1.
Numeral system
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The number the numeral represents is called its value. 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 place-value notation in the 5th century and a century later Brahmagupta introduced the symbol for zero. The Arabs adopted and modified it. The Arabs call the Hindu numeral system. The Arabs spread them with them. The Western world modified them and called them the Arabic numerals, as they learned from them. Hence the western system is the modified version of the Hindu numeral system developed in India. It also exhibits a great similarity to the Sanskrit–Devanagari notation, still used in India and neighboring Nepal. The simplest system is the unary system, in which every natural number is represented by a corresponding number of symbols. If the / is chosen, for example, 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.
Numeral system
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Numeral systems
2.
Arabic numerals
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The symbol for zero is the key to the effectiveness of the system, developed by ancient mathematicians in the Indian subcontinent around AD 500. The system was adopted by Arab mathematicians in Baghdad and passed on to the Arabs farther west. 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 and colonialism. The term Arabic numerals is ambiguous. It most commonly refers to the numerals widely used in Europe and the Americas; to avoid confusion, Unicode calls these European digits. Arabic numerals is also the conventional name for the entire family of related numerals of Arabic and Indian numerals. It may also be intended to mean the numerals used by Arabs, in which case it generally refers to the Eastern Arabic numerals. 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 AD 700. The development was gradual, spanning several centuries, but the decisive step was probably provided by Brahmagupta's formulation of zero as a number in AD 628. The system was revolutionary by including zero in positional notation, thereby limiting the number of individual digits to ten. It is considered an important milestone in the development of mathematics. One may distinguish between this positional system, identical throughout the family, the precise glyphs used to write the numerals, which varied regionally. The glyphs most commonly used in conjunction with the Latin script since modern times are 0 2 3 7 9.
Arabic numerals
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Numeral systems
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
Arabic numerals
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The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.
Arabic numerals
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Woodcut showing the 16th century astronomical clock of Uppsala Cathedral, with two clockfaces, one with Arabic and one with Roman numerals.
3.
Eastern Arabic numerals
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These numbers are known as أرقام هندية in Arabic. They are sometimes also called "Indic numerals" in English. However, 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. Eastern Arabic numerals remain strongly predominant Arabic numerals in many countries to the East of the Arab world, particularly in Afghanistan. In Pakistan, Western Arabic numerals are more extensively used as a considerable majority of the population is anglophone. Eastern numerals still continue to see use in Urdu publications and newspapers, as well as sign boards. In North Africa, only Western Arabic numerals are now commonly used. In medieval times, these areas 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
4.
Bengali numerals
Bengali numerals
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Numeral systems
5.
Indian numerals
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Indian numerals are the symbols representing numbers in India. These numerals are distinct from, though related to Arabic numerals. Since Sanskrit is an Indo-European language, it is obvious that the words for numerals closely resemble those of Greek and Latin. The word "Shunya" for zero was translated as" صفر" meaning ` nothing' which became the term "zero" in many European languages from Medieval Latin, zephirum. For numerals in Tamil language see Tamil numerals. For numerals in Telugu language see Telugu numerals. Tamil and Malayalam scripts also have distinct forms for 10,100,1000 numbers, ௰,௱,௲and ൰,൱,൲ respectively in tamil and scripts. A decimal system has been traced back in India. Before that epoch, the Brahmi numeral system was in use; that system did not encompass the concept of the place-value of numbers. Instead, Brahmi numerals included additional symbols for the tens, well as separate symbols for thousand. The Indian place-system numerals spread to neighboring Persia, where they were picked up by the conquering Arabs. I wish only to say that this computation is done by means of nine signs. In academic circles they are called the Hindu–Arabic or Indo–Arabic numerals. But what was the net achievement in the field of reckoning, the earliest art practiced by man? John Wiley, 2000.
Indian numerals
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Numeral systems
6.
Sinhala numerals
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Sinhalese belongs to the Indo-European language family with its roots deeply associated with Indo-Aryan sub family to which the languages such as Persian and Hindi belong. It is also surmised that Sinhala had evolved from an ancient variant of Apabramsa, known as ‘Elu’. When tracing history of Elu, it was preceded by Hela or Pali Sihala. The Sinhala script had evolved from which all the Southern Indic Scripts such as Telugu and Oriya had evolved. Later Sinhala was influenced by Grantha writing of Southern India. Since 1250 AD, the Sinhala script had remained the same with few changes. Archeologists had found pottery fragments in Anuradhapura Sri Lanka with older Brahmi script inscriptions, carbon dated to 5th century BC. Sinhala letters are round-shaped and are written from left to right and they are the most circular-shaped script found in the Indic scripts. The evolution of the script to the present shapes may have taken place due to writing on Ola leaves. Unlike chiseling on a rock, writing on palm leaves has to be more round-shaped to avoid the stylus ripping the Palm leaf while writing on it. Instead a stylistic stop, known as ‘Kundaliya’ is used. Period and commas were later introduced due to the influence of Western languages. In modern Sinhala, Arabic numerals, which were introduced by Portuguese, Dutch and English, is used for writing numbers and carrying out calculations. It is accepted that Arabic numerals had evolved from Brahmi numerals. This article will also touch upon Brahmi numerals, which were found in Sri Lanka.
Sinhala numerals
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Numeral systems
Sinhala numerals
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Stages
Sinhala numerals
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Archaic Sinhala numerals from ‘Catalogue of Palm leaf manuscripts in the library of Colombo Museum’, Volume I, compiled by W. A. De Silva, published by the Government Printer in 1938.
Sinhala numerals
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A day had sixy Sinhala hora or hours. The watch shows thirty horas or hours in Sinhala Illakkam. Even today Sinhala Astrologers convert time of Birth to Sinhala Hora or Hours for casting horoscopes. This watch was owned by the last of king of Kandy.
7.
Tamil numerals
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Traditionally Vattezhuttu characters were used, but now Arabic numerals have become commonplace. There are two systems which can be used, the ` pure' system used in the Sanskrit-based Hindu-Arabic system. Old Tamil possesses a special numerical character for zero and it is read as andru. But yet Modern Tamil renounces the use of its native character and uses Arabic, 0. Modern Tamil words for zero include சுழியம் or பூச்சியம். For instance, the word for fifty, ஐம்பது is a combination of ஐ and பத்து. The prefix for nine changes with respect to the succeeding base 10. தொ+ the unvoiced consonant of the succeeding base 10 forms the prefix for nine. For instance, 90 is தொ+ண், hence, தொண்ணூறு). Unlike Indian languages, Tamil has distinct digits for 1000. It also has distinct characters for other number-based aspects of day-to-day life. The following are the traditional numbers of the Ancient Tamil Country, Tamizhakam. Sanskrit based multiples like lakhs are also followed just like other Indian languages. You can transcribe any fraction, by affixing -இல் after the denominator followed by the numerator. For instance, 1/41 can be said as நாற்பத்து ஒன்றில் ஒன்று.
Tamil numerals
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Numeral systems
Tamil numerals
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A milestone which uses both Tamil and Indo-Arabic Numerals (Tanjore Palace Museum).
Tamil numerals
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Tamil is written in a non-Latin script. Tamil text used in this article is transliterated into the Latin script according to the ISO 15919 standard.
Tamil numerals
8.
Balinese numerals
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The Balinese language has an elaborate decimal numeral system. The numerals 1–10 have basic, combining, independent forms, many of which are formed through reduplication. The combining forms are used to form higher numbers. In some cases there is more than one word for a numeral, reflecting the Balinese system; forms are listed in italics. Final orthographic -a is a schwa. * A less productive combining form of a- 1 is sa-, as can be seen in many of the numbers below. It, ulung-, sangang- are from Javanese. Dasa 10 is from Sankrit désa. Like English, Balinese has compound forms for the tens; however, it also has a series of 21 -- 29. The teens are based on a root * - the tweens on - the tens are formed by the combining forms above. Hyphens are not used in the orthography, but have been added to the table below to clarify their derivation. The high-register combining forms kalih- 2 and tigang- 3 are used with -likur, -dasa, higher numerals, but not for the teens. The teens are from Javanese, where the -olas forms are regular, apart from pele-kutus 18, suppletive. Cognate with Javanese səlawé 25 and səkət 50. There are additional numerals pasasur ~ sasur 35 and se-timahan ~ se-timan 45, a compound telung-benang for 75.
Balinese numerals
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Numeral systems
9.
Burmese numerals
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Burmese numerals are a set of numerals traditionally used in the Burmese language, although the Arabic numerals are also used. Burmese numerals follow the Hindu-Arabic numeral system commonly used in the rest of the world. 1 Burmese for zero comes from Sanskrit śūnya.2 Can be abbreviated to IPA: in list contexts, such as telephone numbers. Spoken Burmese has a word. Other suffixes such as ထောင်, သောင်း, သိန်း, သန်း all shift to, and; million), respectively. For eight, no shift occurs. These pronunciation shifts are not spelt differently. 1 Shifts to voiced consonant following four, nine. Ten to nineteen are almost always expressed without including တစ်. Another rule shifts numerical name from the low tone to the creaky tone. Number places from 10 up to 107 has increment of 101. Beyond those Number places, larger number places have increment of 107. 1014 up to 10140 has increment of 107. Numbers in the hundreds place: shift from ရာ to ရာ့, except for numbers divisible by 100. Numbers in the thousands place: shift from ထောင် to ထောင့်, except for numbers divisible by 1000.
Burmese numerals
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Numeral systems
Burmese numerals
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Burmese numerals in various script styles
10.
Dzongkha numerals
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Dzongkha, the national language of Bhutan, has two numeral systems, one vigesimal, a modern decimal system. The vigesimal system remains in robust use. Ten is an auxiliary base: the teens are formed with 1 -- 9. *When it appears on its own,'ten' is usually said cu-tʰãm'a full ten'. In combinations it is simply cu. Factors of 20 are formed from kʰe. Intermediate factors of ten are formed with pɟʱe-da ` half to': 400 ɲiɕu is the next unit: ɲiɕu ɲi 800, etc.. Higher powers are 8000 kʰecʰe and jãːcʰe 160,000. The decimal system is the same as the vigesimal system up to 19. Then decades, however, are formed as unit–ten, as in Chinese, the hundreds similarly. 20 is reported to be the vigesimal 400; this may be lexical interference for the expected * ɲi-cu. Mazaudon & Lacito, 2002, "Les principes de construction du nombre dans les langues tibeto-birmanes", in François, ed. La Pluralité, p. 6 ff
Dzongkha numerals
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Numeral systems
11.
Gujarati numerals
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The Gujarati script, which like all Nagari writing systems is an abugida, a type of alphabet, is used to write the Gujarati and Kutchi languages. With a few additional characters, added for this purpose, the Gujarati script is also often used to write Sanskrit and Hindi. Gujarati numerical digits are also different from their Devanagari counterparts. Gujarati script is descended from Brahmi and is part of the Brahmic family. The Gujarati script was adapted from the Devanagari script to write the Gujarati language. Gujarati script developed in three distinct phases -- 10th to 15th century, 17th to 19th century. The first phase is marked by use of Prakrit, Apabramsa and its variants such as Paisaci, Shauraseni, Magadhi and Maharashtri. In second phase, Old Gujarati script was in wide use. The third phase is the use of script developed for ease and fast writing. The use of shirorekha was abandoned. Until the 19th century it was used mainly for writing letters and keeping accounts, while the Devanagari script was used for literature and academic writings. It is also known as the mahājanī script. This script became basis for modern script. Later the same script was adopted by writers of manuscripts. Jain community also promoted its use for copying religious texts by hired writers.
