. For k ≫1, an approximation of the sign function is sgn ≈ tanh . Another approximation is sgn ≈ x x 2 + ε2, which gets sharper as ε →0, note that this is the derivative of √x2 + ε2. This is inspired from the fact that the above is equal for all nonzero x if ε =0. See Heaviside step function – Analytic approximations, the signum function can be generalized to complex numbers as, sgn = z | z | for any complex number z except z =0. The signum of a complex number z is the point on the unit circle of the complex plane that is nearest to z. Then, for z ≠0, sgn = e i arg z and we then have, csgn = z z 2 = z 2 z. At real values of x, it is possible to define a generalized function–version of the function, ε such that ε2 =1 everywhere. This generalized signum allows construction of the algebra of generalized functions, absolute value Heaviside function Negative number Rectangular function Sigmoid function Step function Three-way comparison Zero crossing Modulus function
Sign function
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Signum function y = sgn(x)
44.
Golden mean 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 =
Golden mean base
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Numeral systems
45.
Gigabyte
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The gigabyte is a multiple of the unit byte for digital information. The prefix giga means 109 in the International System of Units, the unit symbol for the gigabyte is GB. However, the term is used in some fields of computer science and information technology to denote 1073741824 bytes. The use of gigabyte may thus be ambiguous, to address this ambiguity, the International System of Quantities standardizes the binary prefixes which denote a series of integer powers of 1024. With these prefixes, a module that is labeled as having the size 1GB has one gibibyte of storage capacity. The term gigabyte is commonly used to mean either 10003 bytes or 10243 bytes, the latter binary usage originated as compromise technical jargon for byte multiples that needed to be expressed in a power of 2, but lacked a convenient name. As 1024 is approximately 1000, roughly corresponding to SI multiples, in 1998 the International Electrotechnical Commission published standards for binary prefixes, requiring that the gigabyte strictly denote 10003 bytes and gibibyte denote 10243 bytes. By the end of 2007, the IEC Standard had been adopted by the IEEE, EU, and NIST and this is the recommended definition by the International Electrotechnical Commission. The file manager of Mac OS X version 10.6 and later versions are an example of this usage in software. The binary definition uses powers of the base 2, as is the principle of binary computers. This usage is widely promulgated by some operating systems, such as Microsoft Windows in reference to computer memory and this definition is synonymous with the unambiguous unit gibibyte. Since the first disk drive, the IBM350, disk drive manufacturers expressed hard drive capacities using decimal prefixes, with the advent of gigabyte-range drive capacities, manufacturers based most consumer hard drive capacities in certain size classes expressed in decimal gigabytes, such as 500 GB. The exact capacity of a given model is usually slightly larger than the class designation. Practically all manufacturers of disk drives and flash-memory disk devices continue to define one gigabyte as 1000000000bytes. Some operating systems such as OS X express hard drive capacity or file size using decimal multipliers and this discrepancy causes confusion, as a disk with an advertised capacity of, for example,400 GB might be reported by the operating system as 372 GB, meaning 372 GiB. The JEDEC memory standards use IEEE100 nomenclature which quote the gigabyte as 1073741824bytes and this means that a 300 GB hard disk might be indicated variously as 300 GB,279 GB or 279 GiB, depending on the operating system. As storage sizes increase and larger units are used, these differences even more pronounced. Some legal challenges have been waged over this confusion such as a lawsuit against drive manufacturer Western Digital, Western Digital settled the challenge and added explicit disclaimers to products that the usable capacity may differ from the advertised capacity
Gigabyte
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This 2.5" hard drive can hold 500 GB of data.
46.
Abacus
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The exact origin of the abacus is still unknown. Today, abaci are often constructed as a frame with beads sliding on wires. The use of the word abacus dates before 1387 AD, when a Middle English work borrowed the word from Latin to describe a sandboard abacus, the Latin word came from Greek ἄβαξ abax which means something without base, and improperly, any piece of rectangular board or plank. Alternatively, without reference to ancient texts on etymology, it has suggested that it means a square tablet strewn with dust. Whereas the table strewn with dust definition is popular, there are those that do not place credence in this at all, Greek ἄβαξ itself is probably a borrowing of a Northwest Semitic, perhaps Phoenician, word akin to Hebrew ʾābāq, dust. The preferred plural of abacus is a subject of disagreement, with both abacuses and abaci in use, the user of an abacus is called an abacist. The period 2700–2300 BC saw the first appearance of the Sumerian abacus, some scholars point to a character from the Babylonian cuneiform which may have been derived from a representation of the abacus. Archaeologists have found ancient disks of various sizes that are thought to have used as counters. However, wall depictions of this instrument have not been discovered, during the Achaemenid Empire, around 600 BC the Persians first began to use the abacus. The earliest archaeological evidence for the use of the Greek abacus dates to the 5th century BC, also Demosthenes talked of the need to use pebbles for calculations too difficult for your head. The Greek abacus was a table of wood or marble, pre-set with small counters in wood or metal for mathematical calculations and this Greek abacus saw use in Achaemenid Persia, the Etruscan civilization, Ancient Rome and, until the French Revolution, the Western Christian world. A tablet found on the Greek island Salamis in 1846 AD, dates back to 300 BC and it is a slab of white marble 149 cm long,75 cm wide, and 4.5 cm thick, on which are 5 groups of markings. Below these lines is a space with a horizontal crack dividing it. Also from this frame the Darius Vase was unearthed in 1851. It was covered with pictures including a holding a wax tablet in one hand while manipulating counters on a table with the other. The earliest known documentation of the Chinese abacus dates to the 2nd century BC. The Chinese abacus, known as the suanpan, is typically 20 cm tall and it usually has more than seven rods. There are two beads on each rod in the deck and five beads each in the bottom for both decimal and hexadecimal computation
Abacus
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A Chinese abacus
Abacus
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Calculating-Table by Gregor Reisch: Margarita Philosophica, 1503. The woodcut shows Arithmetica instructing an algorist and an abacist (inaccurately represented as Boethius and Pythagoras). There was keen competition between the two from the introduction of the Algebra into Europe in the 12th century until its triumph in the 16th.
Abacus
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Copy of a Roman abacus
Abacus
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Japanese soroban
47.
Computer
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A computer is a device that can be instructed to carry out an arbitrary set of arithmetic or logical operations automatically. The ability of computers to follow a sequence of operations, called a program, such computers are used as control systems for a very wide variety of industrial and consumer devices. The Internet is run on computers and it millions of other computers. Since ancient times, simple manual devices like the abacus aided people in doing calculations, early in the Industrial Revolution, some mechanical devices were built to automate long tedious tasks, such as guiding patterns for looms. More sophisticated electrical machines did specialized analog calculations in the early 20th century, the first digital electronic calculating machines were developed during World War II. The speed, power, and versatility of computers has increased continuously and dramatically since then, conventionally, a modern computer consists of at least one processing element, typically a central processing unit, and some form of memory. The processing element carries out arithmetic and logical operations, and a sequencing, peripheral devices include input devices, output devices, and input/output devices that perform both functions. Peripheral devices allow information to be retrieved from an external source and this usage of the term referred to a person who carried out calculations or computations. The word continued with the same meaning until the middle of the 20th century, from the end of the 19th century the word began to take on its more familiar meaning, a machine that carries out computations. The Online Etymology Dictionary gives the first attested use of computer in the 1640s, one who calculates, the Online Etymology Dictionary states that the use of the term to mean calculating machine is from 1897. The Online Etymology Dictionary indicates that the use of the term. 1945 under this name, theoretical from 1937, as Turing machine, devices have been used to aid computation for thousands of years, mostly using one-to-one correspondence with fingers. The earliest counting device was probably a form of tally stick, later record keeping aids throughout the Fertile Crescent included calculi which represented counts of items, probably livestock or grains, sealed in hollow unbaked clay containers. The use of counting rods is one example, the abacus was initially used for arithmetic tasks. The Roman abacus was developed from used in Babylonia as early as 2400 BC. Since then, many forms of reckoning boards or tables have been invented. In a medieval European counting house, a checkered cloth would be placed on a table, the Antikythera mechanism is believed to be the earliest mechanical analog computer, according to Derek J. de Solla Price. It was designed to calculate astronomical positions and it was discovered in 1901 in the Antikythera wreck off the Greek island of Antikythera, between Kythera and Crete, and has been dated to circa 100 BC
Computer
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Computer
Computer
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Suanpan (the number represented on this abacus is 6,302,715,408)
Computer
Computer
48.
Binary numeral system
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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
Binary numeral system
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Numeral systems
Binary numeral system
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Gottfried Leibniz
Binary numeral system
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George Boole
49.