Gujarati numerals
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Gujarati
12.
Javanese numerals
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The Javanese language has a decimal numeral system with distinct words for the'tweens' from 21 to 29, called likuran. The basic 1 -- 10 have the latter derived via a suffix - ng. The combining forms are used to form the tens, millions. The numerals 1–5 and 10 have distinct high-register and low register forms. The halus forms are listed below in italics. Like English, Javanese has compound forms for the teens; however, it also has a series of compound'tweens', 21–29. The teens are based on a root -las, the tweens on -likur, the tens are formed by the combining forms. Hyphens are not used in the orthography, but have been added to the table below to clarify their derivation. Final orthographic -a tends to in many dialects, as does any preceding a. Parallel to the tens are the hundreds; the thousands, the millions, except that the compounds of five and six are formed with limang- and nem-. Balinese a related but yet more complex system.
Javanese numerals
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Numeral systems
13.
Khmer numerals
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Khmer numerals are the numerals used in the Khmer language. Having been derived from the Hindu numerals, modern Khmer numerals also represent a positional system. However, Angkorian Khmer, also possessed separate symbols for 10, 20, 100. 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, either or for the word three. In neighbouring Thailand the number three is thought to bring good luck. 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 Khmer's 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.
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
14.
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 replaced by the Thai script. It has 27 consonants, 7 consonantal ligatures, 4 tone marks. Akson Lao is a sister system to the Thai script, which it shares many roots. However, Lao is formed in a more curvilinear fashion than Thai. Lao is traditionally written to right. Lao is considered an abugida, in which certain'implied' vowels are unwritten. However, due to spelling reforms by the communist Lao People's Revolutionary Party, it is less apparent. For the old spelling of ສເລີມ ` to hold a ceremony, celebrate' contrasts with the new ສະເຫລີມ. Vowels can be written below, in front of, or behind consonants, with some vowel combinations written before, over and after. A space is used and functions in place of a comma or period. The letters have no minuscule differentiation. This script, sometimes known as Tai Noi, has changed little since continued use in the Lao-speaking regions of modern-day Laos and Isan. The scripts still share similarities. Traditionally, only literature was written with the Lao alphabet.
Lao alphabet
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Numeral systems
Lao alphabet
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Lao
15.
Mongolian numerals
Mongolian numerals
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Numerals reading "2013" in Ulaanbaatar
Mongolian numerals
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Numeral systems
16.
Thai numerals
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The Thai language lacks grammatical number. A count is usually expressed in the form of an uninflected noun followed by a number and a classifier. In Thai, counting is kannap; the classifier, laksananam Variations to this pattern do occur, there really is no hierarchy among Thai classifiers. A partial list of Thai words that also classify nouns can be found in Wiktionary category: Thai classifiers. It is from Sanskrit śūnya, as are the alternate names for numbers one to four given below; but not the counting 1. 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. 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. For the numbers twenty-one through twenty-nine, the part signifying twenty: yi sip, may be colloquially shortened to yip. See the alternate numbers section below. The hundreds are formed by combining roi with the tens and ones values. For example, thirty-two is roi sam song.
Thai numerals
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Numeral systems
17.
Chinese numerals
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Chinese numerals are words and characters used to denote numbers in Chinese. Today speakers of Chinese use three written numeral systems: the system of Arabic numerals used world-wide, two indigenous systems. The more familiar indigenous system is based on Chinese characters that correspond to numerals in the spoken language. These are shared with other languages of the Chinese cultural sphere such as Japanese, Korean and Vietnamese. The other indigenous system is the only surviving form of the rod numerals. The Chinese system consists of the Chinese characters used by the written language to write spoken numerals. Similar to spelling-out numbers in English, it is not an independent system per se. There are other characters representing larger numbers such as tens, so on. There are two sets of characters for Chinese numerals: one for everyday writing and one for use in commercial or financial contexts known as dàxiě. A forger could easily change the everyday characters 三十 to 五千 just by adding a few strokes. That would not be possible when writing using the financial characters 參拾 and 伍仟. They are also referred to as "banker's numerals", "anti-fraud numerals", or "banker's anti-fraud numerals". For the same reason, rod numerals were never used in commercial records. T denotes Traditional Chinese characters, S denotes Simplified Chinese characters. In the PLA, some numbers will have altered names when used for clearer radio communications.
Chinese numerals
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Numeral systems
Chinese numerals
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Chinese and Arabic numerals may coexist, as on this kilometer marker: 1620 km on Hwy G209 (G二〇九)
Chinese numerals
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Counting rod numerals
Chinese numerals
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Shang oracle bone numerals of 14th century B.C.
18.
Suzhou numerals
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The Suzhou numerals or huama is a numeral system used in China before the introduction of Arabic numerals. The Suzhou system is the only surviving variation of the rod numeral system. The rod system is a positional numeral system used by the Chinese in mathematics. Suzhou numerals are a variation of the Southern rod numerals. Suzhou numerals were used in number-intensive areas of commerce such as accounting and bookkeeping. At the same time, Chinese numerals were used in formal writing, akin to spelling out the numbers in English. This is similar to what had happened with Roman numerals used in ancient and medieval Europe for mathematics and commerce. Nowadays, the Suzhou system is only used for displaying prices in Chinese markets or on traditional handwritten invoices. In the Suzhou system, special symbols are used for digits instead of the Chinese characters. The digits of the Suzhou numerals are defined in Unicode. An additional three code points starting from U+3038 were added later. One, two, three are all represented by vertical bars. This can cause confusion when they appear next to each other. Standard Chinese ideographs are often used in this situation to avoid ambiguity. For example, "21" is written as "〢一" instead of "〢〡" which can be confused with "3".
Suzhou numerals
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Numeral systems
Suzhou numerals
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The Suzhou numerals for 5 and 9 come from their respective horizontal forms of the rod numerals.
Suzhou numerals
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Suzhou numerals on banquet invoices issued by restaurants in the 1910 - 1920s. Although the invoices use traditional right-to-left vertical writing, the Suzhou numerals recording the amounts are written horizontally from left to right.
Suzhou numerals
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Suzhou numerals on a market in Wan Chai
19.
Japanese numerals
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The system of Japanese numerals is the system of number names used in the Japanese language. The Japanese numerals in writing are entirely based on the Chinese numerals and the grouping of large numbers follow the Chinese tradition of grouping by 10,000. There are two ways of writing the numbers in Japanese, in Hindu-Arabic numerals or in Chinese numerals. The Hindu-Arabic numerals are more often used in horizontal writing, the Chinese numerals are more common in vertical writing. Numerals with multiple On readings use the Go-on and Kan-on variants respectively. * The special reading 〇 maru is also found. It may be optionally used when reading individual digits of a number one after another, instead of as a full number. A popular example is the famous 109 store in Shibuya, Tokyo, read as ichi-maru-kyū. This usage of maru for numerical 0 is similar to reading numeral 0 in English as oh. It literally means a circle. However, as a number, it is only written as 0 or rei. Additionally, two and five are pronounced with a long vowel in phone numbers Starting at 万, numbers begin with 一 if no digit would otherwise precede. That is, 100 is just 百 hyaku, 1000 is just 千 sen, but 10,000 is 一万 ichiman, not just *man. This differs from Chinese as numbers begin with 一 if no digit would otherwise precede starting at 百. And, if 千 sen directly precedes the name of powers of myriad, 一 ichi is normally attached before 千 sen, which yields 一千 issen.
Japanese numerals
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Numeral systems
20.
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 and the ones places. For instance, 15 would be sib-o, but not usually il-sib-o in the Sino-Korean system, yeol-daseot in native Korean. 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 numeral 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, se for Sino-Korean. 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.
Korean numerals
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Numeral systems
21.
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 Vietnamese vocabulary is used for 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 cultural sphere, Japanese and Korean both use two numerical systems, 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, only a few numbers are based on Sino-Vietnamese vocabulary. In the Vietnamese 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, using Chinese characters to write Sino-Vietnamese numbers. 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 and native vocabulary was written in a system of modified Chinese characters known as Chữ Nôm. Basic features of the Vietnamese system include the following: Unlike sinoxenic numbering systems, Vietnamese separates place values in thousands rather than myriads. The Sino-Vietnamese numbers are not in frequent use in modern Vietnamese.
Vietnamese numerals
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Numeral systems
22.
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, Vietnam. They are placed horizontally or vertically to represent any rational number. The written forms based on them are called rod numerals. They are a positional 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 than two thousand years. In 1954, forty-odd counting rods of the Warring States period were found in Zuǒjiāgōngshān Chǔ Grave No.15 in Changsha, Hunan. In 1973, archeologists unearthed a number of wood scripts from a Han dynasty tomb in Hubei. On one of the wooden 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 West Han counting rods made of bones was unearthed from Qian Yang county in Shanxi. The use of counting rods must predate it; Laozi said "a good calculator doesn't use counting rods". The Book of Han recorded: "they calculate with bamboo, length six cun, arranged into a hexagonal bundle of hundred seventy one pieces". After the abacus flourished, counting rods were abandoned in Japan, where rod numerals developed for algebra. The rod represents five. To avoid confusion, vertical and horizontal forms are alternately used.
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
23.
Abjad numerals
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The Abjad numerals are a decimal numeral system in which the 28 letters of the Arabic alphabet are assigned numerical values. They have been used in the Arabic-speaking world since before the eighth century when Arabic numerals were adopted. In modern Arabic, the word abjadīyah means'alphabet' in general. Individual letters also represent 100s: kāf for 20, qāf for etc.. The word abjad itself derives in the Phoenician alphabet, other scripts for Semitic languages. These older alphabets contained only 22 letters, stopping at taw, numerically equivalent to 400. The Arabic Abjad system continues at this point with letters not found in other alphabets: ṯāʾ=500, etc. The Abjad order of the Arabic alphabet has two slightly different variants. The most common Abjad sequence, read from right to left, is: This is commonly vocalized as follows: abjad hawwaz ḥuṭṭī kalaman saʻfaṣ qarashat thakhadh ḍaẓagh. Before the introduction of the Hindu–Arabic numeral system, the abjad numbers were used for all mathematical purposes. In modern Arabic, they are primarily used for numbering outlines, items in lists, points of information. The abjad numbers are also used to assign numerical values to Arabic words for purposes of numerology. The common Islamic phrase بسم الله الرحمن الرحيم bismillāh al-Raḥmān al-Raḥīm has a numeric value of 786. The name Allāh الله by itself has the value 66. A few of the numerical values are different in the alternative Abjad order.
Abjad numerals
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Numeral systems
24.
Armenian numerals
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The system of Armenian numerals is a historic numeral system created using the majuscules of the Armenian alphabet. The numeric values for individual letters were added together. The principles behind this system are the same as for Hebrew numerals. In modern Armenia, the 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. գլուխ means Chapter III. Since not all browsers can render Armenian letters, the classical transliteration is given. Thus, they do not have a numerical value assigned to them. Numbers in the Armenian system are obtained by simple addition. Armenian numerals are written left-to-right. Although the order of the numerals is irrelevant since only addition is performed, the convention is to write them in decreasing order of value.
Armenian numerals
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Numeral systems
25.
Cyrillic numerals
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The Cyrillic numerals are a numeral system derived from the Cyrillic script, developed in the First Bulgarian Empire in the late tenth century. It was used by the First Bulgarian Empire and by South and East Slavic peoples. The system was used in Russia as late as the early 18th century when Peter the Great replaced it with Arabic numerals. The Cyrillic numerals may still be found in books written in the Church Slavonic language. The system is quasidecimal, basically the Ionian numeral system written with the corresponding graphemes of the Cyrillic script. The order is based on the original Greek alphabet and does not correspond to the different standard Cyrillic alphabetical orders. A separate letter is assigned to each multiple of hundred. For example, 17 is "семнадсять", "s'em'-na-d's'at'". To cipher a Cyrillic number, one has to add all the figures. To distinguish numbers from text, a titlo is drawn over the numbers. If the number exceeds 1000, the thousands sign is drawn before the figure, the thousands figure are written with a letter assigned to the units. Early Cyrillic alphabet Glagolitic alphabet Greek numerals Combining Cyrillic Millions Гаманович, Алипий. Grammar of the Church Slavonic Language. Jordanville, NY: Printshop of St. Job of Pochaev. P. 271.
Cyrillic numerals
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Numeral systems
Cyrillic numerals
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Tower clock with Cyrillic numerals in Suzdal
Cyrillic numerals
Cyrillic numerals
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– 1706
26.
Ge'ez script
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Ge'ez is a script used as an abugida for several languages of Ethiopia and Eritrea. In Amharic and Tigrinya, the script is often called fidäl, meaning "script" or "alphabet". The Ge'ez script has been adapted to write other, mostly Semitic, languages, particularly Amharic in Ethiopia, Tigrinya in both Eritrea and Ethiopia. It is also used for Sebatbeit, Me'en, most other languages of Ethiopia. In Eritrea it is used for Tigre, it has traditionally been used for Blin, a Cushitic language. Tigre, spoken in western and northern Eritrea, is considered to resemble Ge'ez more than do the other derivative languages. Some other languages in the Horn of Africa, such as Oromo, used to be written using Ge'ez, but have migrated to Latin-based orthographies. For the representation of sounds, this article uses a system, 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. After the 7th and 6th centuries BC, however, variants of the script arose, evolving in the direction of the Ge'ez abugida. This evolution can be seen most clearly in evidence from inscriptions in Tigray region in northern Ethiopia and the former province of Akkele Guzay in Eritrea. At least one of Wazeba's 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 vowel marking pattern of the script reflects a South Asian system, such as would have been known by Frumentius. Ge'ez has 26 consonantal letters.
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
27.
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. An older method for writing numerals exists in which most of letters of the Georgian alphabet are each assigned a numeric value. The cardinal numerals up to ten are primitives, as are etc.. Cardinal numbers are formed from these primitives via a mixture of decimal and structural principles. The following chart shows the nominative forms of the primitive numbers. Except for tskhra, these words are all consonant-final may lose the final i in certain situations. Numbers from 11 to 19 are formed from 1 respectively, by adding met ` i. In some cases, the prefixed t coalesces with the initial consonant of the root word to form a single consonant, or induces metathesis in the root. Numbers between 20 and 99 use a vigesimal system. . . 10 directly to the word for 100. 1000 is expressed as atasi, multiples of 1000 are expressed using atasi — so, for example, 2000 is ori atasi. 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. 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).
28.
Greek numerals
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Greek numerals is a system of representing numbers using the letters of the Greek alphabet. These alphabetic numerals are also known by names Ionic or Ionian numerals, Milesian numerals, Alexandrian numerals. 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. They ran = 1, = 10000. 50000 were represented by the letter with minuscule powers 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 position of those 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, from rho to sampi. This alphabetic system operates on the additive principle in which the numeric values of the letters are added together to obtain the total.
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.
29.
Hebrew numerals
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The system of Hebrew numerals is a quasi-decimal alphabetic numeral system using the letters of the Hebrew alphabet. The system was adapted from that of the Greek numerals in the late 2nd century BC. The current numeral system is also known as the Hebrew alphabetic numerals to contrast with earlier systems of writing numerals used in classical antiquity. The Greek system was adopted in Hellenistic Judaism and had been in use in Greece since about the 5th century BC. In this system, there is no notation for zero, the numeric values for individual letters are added together. Each unit is assigned a separate letter, each tens a separate letter, the first four hundreds a separate letter. The later hundreds are represented by the sum of two or three letters representing the first four hundreds. To represent numbers from 1,000 to 999,999, the same letters are reused to serve as thousands, tens of thousands, hundreds of thousands. Gematria uses these transformations extensively. In Israel today, the decimal system of Arabic numerals is used in almost all cases. Numbers in Hebrew from zero to one million. Hebrew alphabet are used to a limited extent to represent numbers, widely used on calendars. For other uses Arabic numerals are included. Cardinal and ordinal numbers must agree in gender with the noun they are describing. If there is no such noun, the feminine form is used.
Hebrew numerals
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Numeral systems
Hebrew numerals
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The lower clock on the Jewish Town Hall building in Prague, with Hebrew numerals in counterclockwise order.
Hebrew numerals
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Early 20th century pocket watches with Hebrew numerals in clockwise order (Jewish Museum, Berlin).
30.
Roman numerals
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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, X. Numbers are formed by combining symbols and adding the values, so II is two and XIII is thirteen. Symbols are placed from left to right in order of value, starting with the largest. Usage in ancient Rome varied greatly and remained inconsistent in medieval and modern times. 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 instances such as IIIIII and XXXXXX rather than VI or LX. Such variation and inconsistency continued through the medieval period and into modern times, even becoming conventional. 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. Although Roman numerals came to be written with letters of the Roman alphabet, they were originally independent symbols. Thus, ⟨I⟩ descends not from the letter ⟨I⟩ but from a notch scored across the stick. Every fifth notch was double cut i.e. ⋀, ⋁, ⋋, ⋌, etc.), every tenth was cross cut, IIIIΛIIIIXIIIIΛIIIIXII...), much like European tally marks today.
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
31.
Aegean numerals
Aegean numerals
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Numeral systems
32.
Attic numerals
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Attic numerals were used by the ancient Greeks, possibly from the 7th century BC. They were also known as Herodianic numerals because they were first described by Herodian. See Greek numerals and acrophony. The use of Η for 100 reflects the early date of this system: Η in the early Attic alphabet represented the sound / h /. It wasn't until Aristophanes of Byzantium introduced the various accent markings during the Hellenistic period that the spiritus asper began to represent /h/. Thus the word for a hundred would originally have been written ΗΕΚΑΤΟΝ, as compared to the now more familiar ἑκατόν. This has had no effect on the basic spelling. Unlike the more familiar Modern Roman numeral system, the Attic system contains only additive forms. Thus, 4 is written ΙΙΙΙ, not ΙΠ. The numerals representing 50, 500, 5,000 were composites of pi and a tiny version of the applicable power of ten. For example, is five times thousand. Example: 1982 = ΧΗΗΗΗ. ΔΔΔΙΙ = MCM. LXXXII. Specific numeral symbols were used to represent talents and staters, to represent ten mnas and to represent one half and one quarter.
Attic numerals
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Numeral systems
33.
Babylonian numerals
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The Babylonians, who were famous for their astronomical calculations, used a positional numeral system inherited from either the Sumerian or the Eblaite civilizations. Neither of the predecessors was a positional system. This system first appeared around 2000 BC; its structure reflects the lexical numerals of Semitic languages rather than lexical numbers. However, the use of a special Sumerian sign for 60 attests to a relation with the Sumerian system. This was an extremely important development, because non-place-value systems require unique symbols to represent each power of a base, which can make calculations more difficult. Only two symbols were used to notate the 59 non-zero digits. A space was left to indicate a place without value, similar to the modern-day zero. Babylonians later devised a sign to represent this empty place. Fractions were represented identically -- a point was not written but rather made clear by context. The Babylonians did not technically have a digit for, nor a concept of, the number zero. Although they understood the idea of nothingness, it was not seen as a number—merely the lack of a number. What the Babylonians had instead was a space to mark the nonexistence of a digit in a certain place value. Babylon Babylonia History of zero Numeral system Menninger, Karl W.. Number Words and Number Symbols: A Cultural History of Numbers. MIT Press.
Babylonian numerals
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Numeral systems
Babylonian numerals
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Babylonian numerals
34.
Brahmi numerals
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The Brahmi numerals are a numeral system attested from the 3rd century BCE. They are the direct graphic ancestors of the modern Indian and Hindu–Arabic numerals. However, they were conceptually 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 1000 which were combined to signify 200, 300, 2000, 3000, etc.. In the oldest inscriptions, 4 is a +, reminiscent of the X of neighboring Kharoṣṭhī, perhaps a representation of 4 lines or 4 directions. However, the other unit numerals appear to be arbitrary symbols in even the oldest inscriptions. Another possibility is that the numerals were acrophonic, like the Attic numerals, based on the Kharoṣṭhī alphabet. 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 they're alphabetic, are purely speculative 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 pub. J. Wiley, 2000. 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)
35.
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. The hieratic form of numerals stressed an exact finite series notation, ciphered one to one onto the Egyptian alphabet. For instance, a stone carving from Karnak shows the number 4622 as Egyptian hieroglyphs could be written in both directions. Rational numbers could also be expressed, but only as sums of unit fractions, i.e. sums of reciprocals of positive integers, except for 2⁄3 and 3⁄4. Instances of numerals written in hieratic can be found as far back as the Early Dynastic Period. The Old Kingdom Abusir Papyri are a particularly important corpus of texts that utilize hieratic numerals. A large number like 9999 could thus be written with only four signs—combining the signs for 9000, 900, 90, 9—as opposed to 36 hieroglyphs. 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 clearly 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, repeated as Roman numerals practiced. 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; this process continued into Demotic as well. 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 vowel remains uncertain: Ancient Egypt Egyptian language Egyptian mathematics Allen, James Paul.
Egyptian numerals
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Numeral systems
36.