ENIAC
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ENIAC was amongst the earliest electronic general-purpose computers made. It was Turing-complete, digital, and could solve a large class of problems through reprogramming. ENIAC was formally dedicated at the University of Pennsylvania on February 15,1946 and was heralded as a Giant Brain by the press. This combination of speed and programmability allowed for thousands more calculations for problems, ENIACs design and construction was financed by the United States Army, Ordnance Corps, Research and Development Command, led by Major General Gladeon M. Barnes. The total cost was about $487,000, equivalent to $6,740,000 in 2016, ENIAC was designed by John Mauchly and J. Presper Eckert of the University of Pennsylvania, U. S. The team of design engineers assisting the development included Robert F. Shaw, Jeffrey Chuan Chu, Thomas Kite Sharpless, Frank Mural, Arthur Burks, Harry Huskey, in 1946, the researchers resigned from the University of Pennsylvania and formed the Eckert-Mauchly Computer Corporation. ENIAC was a computer, composed of individual panels to perform different functions. Twenty of these modules were accumulators which could not only add and subtract, numbers were passed between these units across several general-purpose buses. In order to achieve its high speed, the panels had to send and receive numbers, compute, save the answer and trigger the next operation, all without any moving parts. Key to its versatility was the ability to branch, it could trigger different operations, depending on the sign of a computed result. By the end of its operation in 1955, ENIAC contained 17,468 vacuum tubes,7200 crystal diodes,1500 relays,70,000 resistors,10,000 capacitors and approximately 5,000,000 hand-soldered joints. It weighed more than 30 short tons, was roughly 2.4 m ×0.9 m ×30 m in size, occupied 167 m2 and this power requirement led to the rumor that whenever the computer was switched on, lights in Philadelphia dimmed. Input was possible from an IBM card reader and an IBM card punch was used for output and these cards could be used to produce printed output offline using an IBM accounting machine, such as the IBM405. While ENIAC had no system to store memory in its inception, in 1953, a 100-word magnetic-core memory built by the Burroughs Corporation was added to ENIAC. ENIAC used ten-position ring counters to store digits, each digit required 36 vacuum tubes,10 of which were the dual triodes making up the flip-flops of the ring counter. ENIAC had 20 ten-digit signed accumulators, which used tens complement representation and it was possible to connect several accumulators to run simultaneously, so the peak speed of operation was potentially much higher, due to parallel operation. The other 9 units in ENIAC were the Initiating Unit, the Cycling Unit, the Master Programmer, the Reader, the Printer, the references by Rojas and Hashagen give more details about the times for operations, which differ somewhat from those stated above. The basic machine cycle was 200 microseconds, or 5,000 cycles per second for operations on the 10-digit numbers, in one of these cycles, ENIAC could write a number to a register, read a number from a register, or add/subtract two numbers
ENIAC
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ENIAC
ENIAC
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Glen Beck (background) and Betty Snyder (foreground) program ENIAC in BRL building 328. (U.S. Army photo)
ENIAC
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Cpl. Irwin Goldstein (foreground) sets the switches on one of ENIAC's function tables at the Moore School of Electrical Engineering. (U.S. Army photo) This photo has been artificially darkened, obscuring details such as the women who were present and the IBM equipment in use.
ENIAC
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A function table from ENIAC on display at Aberdeen Proving Ground museum.
50.
IBM 650
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The IBM650 Magnetic Drum Data-Processing Machine is one of IBMs early computers, and the world’s first mass-produced computer. It was announced in 1953 and in 1956 enhanced as the IBM650 RAMAC with the addition of up to four disk storage units, almost 2,000 systems were produced, the last in 1962. Support for the 650 and its component units was withdrawn in 1969, the 650 was a two-address, bi-quinary coded decimal computer, with memory on a rotating magnetic drum. Character support was provided by the units converting punched card alphabetical and special character encodings to/from a two-digit decimal code. The 650 was marketed to business, scientific and engineering users as well as to users of punched card machines who were upgrading from calculating punches, such as the IBM604, to computers. The IBM7070, announced 1958, was expected to be a successor to at least the 650. The IBM1620, introduced in 1959, addressed the lower end of the market, the UNIVAC Solid State was announced by Sperry Rand in December 1958 as a response to the 650. None of these had a 650 compatible instruction set, the basic 650 system consisted of three units, IBM650 Console Unit housed the magnetic drum storage, arithmetical device and the operators console. IBM655 Power Unit IBM533 or IBM537 Card Read Punch Unit The IBM533 had separate feeds for reading and punching, the IBM537 had one feed, thus could read and then punch into the same card. Words on the drums were organized in bands around the drum, fifty words per band, a word could be accessed when its location on the drum surface passed under the read/write heads during rotation. Because of this timing, the address in each instruction was the address of the next instruction. Instructions could then be interleaved, placing many at addresses that would be accessible when execution of the previous instruction was completed. Instructions read from the drum went to a program register, Data read from the drum went through a 10-digit distributor. The 650 had a 20-digit accumulator, divided into 10-digit lower and upper accumulators with a common sign, arithmetic was performed by a one-digit adder. The console, distributor, lower and upper accumulators were all addressable,8000,8001,8002,8003 respectively. Three four-digit index registers at addresses 8005 to 8007, drum addresses were indexed by adding 2000,4000 or 6000 to them, the 4000-word systems required transistorized read/write circuitry for the drum memory and were available before 1963. Floating point – arithmetic instructions supported an eight-digit mantissa and two-digit characteristic – MMMMMMMMCC, the 650 instructions consisted of a two-digit operation code, a four-digit data address and the four-digit address of the next instruction. The sign was ignored on the machine, but was used on machines with optional features
IBM 650
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Part of the first IBM 650 computer in Norway (1959), known as "EMMA". 650 Console Unit (right, an exterior side panel is missing), 533 Card Read Punch unit (middle, input-output). 655 Power Unit is missing. Punched card sorter (left, not part of the 650). Now at Norwegian Museum of Science and Technology in Oslo.
IBM 650
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An IBM 650 at Texas A&M University. The IBM 533 Card Read Punch unit is on the right.
IBM 650
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IBM 650 console panel, showing bi-quinary indicators. (At House for the History of IBM Data Processing(closed), Sindelfingen)
IBM 650
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Close-up of bi-quinary indicators
51.
Warring States
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The Warring States Period derives its name from the Record of the Warring States, a work compiled early in the Han dynasty. The political geography of the era was dominated by the Seven Warring States, namely, Qin, The State of Qin was in the far west, with its core in the Wei River Valley and Guanzhong. This geographical position offered protection from the states of the Central Plains, the Three Jins, Northeast of Qin, on the Shanxi plateau, were the three successor states of Jin. These were, Han, south, along the Yellow River, Zhao, the northernmost of the three. Qi, located in the east of China, centred on the Shandong Peninsula, described as east of Mount Tai, Chu, located in the south of China, with its core territory around the valleys of the Han River and, later, the Yangtze River. Yan, located in the northeast, centred on modern-day Beijing, late in the period Yan pushed northeast and began to occupy the Liaodong Peninsula Besides these seven major states, some minor states also survived into the period. Yue, On the southeast coast near Shanghai was the State of Yue, Sichuan, In the far southwest were the States of Ba and Shu. These were non-Zhou states that were conquered by Qin late in the period, in the Central Plains comprising much of modern-day Henan Province, many smaller city states survived as satellites of the larger states, though they were eventually to be absorbed as well. Zhongshan, Between the states of Zhao and Yan was the state of Zhongshan, the Spring and Autumn period was initiated by the eastward flight of the Zhou court. There is no one single incident or starting point for the Warring States era, some proposed starting points are as follows,481 BC, Proposed by Song-era historian Lü Zuqian, since it is the end of the Spring and Autumn Annals. 476–475 BC, The author, Sima Qian, of Records of the Grand Historian who chose the year of King Yuan of Zhou. 403 BC, The year when Han, Zhao and Wei were officially recognised as states by the Zhou court, author Sima Guang of Zizhi Tongjian tells us that the symbol of eroded Zhou authority should be taken as the start of the Warring States era. The Spring and Autumn period led to a few states gaining power at the expense of many others, during the Warring States period, many rulers claimed the Mandate of Heaven to justify their conquest of other states and spread their influence. Other major states also existed, such as Wu and Yue in the southeast, the last decades of the Spring and Autumn era were marked by increased stability, as the result of peace negotiations between Jin and Chu which established their respective spheres of influence. This situation ended with the partition of Jin, whereby the state was divided between the houses of Han, Zhao and Wei, and thus enabled the creation of the seven major warring states. This allowed other clans to gain fiefs and military authority, and decades of struggle led to the establishment of four major families. The Battle of Jinyang saw the allied Han, Zhao and Wei destroy the Zhi family, with this, they became the de facto rulers of most of Jins territory, though this situation would not be officially recognised until half a century later. The Jin division created a vacuum that enabled during the first 50 years expansion of Chu and Yue northward
Warring States
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History of China
Warring States
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Warring States about 350 BC
Warring States
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Tomb Guardian held at Birmingham Museum of Art
Warring States
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A jade -carved dragon garment ornament from the Warring States period
52.