Etruscan numerals
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The Etruscan numerals were used by the ancient Etruscans. The system was adapted from the Greek Attic numerals and formed the inspiration for the later Roman numerals via the Old Italic script. There is very little surviving evidence of these numerals. Examples are known of the symbols for larger numbers, but it is unknown which symbol represents which number. The assignment depended on the answer to the question whether the numbers on opposite faces on Etruscan dice add up to seven, like nowadays. Some dice found did not show this proposed pattern. An interesting aspect of the Etruscan numeral system is that some numbers, as in the Roman system, are represented as partial subtractions. So "17" is not written *semφ-śar as users of the Hindu-Arabic numerals might reason. We instead find ci-em zaθrum, literally "three from twenty". The numbers 17, 18 and 19 are all written in this way. The general agreement among Etruscologists nowadays is the following: Archaeological evidence strongly supports the correspondence 4/huth and 6/sa. However, other scholars disagree with this attribution. In 2006, S. A. Yatsemirsky presented evidence that zar = śar meant ‘12’ while halχ meant ‘10’. According to his interpretation, the attested form huθzar was used for ‘sixteen’, not ‘fourteen’, assuming huθ meant four. Much debate has been carried out about a possible Indo-European origin of the Etruscan cardinals.
Etruscan numerals
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Numeral systems
Etruscan numerals
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History
37.
Inuit numerals
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Inuit, like other Eskimo languages, uses a vigesimal counting system. Inuit counting has sub-bases at 5, 15. The picture below shows 1 -- 19 and then 0. Twenty is written with a one and a zero, forty with a one and two zeros. The corresponding spoken forms are: In Greenlandic Inuit language:
Inuit numerals
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Numeral systems
38.
Maya numerals
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The Maya numeral system is a vigesimal positional numeral system used by the Pre-Columbian Maya civilization. The numerals are made up of three symbols; five. For example, thirteen is written as three dots in a horizontal row above two horizontal lines stacked above each other. Numbers after 19 were written vertically in powers of twenty. For example, thirty-three would be written as one dot above three dots, which are in turn atop two lines. The first dot represents "one twenty" or "1×20", added to three dots and two bars, or thirteen. Therefore, + 13 = 33. Upon reaching 202 or 400, another row is started. The 429 would + + 9 = 429. The powers of twenty are numerals, just as the Hindu-Arabic numeral system uses powers of tens. Other than Maya numerals can be illustrated by type glyphs or pictures. The face glyph for a number represents the deity associated with the number. These number glyphs are mostly seen on some of the most elaborate monumental carving. Addition and subtraction: Adding and subtracting numbers below 20 using Maya numerals is very simple. If four or more bars result, four bars are removed and a dot is added to the next higher row.
Maya numerals
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Numeral systems
Maya numerals
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Maya numerals
Maya numerals
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Detail showing three columns of glyphs from La Mojarra Stela 1. The left column uses Maya numerals to show a Long Count date of 8.5.16.9.7, or 156 CE.
39.
Quipu
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A quipu usually plied thread or strings made from cotton or camelid fiber. For the Inca, the system aided in collecting data and properly collecting census records, calendrical information, military organization. The cords contained numeric and other values encoded by knots in a base ten positional system. A quipu could have only a few or up to 2,000 cords. The configuration of the quipus have also been "compared to string mops." A relatively small number have survived. Objects that can be identified unambiguously as quipus first appear in the archaeological record in the first millennium AD. As the region was subsumed under the invading Spanish Empire, the use of the quipu faded from use, to be replaced by European writing systems. However, in several villages, quipu continued to be important items for the local community, albeit for ritual rather than recording use. It is unclear as to how intact quipus still exist, as many have been stored away in mausoleums, ` along with the dead.' Quipu is the Spanish spelling and the most common spelling in English. Khipu is the word for "knot" in Cusco Quechua; the kh is an aspirated k. In most Quechua varieties, the term is kipu. The word "khipu", meaning "knot" or "to knot", comes from the Quechua word: the "lingua franca and language of administration" of Tahuantinsuyu. Most information recorded on the quipus consists of numbers in a decimal system.
Quipu
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An example of a quipu from the Inca Empire, currently in the Larco Museum Collection.
Quipu
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Numeral systems
40.
Prehistoric numerals
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Counting in prehistory was first assisted by using body parts, primarily the fingers. Early systems of counting using tally marks appear in the Upper Paleolithic. The first more complex systems develop in the Ancient Near East together with the development of early writing out of proto-writing systems. Numerals originally developed with the oldest examples being about 35,000 to 25,000 years old. Counting aids like tally marks become developing into various types of proto-writing. The Cuneiform script develops out of proto-writing associated with keeping track of goods during the Chalcolithic. The Moksha people, whose existence dates to about the beginning of the 1st millennium BC, had a system. The numerals were tally marks carved on wood, drawn on clay or bark. In some places they were preserved mostly among small traders, bee-keepers, village elders. These numerals still can be found on old shepherd and tax-gatherer staffs, pottery. Http://www.thocp.net/timeline/0000.htm http://members.fortunecity.com/jonhays/tallying.htm
Prehistoric numerals
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Numeral systems
Prehistoric numerals
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Old Mokshan numerals
41.
Positional notation
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Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the same symbol for the different orders of magnitude. This greatly simplified arithmetic, leading to the rapid spread of the notation across the world. With the use of a point, the notation can be extended to include the numeric expansions of real numbers. The Babylonian system, base-60, was the positional system developed, is still used today to count time and angles. The Hindu–Arabic numeral system, base-10, is the most commonly used system in the world today for most calculations. Today, the base-10 system, likely motivated by counting with the ten fingers, is ubiquitous. Other bases have been used in the past however, some continue to be used today. For example, the Babylonian numeral system, credited as the first positional numeral system, was base-60, but it lacked a real 0 value. Zero was indicated by a space between sexagesimal numerals. By 300 BC, a punctuation symbol was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish, the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges. The Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number. Thus numbers like 120, 4 and 240, looked the same because the larger numbers lacked a final sexagesimal placeholder.
Positional notation
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Numeral systems
42.
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 the decimal system the radix is ten, because it uses the ten digits from 0 through 9. For example, 10 represents the number 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. Commonly used numeral systems include: For a larger list, see List of numeral systems. The hexadecimal systems are often used in computing because of their ease as shorthand for binary. Every hexadecimal digit corresponds to a sequence of four binary digits, since sixteen is the fourth power of two; for example, hexadecimal 7816 is binary 11110002. A similar relationship holds between every possible sequence of three binary digits, since eight is the cube of two. Radices are usually natural numbers. However, positional systems are possible, e.g. golden ratio base, negative 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
43.
Binary number
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The base-2 system is a positional notation with a radix of 2. Of its straightforward implementation in electronic circuitry using logic gates, the binary system is used internally by almost all modern computers and computer-based devices. Each digit is referred to as a bit. The binary system was devised by Gottfried Leibniz in 1679 and appears in his article Explication de l'Arithmétique Binaire. Systems related to binary numbers have appeared earlier in multiple cultures including India. Leibniz was specifically inspired by the Chinese I Ching. The scribes of ancient Egypt used two different systems for Horus-Eye fractions. The method used for ancient Egyptian multiplication is also closely related to binary numbers. This method can be seen in the Rhind Mathematical Papyrus, which dates to around 1650 BC. 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. It is based on taoistic duality of yin and yang. The Indian scholar Pingala developed a binary system for describing prosody. He used binary numbers in the form of short and long syllables, making it similar to Morse code. Pingala's Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter.
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
44.
Ternary numeral system
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The ternary numeral system has three as its base. Analogous to a bit, a ternary digit is a trit. One trit is equivalent to log23 bits of information. Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary. For example, decimal 365 corresponds to binary 101101101 and to ternary 111112. The value of a binary number with n bits that are all 1 is 2n − 1. Then N = M, N = /, N = bd − 1. Nonary or septemvigesimal can be used for compact representation of ternary, similar to how octal and hexadecimal systems are used in place of binary. A base-three system is used in Islam to keep track of counting Tasbih to 99 or to 100 on a single hand for counting prayers. In certain analog logic, the state of the circuit is often expressed ternary. This is most commonly seen in Transistor–transistor logic using 7406 open collector logic. The output is said to either be low, high, or open. In this configuration the output of the circuit is actually not connected to any voltage reference at all. Thus, the actual voltage level is sometimes unpredictable. A rare "ternary point" is used to denote fractional parts of an inning in baseball.
Ternary numeral system
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Numeral systems
45.
Quaternary numeral system
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Quaternary is the base-4 numeral system. It uses the digits 0, 3 to represent any real number. Despite being as large, its economy is equal to that of binary. However, it fares no better in the localization of prime numbers. See decimal and binary for a discussion of these properties. As with the hexadecimal numeral systems, quaternary has a special relation to the binary system. In base 4, 302104 = 01 002. By analogy with byte and nybble, a quaternary digit is sometimes called a crumb. There is a surviving list of Ventureño number words up to 32 written down by a Spanish priest 1819. 1819. 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.
Quaternary numeral system
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Numeral systems
46.
Quinary
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Quinary is a numeral system with five as the base. A possible origination of a quinary system is that there are five fingers on either hand. The base five is stated from 0–4. In the quinary 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 and sixty is written as 220. The main usage of base 5 is as a biquinary system, decimal using five as a sub-base. Another example of a sub-base system, is base 60, which used 10 as a sub-base. Each quinary digit has log25 bits of information. Many languages use quinary number systems, including Gumatj, Nunggubuyu, Kuurn Kopan Noot, Saraveca. Gumatj is a true "5–25" language, in which 25 is the higher group of 5. The Gumatj numerals are shown below: In subsequent games of the Myst franchise, the D'ni language uses a quinary numeral system. A decimal system as a sub-bases is called biquinary, is found in Wolof and Khmer. Roman numerals are a biquinary system. Numbers 1, 5, 10, 50 are written as I, V, X, L respectively. Eight is seventy is LXX.
Quinary
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Numeral systems
47.
Senary
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The senary numeral system has six as its base. It has been adopted independently by a small number of cultures. As six is a superior highly composite number, many of the arguments made in favor of the duodecimal system also apply to this base. This is proved by contradiction. This property maximizes the probability that the result of an multiplication will end in zero, given that neither of its factors do. E.g. if three fingers are extended on the right, 34senary is represented. This is equivalent to 3 × 6 + 4, 22decimal. Flipping the ` sixes' hand around to its backside may help to further disambiguate which represents the units. More abstract finger counting systems, such as chisanbop or binary, allow counting to 99, 1,023, or even higher depending on the method. The Ndom language of Papua New Guinea is reported to have senary numerals. Mer means 6, mer an thef means 6 × 2 = 12, nif thef means 36 × 2 = 72. Another example from Papua New Guinea are the Morehead-Maro languages. In these languages, counting is connected to ritualized yam-counting. These languages count from a base six, employing words for the powers of six; running up for some of the languages. One example is Kómnzo with the following numerals: nimbo, féta, tarumba, wärämäkä, wi.
Senary
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Numeral systems
Senary
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34 senary = 22 decimal, in senary finger counting
Senary
48.