Egyptian hieroglyphs
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Egyptian hieroglyphs were the formal writing system used in Ancient Egypt. It combined logographic, syllabic and alphabetic elements, with a total of some 1,000 distinct characters, cursive hieroglyphs were used for religious literature on papyrus and wood. The later hieratic and demotic Egyptian scripts are derived from hieroglyphic writing, the writing system continued to be used throughout the Late Period, as well as the Persian and Ptolemaic periods. Late survivals of hieroglyphic use are found well into the Roman period, with the closing of pagan temples in the 5th century, knowledge of hieroglyphic writing was lost, and the script remained undeciphered throughout the medieval and early modern period. The decipherment of hieroglyphs would only be solved in the 1820s by Jean-François Champollion, the word hieroglyph comes from the Greek adjective ἱερογλυφικός, a compound of ἱερός and γλύφω, supposedly a calque of an Egyptian phrase mdw·w-nṯr gods words. The glyphs themselves were called τὰ ἱερογλυφικὰ γράμματα the sacred engraved letters, the word hieroglyph has become a noun in English, standing for an individual hieroglyphic character. As used in the sentence, the word hieroglyphic is an adjective. Hieroglyphs emerged from the artistic traditions of Egypt. For example, symbols on Gerzean pottery from c.4000 BC have been argued to resemble hieroglyphic writing, proto-hieroglyphic symbol systems develop in the second half of the 4th millennium BC, such as the clay labels of a Predynastic ruler called Scorpion I recovered at Abydos in 1998. The first full sentence written in hieroglyphs so far discovered was found on a seal found in the tomb of Seth-Peribsen at Umm el-Qaab. There are around 800 hieroglyphs dating back to the Old Kingdom, Middle Kingdom, by the Greco-Roman period, there are more than 5,000. However, given the lack of evidence, no definitive determination has been made as to the origin of hieroglyphics in ancient Egypt. Since the 1990s, and discoveries such as the Abydos glyphs, as writing developed and became more widespread among the Egyptian people, simplified glyph forms developed, resulting in the hieratic and demotic scripts. These variants were more suited than hieroglyphs for use on papyrus. Hieroglyphic writing was not, however, eclipsed, but existed alongside the other forms, especially in monumental, the Rosetta Stone contains three parallel scripts – hieroglyphic, demotic, and Greek. Hieroglyphs continued to be used under Persian rule, and after Alexander the Greats conquest of Egypt, during the ensuing Ptolemaic and Roman periods. It appears that the quality of comments from Greek and Roman writers about hieroglyphs came about, at least in part. Some believed that hieroglyphs may have functioned as a way to distinguish true Egyptians from some of the foreign conquerors, another reason may be the refusal to tackle a foreign culture on its own terms, which characterized Greco-Roman approaches to Egyptian culture generally
Egyptian hieroglyphs
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A section of the Papyrus of Ani showing cursive hieroglyphs.
Egyptian hieroglyphs
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Hieroglyphs on a funerary stela in Manchester Museum
Egyptian hieroglyphs
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The Rosetta Stone in the British Museum
Egyptian hieroglyphs
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Hieroglyphs typical of the Graeco-Roman period
53.
Cretan hieroglyphs
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Cretan hieroglyphs are undeciphered hieroglyphs found on artefacts of early Bronze Age Crete, during the Minoan era. It predates Linear A by about a century, but the two writing systems continued to be used in parallel for most of their history, the seals and sealings represent about 307 distinct sign-groups, consisting all together of ±832 signs. The other inscriptions represent about 274 distinct sign-groups, consisting all together of ±723 signs, more documents have been published since then, such as, for example, from the Petras deposit. Definitive edition was published in 2010, the relation of the last three items with the script of the main corpus is uncertain. Some Cretan Hieroglyphic inscriptions were found on the island of Samothrace in the northeastern Aegean. It has been suggested there was an evolution of the hieroglyphs into the linear scripts. Also, some relations to Anatolian hieroglyphs have been suggested, the overlaps between the Cretan script and other scripts, such as the hieroglyphic scripts of Cyprus and the Hittite lands of Anatolia, may suggest. That they all evolved from an ancestor, a now-lost script perhaps originating in Syria. Symbol inventories have been compiled by Evans, Meijer, Olivier/Godart, the known corpus has been edited in 1996 as CHIC, listing a total of 314 items. The glyph inventory as presented by CHIC includes 96 syllabograms, ten of which double as logograms, there are also 23 logograms representing four levels of numerals, numerical fractions, and two types of punctuation. Many symbols have apparent Linear A counterparts, so that it is tempting to insert Linear B sound values, Cretan Writing in the Second Millennium B. C. World Archaeology,17, 377–389, doi,10. 1080/00438243.1986.9979977 Yule, Paul, Early Cretan Seals, marburger Studien zur Vor und Frühgeschichte, ISBN 3-8053-0490-0 W. C. Brice, Notes on the Cretan Hieroglyphic Script, I, the Clay Bar from Malia, H20, Kadmos 29 1-10. Brice, Cretan Hieroglyphs & Linear A, Kadmos 29 171-2, brice, Notes on the Cretan Hieroglyphic Script, III. The Inscriptions from Mallia Quarteir Mu, the Clay Bar from Knossos, P116, Kadmos 30 93-104. Brice, Notes on the Cretan Hieroglyphic Script, Kadmos 31, M. Civitillo, LA SCRITTURA GEROGLIFICA MINOICA SUI SIGILLI. Il messaggio della glittica protopalaziale, Biblioteca di Pasiphae XII, Pisa-Roma 2016, G. M. Facchetti La questione della scrittura «geroglifica cretese» dopo la recente edizione del corpus dei testi. Pasiphae, Rivista di filologia e antichita egee, the Cretan hieroglyphic script of the second millennium BC, description, analysis, function and decipherment perspectives
Cretan hieroglyphs
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A green jasper seal with Cretan hieroglyphs. 1800 BC
54.
Minoans
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The Minoan civilization was an Aegean Bronze Age civilization on the island of Crete and other Aegean islands which flourished from about 2600 to 1100 BC. It preceded the Mycenaean civilization of Ancient Greece, the civilization was rediscovered at the beginning of the 20th century through the work of British archaeologist Arthur Evans. It has been described as the earliest of its kind in Europe, the term Minoan, which refers to the mythical King Minos, originally described the pottery of the period. Minos was associated in Greek mythology with the labyrinth and the Minotaur, according to Homer, Crete once had 90 cities. The Minoan period saw trade between Crete and Aegean and Mediterranean settlements, particularly the Near East, traders and artists, the Minoan cultural influence reached beyond Crete to the Cyclades, Egypts Old Kingdom, copper-bearing Cyprus, Canaan and the Levantine coast, and Anatolia. Some of its best art is preserved in the city of Akrotiri on the island of Santorini, although the Minoan language and writing systems remain undecipherable and are subjects of academic dispute, they apparently conveyed a language entirely different from the later Greek. The reason for the end of the Minoan period is unclear, theories include Mycenaean invasions from mainland Greece, the term Minoan refers to the mythical King Minos of Knossos. Its origin is debated, but it is attributed to archeologist Arthur Evans. Minos was associated in Greek mythology with the labyrinth, which Evans identified with the site at Knossos. However, Karl Hoeck had already used the title Das Minoische Kreta in 1825 for volume two of his Kreta, this appears to be the first known use of the word Minoan to mean ancient Cretan, Evans said that applied it, not invented it. Hoeck, with no idea that the archaeological Crete had existed, had in mind the Crete of mythology, although Evans 1931 claim that the term was unminted before he used it was called a brazen suggestion by Karadimas and Momigliano, he coined its archaeological meaning. Instead of dating the Minoan period, archaeologists use two systems of relative chronology, the first, created by Evans and modified by later archaeologists, is based on pottery styles and imported Egyptian artifacts. Evans system divides the Minoan period into three eras, early, middle and late. These eras are subdivided—for example, Early Minoan I, II and III, another dating system, proposed by Greek archaeologist Nicolas Platon, is based on the development of architectural complexes known as palaces at Knossos, Phaistos, Malia and Kato Zakros. Platon divides the Minoan period into pre-, proto-, neo-, the relationship between the systems in the table includes approximate calendar dates from Warren and Hankey. The Thera eruption occurred during a phase of the LM IA period. Efforts to establish the volcanic eruptions date have been controversial, the eruption is identified as a natural event catastrophic for the culture, leading to its rapid collapse. Although stone-tool evidence exists that hominins may have reached Crete as early as 130,000 years ago, evidence for the first anatomically-modern human presence dates to 10, the oldest evidence of modern human habitation on Crete are pre-ceramic Neolithic farming-community remains which date to about 7000 BC
Minoans
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Minoan civilization
Minoans
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Minoan copper ingot.
Minoans
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Fresco showing three women who were possibly queens. [citation needed]
55.
Linear B
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Linear B is a syllabic script that was used for writing Mycenaean Greek, the earliest attested form of Greek. The script predates the Greek alphabet by several centuries, the oldest Mycenaean writing dates to about 1450 BC. It is descended from the older Linear A, an earlier script used for writing the Minoan language, as is the later Cypriot syllabary. Linear B, found mainly in the archives at Knossos, Cydonia, Pylos, Thebes and Mycenae. The succeeding period, known as the Greek Dark Ages, provides no evidence of the use of writing and it is also the only one of the prehistoric Aegean scripts to have been deciphered, by English architect and self-taught linguist Michael Ventris. Linear B consists of around 87 syllabic signs and over 100 ideographic signs and these ideograms or signifying signs symbolize objects or commodities. They have no value and are never used as word signs in writing a sentence. The application of Linear B appears to have been confined to administrative contexts, in all the thousands of clay tablets, a relatively small number of different hands have been detected,45 in Pylos and 66 in Knossos. It is possible that the script was used only by a guild of professional scribes who served the central palaces, once the palaces were destroyed, the script disappeared. Linear B has roughly 200 signs, divided into syllabic signs with phonetic values, the representations and naming of these signs have been standardized by a series of international colloquia starting with the first in Paris in 1956. Colloquia continue, the 13th occurred in 2010 in Paris, many of the signs are identical or similar to those in Linear A, however, Linear A encodes an as-yet unknown language, and it is uncertain whether similar signs had the same phonetic values. The grid developed during decipherment by Michael Ventris and John Chadwick of phonetic values for syllabic signs is shown below, initial consonants are in the leftmost column, vowels are in the top row beneath the title. The transcription of the syllable is listed next to the sign along with Bennetts identifying number for the sign preceded by an asterisk, in cases where the transcription of the sign remains in doubt, Bennetts number serves to identify the sign. The signs on the tablets and sealings often show considerable variation from each other, discovery of the reasons for the variation and possible semantic differences is a topic of ongoing debate in Mycenaean studies. Many of these were identified by the edition and are shown in the special values below. The second edition relates, It may be taken as axiomatic that there are no true homophones, the unconfirmed identifications of *34 and *35 as ai2 and ai3 were removed. Other values remain unknown, mainly because of scarcity of evidence concerning them, note that *34 and *35 are mirror images of each other but whether this graphic relationship indicates a phonetic one remains unconfirmed. In recent times, CIPEM inherited the authority of Bennett
Linear B
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Linear B
Linear B
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Linear B tablet discovered by Arthur Evans
Linear B
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Tablets
56.
Archimedes
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Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the scientists in classical antiquity. He was also one of the first to apply mathematics to physical phenomena, founding hydrostatics and statics and he is credited with designing innovative machines, such as his screw pump, compound pulleys, and defensive war machines to protect his native Syracuse from invasion. Archimedes died during the Siege of Syracuse when he was killed by a Roman soldier despite orders that he should not be harmed. Cicero describes visiting the tomb of Archimedes, which was surmounted by a sphere and a cylinder, unlike his inventions, the mathematical writings of Archimedes were little known in antiquity. Archimedes was born c.287 BC in the city of Syracuse, Sicily, at that time a self-governing colony in Magna Graecia. The date of birth is based on a statement by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years, in The Sand Reckoner, Archimedes gives his fathers name as Phidias, an astronomer about whom nothing is known. Plutarch wrote in his Parallel Lives that Archimedes was related to King Hiero II, a biography of Archimedes was written by his friend Heracleides but this work has been lost, leaving the details of his life obscure. It is unknown, for instance, whether he married or had children. During his youth, Archimedes may have studied in Alexandria, Egypt and he referred to Conon of Samos as his friend, while two of his works have introductions addressed to Eratosthenes. Archimedes died c.212 BC during the Second Punic War, according to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, the soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives an account of the death of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier. According to this story, Archimedes was carrying mathematical instruments, and was killed because the thought that they were valuable items. General Marcellus was reportedly angered by the death of Archimedes, as he considered him a valuable asset and had ordered that he not be harmed. Marcellus called Archimedes a geometrical Briareus, the last words attributed to Archimedes are Do not disturb my circles, a reference to the circles in the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier. This quote is given in Latin as Noli turbare circulos meos. The phrase is given in Katharevousa Greek as μὴ μου τοὺς κύκλους τάραττε
Archimedes
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Archimedes Thoughtful by Fetti (1620)
Archimedes
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Cicero Discovering the Tomb of Archimedes by Benjamin West (1805)
Archimedes
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Artistic interpretation of Archimedes' mirror used to burn Roman ships. Painting by Giulio Parigi.
Archimedes
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A sphere has 2/3 the volume and surface area of its circumscribing cylinder including its bases. A sphere and cylinder were placed on the tomb of Archimedes at his request. (see also: Equiareal map)
57.
Vedas
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The Vedas are a large body of knowledge texts originating in the ancient Indian subcontinent. Composed in Vedic Sanskrit, the texts constitute the oldest layer of Sanskrit literature, Hindus consider the Vedas to be apauruṣeya, which means not of a man, superhuman and impersonal, authorless. Vedas are also called śruti literature, distinguishing them from religious texts. The Veda, for orthodox Indian theologians, are considered revelations seen by ancient sages after intense meditation, in the Hindu Epic the Mahabharata, the creation of Vedas is credited to Brahma. The Vedic hymns themselves assert that they were created by Rishis, after inspired creativity. There are four Vedas, the Rigveda, the Yajurveda, the Samaveda, each Veda has been subclassified into four major text types – the Samhitas, the Aranyakas, the Brahmanas, and the Upanishads. Some scholars add a fifth category – the Upasanas, the various Indian philosophies and denominations have taken differing positions on the Vedas. Schools of Indian philosophy which cite the Vedas as their authority are classified as orthodox. Other śramaṇa traditions, such as Lokayata, Carvaka, Ajivika, Buddhism and Jainism, despite their differences, just like the texts of the śramaṇa traditions, the layers of texts in the Vedas discuss similar ideas and concepts. The Sanskrit word véda knowledge, wisdom is derived from the root vid- to know and this is reconstructed as being derived from the Proto-Indo-European root *u̯eid-, meaning see or know. The noun is from Proto-Indo-European *u̯eidos, cognate to Greek εἶδος aspect, not to be confused is the homonymous 1st and 3rd person singular perfect tense véda, cognate to Greek οἶδα oida I know. Root cognates are Greek ἰδέα, English wit, etc, the Sanskrit term veda as a common noun means knowledge. The term in some contexts, such as hymn 10.93.11 of the Rigveda, means obtaining or finding wealth, property, a related word Vedena appears in hymn 8.19.5 of the Rigveda. It was translated by Ralph T. H. Griffith as ritual lore, as studying the Veda by the 14th century Indian scholar Sayana, as bundle of grass by Max Müller, Vedas are called Maṛai or Vaymoli in parts of South India. Marai literally means hidden, a secret, mystery, in some south Indian communities such as Iyengars, the word Veda includes the Tamil writings of the Alvar saints, such as Divya Prabandham, for example Tiruvaymoli. The Vedas are among the oldest sacred texts, the Samhitas date to roughly 1700–1100 BC, and the circum-Vedic texts, as well as the redaction of the Samhitas, date to c. 1000-500 BC, resulting in a Vedic period, spanning the mid 2nd to mid 1st millennium BC, or the Late Bronze Age, Michael Witzel gives a time span of c.1500 to c. Witzel makes special reference to the Near Eastern Mitanni material of the 14th century BC the only record of Indo-Aryan contemporary to the Rigvedic period
Vedas
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Rigveda (padapatha) manuscript in Devanagari
58.