Octal
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The octal numeral system, or oct for short, is the base-8 number system, uses the digits 0 to 7. Octal numerals can be made by grouping consecutive binary digits into groups of three. For example, the binary representation for decimal 74 is 1001010. Two zeroes can be added at the left: 001 010, corresponding the octal digits 1 1 2, yielding the octal representation 112. In the decimal system each decimal place is a power of ten. In the octal system each place is a power of eight. It has been suggested that the Proto-Indo-European word for "nine" might be related to the PIE word for "new". Based on this, some have speculated that proto-Indo-Europeans used an octal system, though the evidence supporting this is slim. In 1716 King Charles XII of Sweden asked Emanuel Swedenborg to elaborate a system based on 64 instead of 10. In 1718 Swedenborg wrote a manuscript: "En ny rekenkonst som vexlas wid Thalet 8 i stelle then wanliga wid Thalet 10". The numbers 1-7 are there denoted by the consonants l, s, n, m, t, f, u and zero by the vowel o. Thus 8 = "lo", 16 = "so", 24 = "no", 64 = "loo", 512 = "looo" etc. Numbers with consecutive consonants are pronounced between in accordance with a special rule. Writing under July 1745, Hugh Jones proposed an octal system for British coins, weights and measures. In 1801, James Anderson criticized the French for basing the Metric system on decimal arithmetic.
Octal
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Numeral systems
49.
Duodecimal
–
The duodecimal system is a positional notation numeral system using twelve as its base. In this system, the number ten may be written by a rotated "2" and the number eleven by a rotated "3". This notation was introduced by Sir Isaac Pitman. These digit forms are available as Unicode characters on computerized systems since June 2015 as ↊ and ↋, respectively. "B" or "E" for eleven. The twelve is instead written in duodecimal whereas the digit string "12" means "1 dozen and 2 units". Similarly, in duodecimal "100" means "1 gross", "1000" means "1 great gross", "0.1" means "1 twelfth". As a result, duodecimal has been described as the optimal number system. Of its factors, 2 and 3 are prime, which means the reciprocals of all 3-smooth numbers have a terminating representation in duodecimal. In particular, the five most elementary fractions all have a short terminating representation in duodecimal, twelve is the smallest radix with this feature. In this section, numerals are based on decimal places. For example, 10 means ten, 12 means twelve. Languages using duodecimal number systems are uncommon. Germanic languages have special words in English. However, they are considered to come from Proto-Germanic * twalif, both of which were decimal.
Duodecimal
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Numeral systems
Duodecimal
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A duodecimal multiplication table
50.
Hexadecimal
–
In mathematics and computing, hexadecimal is a positional numeral system with a radix, or base, of 16. A, B, C, D, E, F to represent values ten to fifteen. Hexadecimal numerals are widely used by system designers and programmers. As each hexadecimal digit represents four binary digits, it allows a human-friendly representation of binary-coded values. One hexadecimal digit represents a nibble, half of an byte. Several different notations are used to represent hexadecimal constants in computing languages; the prefix "0x" is widespread due to its use in Unix and C. Alternatively, some authors denote hexadecimal values using a suffix or subscript. For example, one could write 2AF316, depending on the choice of notation. In contexts where the base is not clear, hexadecimal numbers can be confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously. A numerical subscript can give the base explicitly: 15910 is decimal 159; 15916 is hexadecimal 159, equal to 34510. Some authors prefer a subscript, such as 159decimal and 159hex, or 159d and 159h. Thus ’ represents the right single quotation mark, Unicode code point number 2019 in hex, 8217. In the Unicode standard, a value is represented with U + followed by the hex value, e.g. U +20 AC is the Euro sign. Color references in HTML, CSS and X Window can be expressed with six hexadecimal digits prefixed with #: white, for example, is represented #FFFFFF.
Hexadecimal
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Numeral systems
Hexadecimal
–
Bruce Alan Martin's hexadecimal notation proposal
Hexadecimal
–
Hexadecimal finger-counting scheme.
51.
Vigesimal
–
The vigesimal or base 20 numeral system is based on twenty. In a vigesimal system, twenty individual numerals are used, ten more than in the usual decimal system. This is similar to the computer-science practice of writing hexadecimal numerals over 9 with the letters "A -- F". The twenty is written as 1020. 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 digit periods because 9 is the number below ten. The number adjacent to 20, 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, 5, is congruent to 1. The fraction of primes that are cyclic is not necessarily ~ 37.395 %. In European languages, 20 is used as a base, at least with respect to the linguistic structure of the names of certain numbers. Vigesimal systems are common in Africa, for example in Yoruba. 20, is the basic numeric block.
Vigesimal
–
Numeral systems
Vigesimal
–
The Maya numerals are a base-20 system.
52.
Non-standard positional numeral systems
–
The numbers written in superscript represent the powers of the base used. Upon introducing a radix point "." 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. A bijective system with base b uses different numerals to represent all non-negative integers. However, the numerals have 3, etc. up to and including b, whereas zero is represented by an empty 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. Non-standard features of this system include: The value of a digit does not depend on its position. Thus, one can easily 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 0 = b 1. The value 0 cannot be represented.
Non-standard positional numeral systems
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Numeral systems
53.
Bijective numeration
–
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 non-negative 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 positive integer. In particular, adding leading zeroes does not change the value represented, so "1", "01" and "001" all represent the number one. Even though only the first is usual, the fact that the others are possible means that decimal is not bijective. However, unary, with only one digit, is bijective. A base-k numeration is a positional notation. The bijective 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 digit-string anan 1... a1a0 is an kn + an − 1 kn − 1 +... + a1 k1 + a0 k0. For a given k ≥ 1, there are exactly kn base-k numerals of length n ≥ 0. 119A = 1×103 + 1×102 + 9×101 + 10×1 = 1200. The 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 position represents a power of ten, so for example 123 is "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.
Bijective numeration
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Numeral systems
54.
Unary numeral system
–
The unary numeral system is the bijective base-1 numeral system. For examples, the numbers 1, 2, 3, 4, 5... would be represented in this system as 1, 11, 111, 1111, 11111... These numbers should be distinguished from repunits, which are also written as sequences of ones but have their usual decimal numerical interpretation. This system is used in tallying. For example, using 3 is represented as | | |. In East Asian cultures, three is represented as a character, drawn with three strokes. Subtraction are particularly simple in the unary system, as they involve little more than concatenation. However, multiplication has often been used as a case for the design of Turing machines. Compared to standard positional numeral systems, the unary system is inconvenient and hence is not used in practice for large calculations. It occurs in some problem descriptions in theoretical science, where it is used to "artificially" decrease the run-time or space requirements of a problem. However, this is potentially misleading. Therefore, while the requirement in unary looks better as function of the input size, it does not represent a more efficient solution. In computational complexity theory, unary numbering is used to distinguish strongly NP-complete problems from problems that are NP-complete but not strongly NP-complete. For such a problem, there exist hard instances for which all parameter values are at most polynomially large. Unary is used as part of some data compression algorithms such as Golomb coding.
Unary numeral system
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Numeral systems
55.
Signed-digit representation
–
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 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, where typically k = ⌊ b 2 ⌋. For balanced forms, odd base numbers are advantageous. With an odd base number, all the digits except 0 are used in both positive and negative form. The numerals have the values − 1, 0 and +1. Balanced ternary uses the minimum number of digits in a balanced form. Balanced decimal uses digits from −5 to +4. Note that signed-digit representation is not necessarily unique. The written forms of numbers in the Punjabi language use a form of a negative numeral one written as una or un. This negative one is used to form 29... 89 from the root for 20, 30... 90. Similarly, the Sesotho language utilizes negative numerals to 8's and 9's.
Signed-digit representation
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Numeral systems
56.
Balanced ternary
–
Balanced ternary is a non-standard positional numeral system, useful for comparison logic. While it is a ternary system, in the standard ternary system, digits have values 0, 1 and 2. The digits in the balanced system have values − 1, 0, 1. 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. In Setun printings, 1 is represented as overturned 1: "1". The notation has a number of computational advantages over regular binary. Particularly, the rounding-truncation equivalence cuts down the carry rate in rounding on fractions. Balanced ternary also has a number of computational advantages over traditional ternary. Particularly, the addition table has only two symmetric carries instead of three. In the balanced 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 balanced ternary. In the following the strings denoting balanced ternary carry bal3. For instance, − 2/3dec = − + 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.
Balanced ternary
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Numeral systems
57.
Factorial number system
–
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, but as place value of digits. 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", a portmanteau of factorial and mixed radix, appears to be of more recent date. . 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 "!", so for instance 341010! Stands for 364514031201, whose value is = 3×5! + 4×4! + 1×3! + 0×2! + 1×1! + 0×0!
Factorial number system
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Numeral systems
Factorial number system
–
Permutohedron graph showing permutations and their inversion vectors (compare version with factorial numbers) The arrows indicate the bitwise less or equal relation.
58.
Negative base
–
A negative base may be used to construct a non-standard positional numeral system. The need to store the information normally contained by a negative sign often results in a negative-base number being one digit longer than its positive-base equivalent. Numerical bases were first considered by Vittorio Grünwald in his work Giornale di Matematiche di Battaglini, published in 1885. Grünwald gave algorithms for performing addition, subtraction, multiplication, division, extraction, divisibility tests, radix conversion. Negative bases were independently rediscovered by A. J. Kempner in 1936 and Zdzisław Pawlak and A. Wakulicz in 1959. Negabinary was implemented in the Polish computer BINEG, built 1957 -- 59, based on ideas by Z. Pawlak and A. Lazarkiewicz from the Mathematical Institute in Warsaw. Implementations since then have been rare. The − r expansion of a is then given by the string dndn-1... d1d0. Some numbers have the same representation in base −r as in base r. For example, the numbers from 100 to 109 have the same representations in decimal and negadecimal. Similarly, 17 = 4 + 2 0 = 4 + 0 and is represented by 10001 in binary and 10001 in negabinary. Note that if a / b = c, remainder d, then bc + d = a. Note that in most programming languages, the result of dividing a negative number by a negative number is rounded towards 0, usually leaving a negative remainder. In such a case we have a = c + d = c + d − r + r = +. Because |d| < r, is the positive remainder.
Negative base
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Numeral systems
59.
Quater-imaginary base
–
The quater-imaginary numeral system was first proposed by Donald Knuth in 1960. It is a positional system which uses the imaginary number 2i as its base. It is able to uniquely represent every complex number using digits 0, 1, 3. … d 3 d 2 d 1 d 0. The real and imaginary parts of this 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. Finding the quater-imaginary representation of an real number can be done manually by solving a system of simultaneous equations, as shown below. But there are faster methods for both, imaginary, integers, as shown in section base #To Negaquaternary. As an example of an number we can try to find 7. In this case, a string of six digits can be chosen.
Quater-imaginary base
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Numeral systems
60.
Non-integer representation
–
A non-integer representation uses non-integer numbers as the radix, or bases, of a positional numbering system. For a non-integer β > 1, the value of x = d n... 2 d 1 d 0. The numbers di are non-negative integers less than β. This is also known as a notion first studied in detail by Parry. 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 decimal expansions are not unique, all decimal expansions are unique. However, even finite β-expansions are not necessarily unique, for example φ 1 = φ2 for the golden ratio. Let > 1 be x a non-negative real number. There exists an integer 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 =. Thus it chooses the lexicographically largest string representing x. With an integer base, this defines the usual radix expansion for the number x.
Non-integer representation
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Numeral systems
61.
Golden ratio base
–
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 phi-base, or, colloquially, phinary. For instance, 11φ = 100φ. Despite using an irrational 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. These representations are unique, 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. To "standardize" a numeral, we can use the following substitutions: 011φ = 100φ, 0200φ = 1001φ, 110φ = 001φ. We can apply the substitutions in any order we like, as the result is the same. Below, the substitutions applied to the number on the previous line are on the resulting number on the left. Any positive number with a non-standard terminating base-φ representation can be uniquely standardized in this manner. If we get to a point where all digits are "0" or "1", for the first digit being negative, then the number is negative. This can be converted by negating every digit, standardizing the result, then marking it as negative. For example, use some other significance to denote negative numbers. If the arithmetic is being performed on a computer, an message may be returned.
Golden ratio base
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Numeral systems
62.
Mixed radix
–
Mixed radix numeral systems are non-standard positional numeral systems in which the numerical base varies from position to position. In numeral format, the radix point is marked by a full stop or period. The base for each digit is the number of corresponding units that make up the next larger unit. As a consequence there is no base for the first digit, since here the "next larger unit" does not exist. The most familiar example of mixed radix systems is in timekeeping and calendars. Western time radices include decimal centuries, years as well as duodecimal months, trigesimal days, overlapped with base 52 weeks and septenary days. One variant uses tridecimal months, septenary days. Time is further divided by quadrivigesimal hours, then decimal fractions thereof. A mixed radix system can often benefit from a tabular summary. Hoc notations for mixed radix numeral systems are commonplace. The Maya calendar consists of several overlapping cycles of different radices. A short tzolk ` in overlaps vigesimal named days with tridecimal numbered days. A haab' consists of vigesimal days, base-52 years forming a round. In addition, a long count of vigesimal days, then vigesimal tun, k ` atun, b ` ak ` tun, etc. tracks historical dates. Mixed-radix numbers of the same base can be manipulated using a generalization of manual arithmetic algorithms.
Mixed radix
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Numeral systems
63.
List of numeral systems
–
This is a list of numeral systems, writing systems for expressing numbers. Numeral systems are classified as to whether they use positional notation, further categorized by base. The common names are derived arbitrarily in some cases including roots from both languages within a single name. In this Youtube video, Matt Parker jokingly invented a base-1082 system. 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
64.
10 (number)
–
10 is an even natural number following 9 and preceding 11. Ten is the base of the decimal system, by far the most common system of denoting numbers in both spoken and written language. The reason for the choice of ten is assumed to be that humans have ten fingers. A collection of ten items is called a decade. The ordinal adjective is decimal; the distributive adjective is denary. Increasing a quantity by one order of magnitude is most widely understood to mean multiplying the quantity by ten. To reduce something by one tenth is to decimate. Ten is its proper divisors being 1, 2 and 5. Ten is a number that can not be expressed as the difference between any integer and the total number of coprimes below it. Ten is the second member of the discrete semiprime family. Ten is accordingly the first discrete semiprime to be in deficit. All discrete semiprimes are in deficit. The aliquot sequence for 10 comprises five members with this number being the composite member of the 7-aliquot tree. It is the aliquot sum of the discrete semiprime 14. Ten is a semi-meandric number.
10 (number)
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10 playing cards of all four suits
10 (number)
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The tetractys
65.
Hindu-Arabic numeral system
–
It was invented between the 1st and 4th centuries by Indian mathematicians. The system was adopted by Persian mathematicians and Arab 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 still 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, also a symbol for "these digits recur ad infinitum". In modern usage, this latter symbol is usually a vinculum. In this more developed form, the numeral system can symbolize any rational number using only 13 symbols. The requisite changes in reading direction are found in text that mixes left-to-right writing systems with right-to-left systems.
Hindu-Arabic numeral system
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Numeral systems
Hindu-Arabic numeral system
–
Arabic and Western Arabic numerals on a road sign in Abu Dhabi
Hindu-Arabic numeral system
Hindu-Arabic numeral system
66.
Floor and ceiling functions
–
In mathematics and computer science, the floor and ceiling functions map a real number to the largest previous or the smallest following integer, respectively. Carl Friedrich Gauss introduced the square bracket notation for the floor function in his third proof of quadratic reciprocity. Both notations are now used in mathematics; this article follows Iverson. The language APL uses ⌊x; other computer languages commonly use notations like entier, INT, or floor. In mathematics, it can also be written with boldface or double brackets. The ceiling function is usually denoted by ceil or ceiling in non-APL computer languages that have a notation for this function. The J Programming Language, a follow on to APL, designed to use standard keyboard symbols, uses >. for ceiling and <. for floor. In mathematics, there is another notation with reversed boldface or double brackets ] ] x x[\!x[. The fractional part is the sawtooth function, denoted by for real x and defined by the formula = x − ⌊ x ⌋. For all x, 0 ≤ < 1. HTML 4.0 uses the same names: ⌊, ⌋, ⌈, ⌉. Unicode contains codepoints for these symbols at U+2308–U+230B: ⌈x⌉, ⌊x⌋. In the following formulas, x and y are real numbers, k, m, n are integers, Z is the set of integers. Floor and ceiling may be defined by the set equations ⌊ x ⌋ = max, ⌈ x ⌉ = min. Then ⌊ x ⌋ = m and ⌈ x ⌉ = n may also be taken as the definition of floor and ceiling.
Floor and ceiling functions
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Floor function
67.
Decimal mark
–
A decimal mark is a symbol used to separate the integer part from the fractional part of a number written in decimal form. Different countries officially designate different symbols for the decimal mark. In mathematics the decimal mark is a type of radix point, a term that also applies to number systems with bases other than ten. His Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear and quadratic equations in Arabic. A similar notation remains in common use as an underbar to superscript digits, especially for monetary values without a decimal mark, e.g. 9995. Later, a "separatrix" between the units and tenths position became the norm among Arab mathematicians, e.g. 99ˌ95. When this character was typeset, it was convenient to use the existing comma or full stop instead. The separatrix was also used before the popularization of the period and mid decimal points. Gerbert of Aurillac marked triples of columns with an arc when using his Hindu–Arabic numeral-based abacus in the 10th century. Fibonacci followed this convention when writing numbers such as in his influential work Liber Abaci in the 13th century. In France, the full stop was already in use in printing to make Roman numerals more readable, so the comma was chosen. Many other countries, such as Italy, also chose to use the comma to mark the decimal units position. It has been made standard by the ISO for international blueprints. However, English-speaking countries took the comma to separate sequences of three digits. In some countries, dash may be used for mark; this is particularly common in handwriting.
Decimal mark
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Numeral systems
Decimal mark
–
Point "."
68.
Decimal
–
This article aims to be an accessible introduction. For the mathematical definition, see Decimal representation. The decimal system has ten as its base. It is the numerical base most widely used by modern civilizations. Other fractions have repeating decimal representations, whereas irrational numbers have infinite non-repeating decimal representations. Decimal notation is the writing of numbers in a base 10 system. Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Chinese numerals, as well as the Hindu-Arabic numerals used by speakers of many European languages. Roman numerals have secondary symbols for half these values. Brahmi numerals have symbols for the nine numbers 1 -- 9, another for 1000. Chinese numerals have symbols for additional symbols for powers of 10, which in modern usage reach 1072. Decimal systems include a zero and use symbols for the ten values to represent any number, no matter how large or how small. Positional notation uses positions for each power of ten: units, tens, thousands, etc.. There were at least two presumably independent sources of decimal systems in ancient civilization: the Chinese counting rod system and the Hindu-Arabic numeral system. Ten is the number, the count of thumbs on both hands. The English digit as well as its translation in many languages is also the anatomical term for fingers and toes.
Decimal
–
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
–
Ten fingers on two hands, the possible starting point of the decimal counting.
Decimal
–
Diagram of the world's earliest decimal multiplication table (c. 305 BC) from the Warring States period
69.
Irrational number
–
In mathematics, an irrational number is a real number that cannot be expressed as a ratio of integers, i.e. as a fraction. Therefore, irrational numbers, when written as decimal numbers, do not terminate, nor do they repeat. The same can be said for any irrational number. As a consequence of Cantor's proof that the rationals countable, it follows that almost all real numbers are irrational. The first proof of the existence of irrational numbers is usually attributed to a Pythagorean, who probably discovered them while identifying sides of the pentagram. His reasoning is as follows: Start with an isosceles triangle with side lengths of integers a, b, c. The ratio of the hypotenuse to a leg is represented by c:b. Assume a, b, c are in the smallest possible terms. By the Pythagorean theorem: c2 = a2+b2 = b2+b2 = 2b2. . Since c2 = 2b2, c2 is divisible by 2, therefore even. Since c2 is even, c must be even. Since c is even, b must be odd. Since c is even, dividing c by 2 yields an integer. Let y be this integer.
Irrational number
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The number is irrational.
70.
Number
–
A number is a mathematical object used to count, measure, label. The original examples are 1, 2, 3, so forth. A notational symbol that represents a number is called a numeral. In addition to their use in measuring, numerals are often used for labels, for ordering, for codes. In common usage, number may refer to a symbol, a mathematical abstraction. Calculations with numbers are done with the most familiar being addition, subtraction, multiplication, division, exponentiation. Their usage is called arithmetic. The same term may also refer to the study of the properties of numbers. Besides their practical uses, numbers have cultural significance throughout the world. For example, "a million" may signify "a lot." Though it is now regarded as pseudoscience, the belief in a mystical significance of numbers, permeated ancient and medieval thought. Numerology heavily influenced the development of Greek mathematics, stimulating the investigation of many problems in theory which are still of interest today. During the 19th century, mathematicians may be seen as extending the concept. Among the first were the hypercomplex numbers, which consist of various extensions or modifications of the complex number system. Numbers should be distinguished from the symbols used to represent numbers.
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.
71.
Hindu-Arabic numerals
–
The symbol for zero is the key to the effectiveness of the system, developed by ancient mathematicians in the Indian subcontinent around AD 500. The system was adopted by Arab mathematicians in Baghdad and passed on to the Arabs farther west. The current form of the numerals developed in North Africa, distinct in form from the Arabic numerals. The use of Arabic numerals spread around the world through European trade, books and colonialism. The term Arabic numerals is ambiguous. It most commonly refers to the numerals widely used in Europe and the Americas; to avoid confusion, Unicode calls these European digits. Arabic numerals is also the conventional name for the entire family of related numerals of Arabic and Indian numerals. It may also be intended to mean the numerals used by Arabs, in which case it generally refers to the Eastern Arabic numerals. 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 system was developed by AD 700. The decisive step was probably provided as a number in AD 628. The system was revolutionary by including zero in positional notation, thereby limiting the number of individual digits to ten. It is considered an important milestone in the development of mathematics. One may distinguish between this positional system, identical throughout the family, the precise glyphs used to write the numerals, which varied regionally. The glyphs most commonly used in conjunction with the Latin script since modern times are 0 2 3 4 5 6 7 8 9.
Hindu-Arabic numerals
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Numeral systems
Hindu-Arabic numerals
–
Arabic numerals sans-serif
Hindu-Arabic numerals
–
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
72.