Simon Stevin
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Simon Stevin, sometimes called Stevinus, was a Flemish/Dutch/Netherlandish mathematician, physicist and engineer. He was active in a great areas of science and engineering. Very little is known with certainty about Stevins life and what we know is mostly inferred from other recorded facts, the exact birth date and the date and place of his death are uncertain. It is assumed he was born in Bruges since he enrolled at Leiden University under the name Simon Stevinus Brugensis and his name is usually written as Stevin, but some documents regarding his father use the spelling Stevijn. This is a normal spelling shift in 16th century Dutch and he was born around the year 1548 to unmarried parents, Anthonis Stevin and Catelyne van der Poort. His father is believed to have been a son of a mayor of Veurne. While Simons father was not mentioned in the book of burghers, many other Stevins were later mentioned in the Poorterboeken. Simon Stevins mother Cathelijne was the daughter of a family from Ypres. Her father Hubert was a poorter of Bruges, Simons mother Cathelijne later married Joost Sayon who was involved in the carpet and silk trade and a member of the schuttersgilde Sint-Sebastiaan. Through her marriage Cathelijne became a member of a family of Calvinists and it is believed that Stevin grew up in a relatively affluent environment and enjoyed a good education. He was likely educated at a Latin school in his hometown, Stevin left Bruges in 1571 apparently without a particular destination. Stevin was most likely a Calvinist since a Catholic would likely not have risen to the position of trust he later occupied with Maurice, Prince of Orange and it is assumed that he left Bruges to escape the religious persecution of Protestants by the Spanish rulers. Based on references in his work Wisconstighe Ghedaechtenissen, it has been inferred that he must have moved first to Antwerp where he began his career as a merchants clerk. Some biographers mention that he travelled to Prussia, Poland, Denmark, Norway and Sweden and other parts of Northern Europe and it is possible that he completed these travels over a longer period of time. In 1577 Simon Stevin returned to Bruges and was appointed city clerk by the aldermen of Bruges and he worked in the office of Jan de Brune of the Brugse Vrije, the castellany of Bruges. Why he had returned to Bruges in 1577 is not clear and it may have been related to the political events of that period. Bruges was the scene of religious conflict. Catholics and Calvinists alternately controlled the government of the city and they usually opposed each other but would occasionally collaborate in order to counteract the dictates of King Philip II of Spain
Simon Stevin
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Simon Stevin
Simon Stevin
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Statue of Simon Stevin by Eugène Simonis, on the Simon Stevinplein (nl) in Bruges
Simon Stevin
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Statue of Stevin (detail)
Simon Stevin
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Statue (detail): Inclined plane diagram
59.
Al Khwarizmi
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Muḥammad ibn Mūsā al-Khwārizmī, formerly Latinized as Algoritmi, was a Persian mathematician, astronomer, and geographer during the Abbasid Caliphate, a scholar in the House of Wisdom in Baghdad. In the 12th century, Latin translations of his work on the Indian numerals introduced the decimal number system to the Western world. Al-Khwārizmīs The Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear and he is often considered one of the fathers of algebra. He revised Ptolemys Geography and wrote on astronomy and astrology, some words reflect the importance of al-Khwārizmīs contributions to mathematics. Algebra is derived from al-jabr, one of the two operations he used to solve quadratic equations, algorism and algorithm stem from Algoritmi, the Latin form of his name. His name is also the origin of guarismo and of algarismo, few details of al-Khwārizmīs life are known with certainty. He was born in a Persian family and Ibn al-Nadim gives his birthplace as Khwarezm in Greater Khorasan, muhammad ibn Jarir al-Tabari gives his name as Muḥammad ibn Musá al-Khwārizmiyy al-Majūsiyy al-Quṭrubbaliyy. The epithet al-Qutrubbulli could indicate he might instead have come from Qutrubbul and this would not be worth mentioning if a series of errors concerning the personality of al-Khwārizmī, occasionally even the origins of his knowledge, had not been made. Recently, G. J. Toomer. with naive confidence constructed an entire fantasy on the error which cannot be denied the merit of amusing the reader. Regarding al-Khwārizmīs religion, Toomer writes, Another epithet given to him by al-Ṭabarī, al-Majūsī, Ibn al-Nadīms Kitāb al-Fihrist includes a short biography on al-Khwārizmī together with a list of the books he wrote. Al-Khwārizmī accomplished most of his work in the period between 813 and 833, douglas Morton Dunlop suggests that it may have been possible that Muḥammad ibn Mūsā al-Khwārizmī was in fact the same person as Muḥammad ibn Mūsā ibn Shākir, the eldest of the three Banū Mūsā. Al-Khwārizmīs contributions to mathematics, geography, astronomy, and cartography established the basis for innovation in algebra, on the Calculation with Hindu Numerals written about 825, was principally responsible for spreading the Hindu–Arabic numeral system throughout the Middle East and Europe. It was translated into Latin as Algoritmi de numero Indorum, al-Khwārizmī, rendered as Algoritmi, led to the term algorithm. Some of his work was based on Persian and Babylonian astronomy, Indian numbers, al-Khwārizmī systematized and corrected Ptolemys data for Africa and the Middle East. Another major book was Kitab surat al-ard, presenting the coordinates of places based on those in the Geography of Ptolemy but with improved values for the Mediterranean Sea, Asia and he also wrote on mechanical devices like the astrolabe and sundial. He assisted a project to determine the circumference of the Earth and in making a map for al-Mamun. When, in the 12th century, his works spread to Europe through Latin translations, the Compendious Book on Calculation by Completion and Balancing is a mathematical book written approximately 830 CE. The term algebra is derived from the name of one of the operations with equations described in this book
Al Khwarizmi
Al Khwarizmi
Al Khwarizmi
60.
Indo-Aryan languages
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The Indo-Aryan or Indic languages are the dominant language family of the Indian subcontinent. They constitute a branch of the Indo-Iranian languages, itself a branch of the Indo-European language family, Indo-Aryan speakers form about one half of all Indo-European speakers, and more than half of all Indo-European languages recognized by Ethnologue. While the languages are spoken in South Asia, pockets of Indo-Aryan languages are found to be spoken in Europe. The largest in terms of speakers are Hindustani, Bengali, Punjabi. Proto-Indo-Aryan, or sometimes Proto-Indic, is the reconstructed proto-language of the Indo-Aryan languages and it is intended to reconstruct the language of the Proto-Indo-Aryans. Proto-Indo-Aryan is meant to be the predecessor of Old Indo-Aryan which is attested as Vedic. Despite the great archaicity of Vedic, however, the other Indo-Aryan languages preserve a number of archaic features lost in Vedic. Vedic has been used in the ancient preserved religious hymns, the canon of Hinduism known as the Vedas. Mitanni-Aryan is of age to the language of the Rigveda. The language of the Vedas – commonly referred to as Vedic Sanskrit by modern scholars – is only marginally different from Proto-Indo-Aryan the proto-language of the Indo-Aryan languages. From the Vedic, Sanskrit developed as the language of culture, science and religion, as well as the court, theatre. Sanskrit is, by convention, referred to by scholars as Classical Sanskrit in contra-distinction to the so-called Vedic Sanskrit. Outside the learned sphere of Sanskrit, vernacular dialects continued to evolve, the oldest attested Prakrits are the Buddhist and Jain canonical languages Pali and Ardha Magadhi, respectively. By medieval times, the Prakrits had diversified into various Middle Indo-Aryan dialects, Apabhramsa is the conventional cover term for transitional dialects connecting late Middle Indo-Aryan with early Modern Indo-Aryan, spanning roughly the 6th to 13th centuries. Some of these dialects showed considerable literary production, the Sravakachar of Devasena is now considered to be the first Hindi book, the next major milestone occurred with the Muslim conquests on the Indian subcontinent in the 13th–16th centuries. Under the flourishing Turco-Mongol Mughal empire, Persian became very influential as the language of prestige of the Islamic courts due to adoptation of the language by the Mughal emperors. However, Persian was soon displaced by Hindustani and this Indo-Aryan language is a combination with Persian, Arabic, and Turkic elements in its vocabulary, with the grammar of the local dialects. The two largest languages that formed from Apabhramsa were Bengali and Hindustani, others include Sindhi, Gujarati, Odia, Marathi, the Indo-Aryan languages of Northern India and Pakistan form a dialect continuum
Indo-Aryan languages
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Dogri–Kangri region
Indo-Aryan languages
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1978 map showing Geographical distribution of the major Indo-Aryan languages. (Urdu is included under Hindi. Romani, Domari, and Lomavren are outside the scope of the map.)
61.
Hungarian language
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Hungarian is the official language of Hungary and one of the 24 official languages of the European Union. Outside Hungary it is spoken by communities of Hungarian people in neighbouring countries. Like Finnish and Estonian, it belongs to the Uralic language family, its closest relatives being Mansi and it is one of several European languages not part of the Indo-European languages, and the most widely-spoken European language that does not belong to the Indo-European family. The Hungarian name for the language is magyar or magyar nyelv, the word Magyar is used as an English and Hungarian word to refer to Hungarian people as an ethnic group. Hungarian is a member of the Uralic language family, the name of Hungary could be a result of regular sound changes of Ungrian/Ugrian, and the fact that the Eastern Slavs referred to Hungarians as Ǫgry/Ǫgrove seemed to confirm that. Current literature favors the hypothesis that it comes from the name of the Turkic tribe Onogur, there are numerous regular sound correspondences between Hungarian and the other Ugric languages. For example, Hungarian /aː/ corresponds to Khanty /o/ in certain positions, for example, Hungarian ház house vs. Khanty xot house, and Hungarian száz hundred vs. Khanty sot hundred. The distance between the Ugric and Finnic languages is greater, but the correspondences are also regular, during the later half of the 19th century, a competing hypothesis proposed a Turkic affinity of Hungarian. Following an academic debate known as Az ugor-török háború, the Finno-Ugric hypothesis was concluded the sounder of the two, foremost based on work by the German linguist Josef Budenz. The traditional view argues that the Hungarian language separated from its Ugric relatives in the first half of the 1st millennium b. c. e. in western Siberia, east of the southern Urals. The Hungarians gradually changed their lifestyle from settled hunters to nomadic pastoralists, in Hungarian, Iranian loans date back to the time immediately following the breakup of Ugric and probably span well over a millennium. Among these include tehén ‘cow’, tíz ‘ten’, tej ‘milk’, increasing archaeological evidence from present-day southern Bashkortostan found in the previous decades confirms the existence of Hungarian settlements between the Volga River and Ural Mountains. The Onogurs later had a influence on the language, especially between the 5th-9th centuries. This layer of Turkic loans is large and varied, and includes words borrowed from Oghur Turkic, e. g. borjú ‘calf’, dél ‘noon, many words related to agriculture, to state administration or even to family relations have such backgrounds. Hungarian syntax and grammar were not influenced in a dramatic way during these 300 years. After the arrival of the Hungarians into the Carpathian Basin the language came into contact with different speech communities, Turkic loans from this period come mainly from the Pechenegs and Cumanians who settled in Hungary during the 12th-13th centuries, e. g. koboz ‘cobza’, komondor ‘mop dog’. Hungarian borrowed many words from especially the neighbouring Slavic languages, in exchange, these languages also borrowed words from Hungarian, e. g. Serbo-Croatian ašov from Hung ásó ‘spade’. Approximately 1. 6% of the Romanian lexicon is of Hungarian origin, on the basis of the growing genetic evidence, the accepted origin theory is contested by geneticists too
Hungarian language
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Hungarian keyboard
Hungarian language
Hungarian language
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Funeral Sermon and Prayer, 12th century
Hungarian language
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A page from the first book written completely in Hungarian from 1533
62.
Vietnamese language
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Vietnamese /ˌviɛtnəˈmiːz/ is an Austroasiatic language that originated in the north of modern-day Vietnam, where it is the national and official language. It is the language of the Vietnamese people, as well as a first or second language for the many ethnic minorities of Vietnam. As the result of Vietnamese emigration and cultural influence, Vietnamese speakers are found throughout the world, notably in East and Southeast Asia, North America, Australia, Vietnamese has also been officially recognized as a minority language in the Czech Republic. It is part of the Austroasiatic language family of which it has by far the most speakers, Vietnamese vocabulary has borrowings from Chinese, and it formerly used a modified set of Chinese characters called chữ nôm given vernacular pronunciation. The Vietnamese alphabet in use today is a Latin alphabet with diacritics for tones. As the national language, Vietnamese is spoken throughout Vietnam by ethnic Vietnamese, Vietnamese is also the native language of the Gin minority group in southern Guangxi Province in China. A significant number of speakers also reside in neighboring Cambodia. In the United States, Vietnamese is the sixth most spoken language, with over 1.5 million speakers and it is the third most spoken language in Texas, fourth in Arkansas and Louisiana, and fifth in California. Vietnamese is the seventh most spoken language in Australia, in France, it is the most spoken Asian language and the eighth most spoken immigrant language at home. Vietnamese is the official and national language of Vietnam. It is the first language of the majority of the Vietnamese population, in the Czech Republic, Vietnamese has been recognized as one of 14 minority languages, on the basis of communities that have either traditionally or on a long-term basis resided in the country. This status grants Czech citizens from the Vietnamese community the right to use Vietnamese with public authorities, Vietnamese is increasingly being taught in schools and institutions outside of Vietnam. Since the 1980s, Vietnamese language schools have been established for youth in many Vietnamese-speaking communities around the world, furthermore, there has also been a number of Germans studying Vietnamese due to increased economic investment in Vietnam. Vietnamese is taught in schools in the form of immersion to a varying degree in Cambodia, Laos. Classes teach students subjects in Vietnamese and another language, furthermore, in Thailand, Vietnamese is one of the most popular foreign languages in schools and colleges. Vietnamese was identified more than 150 years ago as part of the Mon–Khmer branch of the Austroasiatic language family. Later, Muong was found to be closely related to Vietnamese than other Mon–Khmer languages. The term Vietic was proposed by Hayes, who proposed to redefine Viet–Muong as referring to a subbranch of Vietic containing only Vietnamese and Muong
Vietnamese language
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In the bilingual dictionary Nhật dụng thường đàm (1851), Chinese characters (chữ nho) are explained in chữ Nôm.
Vietnamese language
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Natively Vietnamese-speaking (non-minority) areas of Vietnam
Vietnamese language
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Jean-Louis Taberd 's dictionary Dictionarium anamitico-latinum (1838) represents Vietnamese (then Annamese) words in the Latin alphabet and chữ Nôm.
Vietnamese language
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A sign at the Hỏa Lò Prison museum in Hanoi lists rules for visitors in both Vietnamese and English.
63.
Thai language
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Thai, also known as Siamese or Central Thai, is the national and official language of Thailand and the native language of the Thai people and the vast majority of Thai Chinese. Thai is a member of the Tai group of the Tai–Kadai language family, over half of the words in Thai are borrowed from Pali, Sanskrit and Old Khmer. It is a tonal and analytic language, Thai also has a complex orthography and relational markers. Spoken Thai is mutually intelligible with Laotian, Thai is the official language of Thailand, natively spoken by over 20 million people. Standard Thai is based on the register of the classes of Bangkok. In addition to Central Thai, Thailand is home to other related Tai languages, Isan, the language of the Isan region of Thailand, a collective term for the various Lao dialects spoken in Thailand that show some Siamese Thai influences, which is written with the Thai script. It is spoken by about 20 million people, Thais from both inside and outside the Isan region often simply call this variant Lao when speaking informally. Northern Thai, spoken by about 6 million in the independent kingdom of Lanna. Shares strong similarities with Lao to the point that in the past the Siamese Thais referred to it as Lao. Southern Thai, spoken by about 4.5 million Phu Thai, spoken by half a million around Nakhon Phanom Province. Phuan, spoken by 200,000 in central Thailand and Isan, Shan, spoken by about 100,000 in north-west Thailand along the border with the Shan States of Burma, and by 3.2 million in Burma. Lü, spoken by about 1,000,000 in northern Thailand, and 600,000 more in Sipsong Panna, Burma, nyaw language, spoken by 50,000 in Nakhon Phanom Province, Sakhon Nakhon Province, Udon Thani Province of Northeast Thailand. Song, spoken by about 30,000 in central and northern Thailand, Elegant or Formal Thai, official and written version, includes respectful terms of address, used in simplified form in newspapers. Rhetorical Thai, used for public speaking, religious Thai, used when discussing Buddhism or addressing monks. Royal Thai, influenced by Khmer, this is used when addressing members of the family or describing their activities. Most Thais can speak and understand all of these contexts, street and Elegant Thai are the basis of all conversations. Rhetorical, religious, and royal Thai are taught in schools as the national curriculum, many scholars believe that the Thai script is derived from the Khmer script, which is modeled after the Brahmic script from the Indic family. However, in appearance, Thai is closer to Thai Dam script, the language and its script are closely related to the Lao language and script
Thai language
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Thai
64.
Units of measurement
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A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same quantity. Any other value of quantity can be expressed as a simple multiple of the unit of measurement. For example, length is a physical quantity, the metre is a unit of length that represents a definite predetermined length. When we say 10 metres, we actually mean 10 times the definite predetermined length called metre, the definition, agreement, and practical use of units of measurement have played a crucial role in human endeavour from early ages up to this day. Different systems of units used to be very common, now there is a global standard, the International System of Units, the modern form of the metric system. In trade, weights and measures is often a subject of regulation, to ensure fairness. The International Bureau of Weights and Measures is tasked with ensuring worldwide uniformity of measurements, metrology is the science for developing nationally and internationally accepted units of weights and measures. In physics and metrology, units are standards for measurement of quantities that need clear definitions to be useful. Reproducibility of experimental results is central to the scientific method, a standard system of units facilitates this. Scientific systems of units are a refinement of the concept of weights, science, medicine, and engineering often use larger and smaller units of measurement than those used in everyday life and indicate them more precisely. The judicious selection of the units of measurement can aid researchers in problem solving, in the social sciences, there are no standard units of measurement and the theory and practice of measurement is studied in psychometrics and the theory of conjoint measurement. A unit of measurement is a quantity of a physical property. Units of measurement were among the earliest tools invented by humans, primitive societies needed rudimentary measures for many tasks, constructing dwellings of an appropriate size and shape, fashioning clothing, or bartering food or raw materials. Weights and measures are mentioned in the Bible and it is a commandment to be honest and have fair measures. As of the 21st Century, multiple unit systems are used all over the world such as the United States Customary System, the British Customary System, however, the United States is the only industrialized country that has not yet completely converted to the Metric System. The systematic effort to develop an acceptable system of units dates back to 1790 when the French National Assembly charged the French Academy of Sciences to come up such a unit system. After this treaty was signed, a General Conference of Weights, the CGPM produced the current SI system which was adopted in 1954 at the 10th conference of weights and measures. Currently, the United States is a society which uses both the SI system and the US Customary system
Units of measurement
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The former Weights and Measures office in Seven Sisters, London
Units of measurement
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Units of measurement, Palazzo della Ragione, Padua
Units of measurement
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An example of metrication in 1860 when Tuscany became part of modern Italy (ex. one "libbra" = 339,54 grams)
65.