Numerical digit
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A digit is a numeric symbol used in combinations to represent numbers in positional numeral systems. In a given system, if the base be an integer, the number of digits required would always equal to the absolute value of the base. For example, the decimal system has ten digits, whereas binary has two digits. In a digital system, a numeral is a sequence of digits, which may be of arbitrary length. Each digit has a value. The value of the numeral is computed by summing the results. Each digit in a system represents an integer. For example, in the hexadecimal system, the letter "A" represents the number ten. A positional system must have a digit representing the integers from zero up to, but not including, the radix of the number system. Thus in the decimal system, the numbers 0 to 9 can be expressed using their respective numerals' 0' to' 9' in the rightmost ` units' position. Each successive place to the left of this has a value equal to the place value of the previous digit times the base. The total value of the number is 1 ten, 0 ones, 4 hundredths. Note that the zero, which contributes no value to the number, indicates that the 1 is in the tens place rather than the ones place. And to the right, the digit is multiplied by the base raised by a negative n. Instead of a zero, a dot was left in the numeral as a placeholder.
Numerical digit
73.
Decimal separator
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A decimal mark is a symbol used to separate the integer part from the fractional part of a number written in decimal form. Different countries officially designate different symbols for the decimal mark. In mathematics the decimal mark is a type of radix point, a term that also applies to number systems with bases other than ten. His Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear and quadratic equations in Arabic. A similar notation remains in common use as an underbar to superscript digits, especially for monetary values without a decimal mark, e.g. 9995. Later, a "separatrix" between the units and tenths position became the norm among Arab mathematicians, e.g. 99ˌ95. When this character was typeset, it was convenient to use the existing comma or full stop instead. The separatrix was also used before the popularization of the period and mid decimal points. Gerbert of Aurillac marked triples of columns with an arc when using his Hindu–Arabic numeral-based abacus in the 10th century. Fibonacci followed this convention when writing numbers such as in his influential work Liber Abaci in the 13th century. In France, the full stop was already in use in printing to make Roman numerals more readable, so the comma was chosen. Many other countries, such as Italy, also chose to use the comma to mark the decimal units position. It has been made standard by the ISO for international blueprints. However, English-speaking countries took the comma to separate sequences of three digits. In some countries, dash may be used for mark; this is particularly common in handwriting.
Decimal separator
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Numeral systems
74.
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 Greek alphabets. Latin was originally spoken in the Italian Peninsula. Through the power of the Roman Republic, it became the dominant language, initially in Italy and subsequently throughout the Roman Empire. Vulgar Latin developed such as Italian, Portuguese, Spanish, French, Romanian. Latin, Italian and French have contributed many words to the English language. Ancient Greek roots are used in theology, biology, medicine. By the late Roman Republic, Old Latin had been standardised into Classical Latin. Vulgar Latin was the colloquial form attested in inscriptions and the works of comic playwrights like Plautus and Terence. Later, Early Modern Latin and Modern Latin evolved. Latin was used until well into the 18th century, when it began to be supplanted by vernaculars. Ecclesiastical Latin remains the Roman Rite of the Catholic Church. Many students, scholars and members of the Catholic clergy speak Latin fluently. It is taught around the world. The language has been passed down through various forms.
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.
75.
Unit of measurement
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Any other value of that 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". 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 the International System of the modern form of the metric system. In trade, measures is often a subject of governmental regulation, to ensure transparency. The International Bureau of Measures is tasked with ensuring worldwide uniformity of their traceability to the International System of Units. Metrology internationally accepted units of measures. In physics and metrology, units are standards for measurement of physical 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 measures developed ago for commercial purposes. Engineering often use larger and smaller units of measurement than those used in everyday life and indicate them more precisely.
Unit of measurement
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The former Weights and Measures office in Seven Sisters, London
Unit of measurement
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Units of measurement, Palazzo della Ragione, Padua
Unit 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)
76.
Globe
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GLOBE is the Global Legislators Organisation for a Balanced Environment, founded in 1989. GLOBE's objective is to support political leadership on issues of climate and energy security, ecosystems. Internationally, GLOBE is focused on leadership from the leaders of the emerging economies as well as formal negotiations within the United Nations. GLOBE allows legislators to work together outside the formal international negotiations. Without the burden of governmental negotiating positions, legislators have the freedom to push the boundaries of what can be politically achieved. Also, GLOBE facilitates regional policy dialogues amongst legislators. GLOBE believes that legislators have a critical role to play in holding their own governments to account for the commitments that are made during international negotiations. In addition, both 2008 U.S. Presidential candidates addressed the GLOBE Forum in Tokyo on 28 June 2008. During COP15 in Copenhagen in December 2009, UK Prime Minister Gordon Brown presented Mexican President Calderon with the GLOBE Award for International Leadership on the Environment. GLOBE has legislative members in all of the 16 major economies that have GLOBE chapters. At a regional level, they also have members within several African and Latin American countries. GLOBE International supports two International Commissions on Energy Security; and Land Use Change and Ecosystems. GLOBE’s International Commission on Climate & Energy Security was launched in the US Congress in Washington DC on 30 March 2009. This Commission comprises senior legislators from each of the major economies.
Globe
77.
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 code was originally designed to prevent spurious output from electromechanical switches. Gray codes are widely used to facilitate error correction in digital communications such as digital terrestrial television and some cable TV systems. Bell Labs researcher Frank Gray introduced the term reflected binary code in his 1947 application, remarking that the code had "as yet no recognized name". He derived the name from the fact that it "may be built up by a sort of reflection process". The code was later named by others who used it. A 1954 application refers to "the Bell Telephone Gray code". Many devices indicate position by opening switches. In the transition between the two states 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. 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 "cyclic" property of a Gray code. These codes are also known as single-distance codes, reflecting the Hamming distance between adjacent codes. Reflected binary codes were applied to mathematical puzzles before they became known to engineers.
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"
78.
Fraction (mathematics)
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A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, simple fraction consists of a non-zero denominator, displayed below that line. Denominators are also used in fractions that are not common, including mixed numerals. The picture to the right illustrates 3 4 or ¾ of a cake. Fractional numbers can also be written by using negative exponents. An integer such as the number 7 can be thought of as having an implicit denominator of one: 7 equals 7/1. Other uses for fractions are to represent ratios and to represent division. Thus 3/4 is also used to represent 3 ÷ 4. The test for a number being a rational number is that it can be written in that form. Informally, they may be distinguished by placement alone but in formal contexts they are always separated by a fraction bar. The bar may be diagonal. These marks are respectively known as the fraction slash. The denominators of English fractions are generally expressed as ordinal numbers, in the plural if the numerator is not one. When the numerator is one, it may be omitted.
Fraction (mathematics)
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A cake with one quarter (one fourth) removed. The remaining three fourths are shown. Dotted lines indicate where the cake may be cut in order to divide it into equal parts. Each fourth of the cake is denoted by the fraction ¼.
79.
Denominator
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A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, eight-fifths, three-quarters. A common, simple fraction consists of an integer numerator displayed above a line, a non-zero integer denominator, displayed below that line. Denominators are also used in fractions that are not common, including compound fractions, complex fractions, mixed numerals. The picture to the right illustrates 3/4 of a cake. Fractional numbers can also be written by using decimals, percent signs, or negative exponents. An integer such as the number 7 can be thought of as having an implicit denominator of one: 7 equals 7/1. Other uses for fractions are to represent division. Thus 3/4 is also used to represent the ratio 3:4 and the division 3 ÷ 4. The test for a number being a rational number is that it can be written in that form. Informally, in formal contexts they are always separated by a fraction bar. The bar may be horizontal, oblique, or diagonal. These marks are respectively known as the horizontal bar, the slash or stroke, the fraction slash. The denominators of English fractions are generally expressed in the plural if the numerator is not one. When the numerator is one, it may be omitted.
Denominator
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A cake with one quarter (one fourth) removed. The remaining three fourths are shown. Dotted lines indicate where the cake may be cut in order to divide it into equal parts. Each fourth of the cake is denoted by the fraction ¼.
80.
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. The exponent −1 of b, or 1 / b, is called the reciprocal of b. The definition of exponentiation can be extended to allow any real or complex exponent. Exponentiation by integer exponents can also be defined for a wide variety of algebraic structures, including matrices. 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. Nicolas Chuquet used a form of exponential notation in the 15th century, later used by Henricus Grammateus and Michael Stifel in the 16th 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 terms square, cube, second sursolid, zenzizenzizenzic. Biquadrate has been used to refer to the fourth power as well. Some mathematicians used exponents only for powers greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials, for example, as ax + bxx + cx3 + d.
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.
81.
Integer part
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In mathematics and computer science, the floor and ceiling functions map a real number to the largest previous or the smallest following integer, respectively. Carl Friedrich Gauss introduced the square bracket notation in his third proof of quadratic reciprocity. Both notations are now used in mathematics; this article follows Iverson. The APL uses ⌊ x; other computer languages commonly use notations like entier, INT, or floor. In mathematics, it can also be written with double brackets. The function is usually denoted by ceil or ceiling in non-APL computer languages that have a notation for this function. The J Programming Language, a follow on to APL, designed to use standard keyboard symbols, uses >. for ceiling and <. for floor. In mathematics, there is another notation with reversed boldface or double brackets ] ] x x[\!x[. The fractional part is the sawtooth function, defined by the formula = x − ⌊ x ⌋. For all x, 0 ≤ < 1. HTML 4.0 uses & rceil;. Unicode contains codepoints for these symbols at U +2308 -- U +230 B: ⌊ x ⌋. In the following formulas, Z is the set of integers. Ceiling may be defined by the set equations ⌊ x ⌋ = max, ⌈ x ⌉ = min. Then ⌈ x ⌉ = n may also be taken as the definition of floor and ceiling.
Integer part
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Floor function
82.
Statistics
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Statistics is the study of the collection, analysis, interpretation, presentation, organization of data. 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 collection in terms of the design of surveys and experiments. Statistician Sir Arthur Lyon Bowley defines statistics as "Numerical statements of facts in any department of inquiry placed to each other". When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that 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 theory, which deals with the analysis of random phenomena. Working from a null hypothesis, two basic forms of error are recognized: Type errors and Type II errors. Multiple problems have come to be associated with this framework: ranging from obtaining a sufficient size to specifying an adequate null hypothesis. Measurement processes that generate statistical data are also subject to error. Other types of errors can also be important. Specific techniques have been developed to address these problems. Statistics continues to be an area of active research, for example on the problem of how to analyze Big data. Statistics is a mathematical body of science that pertains as a branch of mathematics.
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.
83.