Base e
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The natural logarithm of a number is its logarithm to the base of the mathematical constant e, where e is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is written as ln x, loge x, or sometimes, if the base e is implicit. Parentheses are sometimes added for clarity, giving ln, loge or log and this is done in particular when the argument to the logarithm is not a single symbol, to prevent ambiguity. The natural logarithm of x is the power to which e would have to be raised to equal x. The natural log of e itself, ln, is 1, because e1 = e, while the natural logarithm of 1, ln, is 0, since e0 =1. The natural logarithm can be defined for any real number a as the area under the curve y = 1/x from 1 to a. The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, like all logarithms, the natural logarithm maps multiplication into addition, ln = ln + ln . However, logarithms in other bases differ only by a constant multiplier from the natural logarithm, for instance, the binary logarithm is the natural logarithm divided by ln, the natural logarithm of 2. Logarithms are useful for solving equations in which the unknown appears as the exponent of some other quantity, for example, logarithms are used to solve for the half-life, decay constant, or unknown time in exponential decay problems. They are important in many branches of mathematics and the sciences and are used in finance to solve problems involving compound interest, by Lindemann–Weierstrass theorem, the natural logarithm of any positive algebraic number other than 1 is a transcendental number. The concept of the natural logarithm was worked out by Gregoire de Saint-Vincent and their work involved quadrature of the hyperbola xy =1 by determination of the area of hyperbolic sectors. Their solution generated the requisite hyperbolic logarithm function having properties now associated with the natural logarithm, the notations ln x and loge x both refer unambiguously to the natural logarithm of x. log x without an explicit base may also refer to the natural logarithm. This usage is common in mathematics and some scientific contexts as well as in many programming languages, in some other contexts, however, log x can be used to denote the common logarithm. Historically, the notations l. and l were in use at least since the 1730s, finally, in the twentieth century, the notations Log and logh are attested. The graph of the logarithm function shown earlier on the right side of the page enables one to glean some of the basic characteristics that logarithms to any base have in common. Chief among them are, the logarithm of the one is zero. What makes natural logarithms unique is to be found at the point where all logarithms are zero. At that specific point the slope of the curve of the graph of the logarithm is also precisely one
Base e
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Graph of the natural logarithm function. The function slowly grows to positive infinity as x increases and slowly goes to negative infinity as x approaches 0 ("slowly" as compared to any power law of x); the y-axis is an asymptote.
66.
Qubit
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In quantum computing, a qubit or quantum bit is a unit of quantum information—the quantum analogue of the classical bit. A qubit is a two-state quantum-mechanical system, such as the polarization of a single photon, in a classical system, a bit would have to be in one state or the other. However, quantum mechanics allows the qubit to be in a superposition of states at the same time, a property that is fundamental to quantum computing. The concept of the qubit was unknowingly introduced by Stephen Wiesner in 1983, in his proposal for quantum money, the coining of the term qubit is attributed to Benjamin Schumacher. The paper describes a way of compressing states emitted by a source of information so that they require fewer physical resources to store. This procedure is now known as Schumacher compression, the bit is the basic unit of information. It is used to represent information by computers, an analogy to this is a light switch—its off position can be thought of as 0 and its on position as 1. A qubit has a few similarities to a bit, but is overall very different. There are two possible outcomes for the measurement of a qubit—usually 0 and 1, like a bit, the difference is that whereas the state of a bit is either 0 or 1, the state of a qubit can also be a superposition of both. It is possible to encode one bit in one qubit. However, a qubit can hold more information, e. g. up to two bits using superdense coding. For a system of n components, a description of its state in classical physics requires only n bits. The two states in which a qubit may be measured are known as basis states, as is the tradition with any sort of quantum states, they are represented by Dirac—or bra–ket—notation. This means that the two basis states are conventionally written as |0 ⟩ and |1 ⟩. A pure qubit state is a superposition of the basis states. When we measure this qubit in the basis, the probability of outcome |0 ⟩ is | α |2. Because the absolute squares of the amplitudes equate to probabilities, it follows that α and β must be constrained by the equation | α |2 + | β |2 =1. It might at first sight seem that there should be four degrees of freedom, as α and β are complex numbers with two degrees of freedom each
Qubit
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Bloch sphere representation of a qubit. The probability amplitudes in the text are given by and.
67.
Yuki tribe
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The Yuki are an indigenous people of California, whose traditional territory is around Round Valley, Mendocino County. Today they are enrolled members of the Round Valley Indian Tribes of the Round Valley Reservation, Yuki tribes are thought to have settled as far south as Hood Mountain in present-day Sonoma County. As European-American settlers began to flock to Northern California in the early 1850s, the Indians suffered deaths in raids by the local ranchers and the authorities, and captives were taken into slavery. In 1856, the US government established the Indian reservation of Nome Cult Farm at Round Valley and it forced thousands of Yuki and other local tribes on to these lands, often without sufficient support for the transition. These events and tensions led to the Mendocino War, where US forces killed hundreds of Yuki, the Yuki language is no longer spoken. It is related to the Wappo language, The Yuki people had a counting system. Scholarly estimates have varied substantially for the populations of most native groups in California, as historians. Alfred L. Kroeber estimated the 1770 population of the Yuki proper, Huchnom, and Coast Yuki as 2,000,500, Sherburne F. Cook initially raised this total slightly to 3,500. Subsequently, he proposed an estimate of 9,730 Yuki. In the 2010 census,569 people claimed Yuki ancestry, Yuki traditional narratives Cook, Sherburne F.1956. The Aboriginal Population of the North Coast of California, Anthropological Records,16, the Conflict between the California Indian and White Civilization. Handbook of the Indians of California, bureau of American Ethnology Bulletin No.78. Four Directions Institute Round Valley history Central California culture, Four Directions Institute
Yuki tribe
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Yuki men at the Nome Cult Farm, ca. 1858
68.
California
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California is the most populous state in the United States and the third most extensive by area. Located on the western coast of the U. S, California is bordered by the other U. S. states of Oregon, Nevada, and Arizona and shares an international border with the Mexican state of Baja California. Los Angeles is Californias most populous city, and the second largest after New York City. The Los Angeles Area and the San Francisco Bay Area are the nations second- and fifth-most populous urban regions, California also has the nations most populous county, Los Angeles County, and its largest county by area, San Bernardino County. The Central Valley, an agricultural area, dominates the states center. What is now California was first settled by various Native American tribes before being explored by a number of European expeditions during the 16th and 17th centuries, the Spanish Empire then claimed it as part of Alta California in their New Spain colony. The area became a part of Mexico in 1821 following its war for independence. The western portion of Alta California then was organized as the State of California, the California Gold Rush starting in 1848 led to dramatic social and demographic changes, with large-scale emigration from the east and abroad with an accompanying economic boom. If it were a country, California would be the 6th largest economy in the world, fifty-eight percent of the states economy is centered on finance, government, real estate services, technology, and professional, scientific and technical business services. Although it accounts for only 1.5 percent of the states economy, the story of Calafia is recorded in a 1510 work The Adventures of Esplandián, written as a sequel to Amadis de Gaula by Spanish adventure writer Garci Rodríguez de Montalvo. The kingdom of Queen Calafia, according to Montalvo, was said to be a land inhabited by griffins and other strange beasts. This conventional wisdom that California was an island, with maps drawn to reflect this belief, shortened forms of the states name include CA, Cal. Calif. and US-CA. Settled by successive waves of arrivals during the last 10,000 years, various estimates of the native population range from 100,000 to 300,000. The Indigenous peoples of California included more than 70 distinct groups of Native Americans, ranging from large, settled populations living on the coast to groups in the interior. California groups also were diverse in their organization with bands, tribes, villages. Trade, intermarriage and military alliances fostered many social and economic relationships among the diverse groups, the first European effort to explore the coast as far north as the Russian River was a Spanish sailing expedition, led by Portuguese captain Juan Rodríguez Cabrillo, in 1542. Some 37 years later English explorer Francis Drake also explored and claimed a portion of the California coast in 1579. Spanish traders made unintended visits with the Manila galleons on their trips from the Philippines beginning in 1565
California
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A forest of redwood trees in Redwood National Park
California
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Flag
California
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Mount Shasta
California
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Aerial view of the California Central Valley
69.
Papua New Guinea
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Its capital, located along its southeastern coast, is Port Moresby. The western half of New Guinea forms the Indonesian provinces of Papua, Papua New Guinea is one of the most culturally diverse countries in the world. There are 852 known languages in the country, of which 12 have no known living speakers, most of the population of more than 7 million people live in customary communities, which are as diverse as the languages. It is also one of the most rural, as only 18 percent of its live in urban centres. The country is one of the worlds least explored, culturally and geographically and it is known to have numerous groups of uncontacted peoples, and researchers believe there are many undiscovered species of plants and animals in the interior. Papua New Guinea is classified as an economy by the International Monetary Fund. Strong growth in Papua New Guineas mining and resource sector led to the becoming the sixth fastest-growing economy in the world in 2011. Growth was expected to slow once major resource projects came on line in 2015, mining remains a major economic factor, however. Local and national governments are discussing the potential of resuming mining operations in Panguna mine in Bougainville Province, nearly 40 percent of the population lives a self-sustainable natural lifestyle with no access to global capital. Most of the still live in strong traditional social groups based on farming. Their social lives combine traditional religion with modern practices, including primary education, at the national level, after being ruled by three external powers since 1884, Papua New Guinea established its sovereignty in 1975. This followed nearly 60 years of Australian administration, which started during the Great War and it became an independent Commonwealth realm with Queen Elizabeth II as its head of state and became a member of the Commonwealth of Nations in its own right. Archaeological evidence indicates that humans first arrived in Papua New Guinea around 42,000 to 45,000 years ago and they were descendants of migrants out of Africa, in one of the early waves of human migration. Agriculture was independently developed in the New Guinea highlands around 7000 BC, a major migration of Austronesian-speaking peoples to coastal regions of New Guinea took place around 500 BC. This has been correlated with the introduction of pottery, pigs, in the 18th century, traders brought the sweet potato to New Guinea, where it was adopted and became part of the staples. Portuguese traders had obtained it from South America and introduced it to the Moluccas, the far higher crop yields from sweet potato gardens radically transformed traditional agriculture and societies. Sweet potato largely supplanted the previous staple, taro, and resulted in a significant increase in population in the highlands. In 1901, on Goaribari Island in the Gulf of Papua, missionary Harry Dauncey found 10,000 skulls in the islands Long Houses, traders from Southeast Asia had visited New Guinea beginning 5,000 years ago to collect bird of paradise plumes
Papua New Guinea
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Kerepunu villagers, British New Guinea, 1885.
Papua New Guinea
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Flag
Papua New Guinea
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Lime container, late 19th or early 20th century. The container is decorated with wood carving of crocodile and bird. Punctuation is emphasised with a white paint. The central portion, hollow to hold the lime, is made of bamboo. The joints are covered with basketry work.
Papua New Guinea
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Australian forces attack Japanese positions during the Battle of Buna–Gona. 7 January 1943.
70.
Base-6
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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
Base-6
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Numeral systems
Base-6
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34 senary = 22 decimal, in senary finger counting
Base-6
71.
Algorism
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Algorism is the technique of performing basic arithmetic by writing numbers in place value form and applying a set of memorized rules and facts to the digits. One who practices algorism is known as an algorist and this title means Algoritmi on the numbers of the Indians, where Algoritmi was the translators Latinization of Al-Khwarizmis name. Al-Khwarizmi was the most widely read mathematician in Europe in the late Middle Ages, primarily through his other book, in late medieval Latin, algorismus, the corruption of his name, simply meant the decimal number system that is still the meaning of modern English algorism. In 17th century French the words form, but not its meaning, changed to algorithm, following the model of the word logarithm, this form alluding to the ancient Greek arithmos = number. English adopted the French very soon afterwards, but it wasnt until the late 19th century that took on the meaning that it has in modern English. In English, it was first used about 1230 and then by Chaucer in 1391, another early use of the word is from 1240, in a manual titled Carmen de Algorismo composed by Alexandre de Villedieu. It begins thus, Haec algorismus ars praesens dicitur, in qua / Talibus Indorum fruimur bis quinque figuris, which translates as, This present art, in which we use those twice five Indian figures, is called algorismus. The word algorithm also derives from algorism, a generalization of the meaning to any set of rules specifying a computational procedure, occasionally algorism is also used in this generalized meaning, especially in older texts. These included the concept of the decimal fractions as an extension of the notation and this system was popularized in Europe by Leonardo of Pisa, now known as Fibonacci. Algorithmic art Positional notation Hindu–Arabic numeral system History of the Hindu–Arabic numeral system
Algorism
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Calculating-Table by Gregor Reisch: Margarita Philosophica, 1508
72.
Scientific notation
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Scientific notation is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. It is commonly used by scientists, mathematicians and engineers, in part because it can simplify certain arithmetic operations, on scientific calculators it is known as SCI display mode. In scientific notation all numbers are written in the form m × 10n, where the exponent n is an integer, however, the term mantissa may cause confusion because it is the name of the fractional part of the common logarithm. If the number is then a minus sign precedes m. In normalized notation, the exponent is chosen so that the value of the coefficient is at least one. Decimal floating point is an arithmetic system closely related to scientific notation. Any given integer can be written in the form m×10^n in many ways, in normalized scientific notation, the exponent n is chosen so that the absolute value of m remains at least one but less than ten. Thus 350 is written as 3. 5×102 and this form allows easy comparison of numbers, as the exponent n gives the numbers order of magnitude. In normalized notation, the exponent n is negative for a number with absolute value between 0 and 1, the 10 and exponent are often omitted when the exponent is 0. Normalized scientific form is the form of expression of large numbers in many fields, unless an unnormalized form. Normalized scientific notation is often called exponential notation—although the latter term is general and also applies when m is not restricted to the range 1 to 10. Engineering notation differs from normalized scientific notation in that the exponent n is restricted to multiples of 3, consequently, the absolute value of m is in the range 1 ≤ |m| <1000, rather than 1 ≤ |m| <10. Though similar in concept, engineering notation is rarely called scientific notation, engineering notation allows the numbers to explicitly match their corresponding SI prefixes, which facilitates reading and oral communication. A significant figure is a digit in a number that adds to its precision and this includes all nonzero numbers, zeroes between significant digits, and zeroes indicated to be significant. Leading and trailing zeroes are not significant because they exist only to show the scale of the number. Therefore,1,230,400 usually has five significant figures,1,2,3,0, and 4, when a number is converted into normalized scientific notation, it is scaled down to a number between 1 and 10. All of the significant digits remain, but the place holding zeroes are no longer required, thus 1,230,400 would become 1.2304 ×106. However, there is also the possibility that the number may be known to six or more significant figures, thus, an additional advantage of scientific notation is that the number of significant figures is clearer
Scientific notation
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A calculator display showing the Avogadro constant in E notation
73.
Hermann Schmid (computer scientist)
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Some also had a variable wordlength, which enabled operations on numbers with a large number of digits. Early computers that were exclusively decimal include the ENIAC, IBM NORC, IBM650, IBM1620, IBM7070. In these machines the basic unit of data was the decimal digit, encoded in one of several schemes, including binary-coded decimal or BCD, bi-quinary, excess-3, except for the 1620, these machines used word addressing. When non-numeric characters were used in machines, they were encoded as two decimal digits. Other early computers were character oriented, providing instructions for performing arithmetic on character strings of decimal numerals, on these machines the basic data element was an alphanumeric character, typically encoded in six bits. UNIVAC I and UNIVAC II used word addressing, with 12-character words, IBM examples include IBM702, IBM705, the IBM1400 series, IBM7010, and the IBM7080. It used 8-bit characters and introduced EBCDIC encoding, though ASCII was also supported, several microprocessor families offer limited decimal support. For example, the 80x86 family of microprocessors provide instructions to convert one-byte BCD numbers to binary format before or after arithmetic operations and these operations were not extended to wider formats and hence are now slower than using 32-bit or wider BCD tricks to compute in BCD. The x87 FPU has instructions to convert 10-byte packed decimal data, Decimal arithmetic is now becoming more common, for instance, three decimal types with two binary encodings were added to the 2008 IEEE 754r standard, with 7-, 16-, and 34-digit decimal significands. The IBM Power6 processor and the IBM System z9 have implemented these types using the Densely Packed Decimal binary encoding, the first in hardware, binghamton, New York, USA, John Wiley & Sons, Inc. Malabar, Florida, USA, Robert E. Krieger Publishing Company, Iowa City, Iowa, USA, The University of Iowa, Department of Computer Science
Hermann Schmid (computer scientist)
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IBM 650 front panel with bi-quinary coded decimal displays