Rational number
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Since q may be equal to 1, every integer is a rational number. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true just for base 10, but also for any other integer base. A real number, not rational is called irrational. Irrational numbers include √ 2, π, φ. The decimal expansion of an irrational number continues without repeating. Since the set of real numbers is uncountable, almost all real numbers are irrational. In abstract algebra, the rational numbers together with certain operations of multiplication form the archetypical field of characteristic zero. As such, it is characterized as having no proper subfield or, alternatively, being the field of fractions for the ring of integers. The algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the rational numbers form a dense subset of the real numbers. The real numbers can be constructed by completion using Cauchy sequences, Dedekind cuts, or infinite decimals. Zero divided by any other integer equals zero; therefore, zero is a rational number. The term rational to the set Q refers to the fact that a rational number represents a ratio of two integers. In mathematics, "rational" is often used as a noun "rational number".
Rational number
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A diagram showing a representation of the equivalent classes of pairs of integers
84.
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 fundamental theorem of arithmetic says that every positive integer has a single prime factorization. To shorten prime factorizations, factors are often expressed in powers. For a prime p of n, the multiplicity of p is the largest exponent a for which pa divides n exactly. Perfect square numbers can be recognized by the fact that all of their prime factors have even multiplicities. Because every prime factor appears an even 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, 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 coprime if their greatest common divisor gcd = 1. 1 is coprime to every positive integer, including itself. This is because it has no prime factors; it is the empty product. This implies that gcd = 1 for any b 1. ω, represents the number of distinct prime factors of n, while the function, Ω, represents the total number of prime factors of n. If n = ∏ i = 1 ω p i α i, then Ω = ∑ i = 1 ω 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
85.
Long division
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In arithmetic, long division is a standard division algorithm suitable for dividing multidigit numbers, simple enough to perform by hand. It breaks down a 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, 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 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 but instead constructs a tableau or table. 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 these two symbols is sometimes known as a long division symbol or bracket. It developed 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 remainder is calculated. This remainder carries forward when the process is repeated on the following digit of the dividend.
Long division
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An example of long division performed without a calculator.
86.
Algorithm
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In mathematics and computer science, an algorithm is a self-contained step-by-step set of operations to be performed. Algorithms perform calculation, data processing, and/or automated reasoning tasks. The transition from one state to the next is not necessarily deterministic; some algorithms, known as randomized algorithms, incorporate random input. Giving a formal definition of algorithms, corresponding to the intuitive notion, remains a challenging problem. In English, it was first used in about 1230 and then by Chaucer in 1391. English adopted the French term, but it wasn't until the late 19th century that "algorithm" took on the meaning that it has in modern English. 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: Algorism is the art by which at present we use those Indian figures, which number two times five. An informal definition could be "a set of rules that precisely defines a sequence of operations." Which would include all computer programs, including programs that do not perform numeric calculations. Generally, a program is only an algorithm if it stops eventually. An "enumerably infinite set" is one whose elements can be put into one-to-one correspondence with the integers. The concept of algorithm is also used to define the notion of decidability. That notion is central for explaining how formal systems come into being starting from a small set of axioms and rules.
Algorithm
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Alan Turing's statue at Bletchley Park.
Algorithm
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Logical NAND algorithm implemented electronically in 7400 chip
87.
Geometric series
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In mathematics, a geometric series is a series with a constant ratio between successive terms. Geometric series are one of the simplest examples of infinite series with finite sums, although not all of them have this property. Historically, they continue to be central in the study of convergence of series. They have important applications in physics, engineering, biology, economics, computer science, queueing theory, finance. 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 geometric series using only two terms, a. A is the first term of the series. In the case above, where r is one half, the series has the one. If r is less than minus one the terms of the series become larger and larger in magnitude. The series has no sum. If r is equal to one, all of the terms of the series are the same. The series diverges. If r is minus one the terms take two values alternately. The sum of the terms oscillates between two values. This again the series has no sum.
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.
88.
Real number
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In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced by Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the fraction 4/3, all the irrational numbers, such as √ 2. Included within the irrationals are the transcendental numbers, such as π. Complex numbers include real numbers. These descriptions of the real numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions are thus equivalent. Around 500 BC, the Greek mathematicians led by Pythagoras realized the need in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of "number" and "magnitude" into a more general idea of real numbers. In the 17th century, Descartes introduced the term "real" to describe roots of a polynomial, distinguishing them from "imaginary" ones. In the 19th centuries, there was much work on irrational and transcendental numbers. Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, Ferdinand von Lindemann, showed that π is transcendental. Lindemann's proof has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the entire set of real numbers without having defined them cleanly.
Real number
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A symbol of the set of real numbers (ℝ)
89.
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 function is often represented as sgn. Any real number can be expressed as its sign function: x = sgn ⋅ | x |. The numbers cancel and all we are left with is the sign of x. D | x | d x = sgn for x ≠ 0. The 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 written using the Iverson notation: sgn = − +. For k ≫ 1, a smooth approximation of the function is sgn ≈ tanh. Another approximation is sgn ≈ x x 2 + ε 2. Which gets sharper as ε → 0; note that this is the derivative of x2 + ε2. See Heaviside step function – Analytic approximations. The function can be generalized to complex numbers as: sgn = z | z | for any complex number z except z = 0. The signum of a given complex number z is the point on the unit circle of the complex plane, nearest to z. Then, for z ≠ 0, sgn = e i arg z, where arg is the complex argument function.
Sign function
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Signum function y = sgn(x)
90.
Common logarithm
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In mathematics, the common logarithm is the logarithm with base 10. It is indicated by log10, or sometimes Log with a capital L. On calculators it is usually "log", but mathematicians usually mean natural logarithm rather than common logarithm when they write "log". To mitigate this ambiguity the ISO 80000 specification recommends that log10 should be written lg and loge should be ln. Before the early 1970s, mechanical calculators capable of multiplication were expensive and not widely available. Instead, tables of base-10 logarithms were used in navigation when calculations required greater accuracy than could be achieved with a slide rule. Use of logarithms avoided laborious and error pencil multiplications and divisions. Because logarithms were so useful, tables of base-10 logarithms were given in appendices of many text books. Navigation handbooks included tables of the logarithms of trigonometric functions well. See log table for the history of such tables. The fractional part is known as the mantissa. Thus log tables need only show the fractional part. Tables of common logarithms typically listed the mantissa, to 4 or 5 decimal places or more, of each number in a range, e.g. 1000 to 9999. Such a range would cover all possible values of the mantissa. The last number —the fractional part or the mantissa of the common logarithm of 120—can be found in the table shown.
Common logarithm
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The logarithm keys (log for base-10 and ln for base- e) on a typical scientific calculator. The advent of hand-held calculators largely eliminated the use of common logarithms as an aid to computation.
Common logarithm
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A graph of the common logarithm function for positive real numbers.
91.
0 (number)
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0 is both a number and the numerical digit used to represent that number in numerals. The 0 fulfills a central role in mathematics as the additive identity of the integers, real numbers, many other algebraic structures. As a digit, 0 is used as a placeholder in value systems. Slang terms for zero include zilch and zip. Aught, as well as cipher, have also been used historically. The zero came into the English language via French zéro from Italian zero, Italian contraction of Venetian zevero form of ` Italian zefiro via ṣafira or ṣifr. In pre-Islamic time the ṣifr had the meaning'em pty'. Sifr evolved to mean zero when it was used to translate śūnya from India. The first English use of zero was in 1598. The Italian mathematician Fibonacci, credited with introducing the decimal system to Europe, used the term zephyrum. This was then contracted to zero in Venetian. The Italian zefiro was already in existence and may have influenced the spelling when transcribing Arabic ṣifr. Modern usage There are different words used depending on the context. For the simple notion of lacking, the words none are often used. Sometimes the words nought, aught are used.
0 (number)
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Example of the early Greek symbol for zero (lower right corner) from a 2nd-century papyrus
0 (number)
0 (number)
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The number 605 in Khmer numerals, from the Sambor inscription (Saka era 605 corresponds to AD 683). The earliest known material use of zero as a decimal figure.
0 (number)
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The back of Olmec stela C from Tres Zapotes, the second oldest Long Count date discovered. The numerals 7.16.6.16.18 translate to September, 32 BC (Julian). The glyphs surrounding the date are thought to be one of the few surviving examples of Epi-Olmec script.
92.
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 phi-base, or, colloquially, phinary. For instance, 11φ = 100φ. Despite using an irrational 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. These representations are unique, 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. To "standardize" a numeral, we can use the following substitutions: 011φ = 100φ, 0200φ = 1001φ, 110φ = 001φ. We can apply the substitutions in any order we like, as the result is the same. Below, the substitutions applied to the number on the previous line are on the resulting number on the left. Any positive number with a non-standard terminating base-φ representation can be uniquely standardized in this manner. If we get to a point where all digits are "0" or "1", for the first digit being negative, then the number is negative. This can be converted by negating every digit, standardizing the result, then marking it as negative. For example, use some other significance to denote negative numbers. If the arithmetic is being performed on a computer, an message may be returned.
Golden mean base
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Numeral systems
93.
Metric prefix
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A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or fraction of the unit. While all metric prefixes in common today are decadic, historically there have been a number of binary metric prefixes as well. Each prefix has a unique symbol, prepended to the symbol. The prefix -, for example, may be added to gram to indicate multiplication by one thousand: one kilogram is equal to one thousand grams. The prefix -, likewise, may be added to metre to indicate division by one thousand; one millimetre is equal to one thousandth of a metre. Decimal multiplicative prefixes have been a feature of all forms of the metric system with six dating back in the 1790s. Metric prefixes have even been prepended to non-metric units. Since 2009, they have formed part of the International System of Quantities. The BIPM specifies twenty prefixes for the International System of Units. Each name has a symbol, used in combination with the symbols for units of measure. For example, the symbol for kilo - is used to produce ` km', ` kg', ` kW', which are the SI symbols for kilometre, kilogram, kilowatt, respectively. Prefixes may not be used in combination. This also applies to mass, for which the SI unit already contains a prefix. For example, milligram is used instead of microkilogram. In arithmetic of measurements having prefixed units, the prefixes must be expanded except when adding or subtracting values with identical units.
Metric prefix
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Distance marker on the Rhine: 36 (XXXVI) myriametres from Basel. Note that the stated distance is 360 km; comma is the decimal mark in Germany.
94.
Megabyte
–
The megabyte is a multiple of the unit byte for digital information. Sometimes MByte is used. The unit prefix mega is a multiplier in the International System of Units. Therefore, one megabyte is million bytes of information. This definition has been incorporated into the International System of Quantities. However, in the computer and technology fields, several other definitions are used that arose for historical reasons of convenience. A common usage has been to designate one megabyte as 1048576bytes, a measurement conveniently expresses the binary multiples inherent in digital computer memory architectures. However, most bodies have deprecated this usage in favor of a set of binary prefixes, in which this measurement is designated by the unit mebibyte. Less common is a measurement that used the megabyte to mean 1000×1024 bytes. The megabyte is commonly used to measure 10242 bytes. As roughly corresponding to the SI prefix kilo -, it was a convenient term to denote the binary multiple. By the end of 2009, the IEC Standard had been adopted by EU, ISO and NIST. The Mac OS X 10.6 manager is a notable example of this usage in software. Since Snow Leopard, file sizes are reported in decimal units. 1048576 bytes is the definition used by Microsoft Windows in reference to computer memory, such as RAM.
Megabyte
–
1.44 MB floppy disks can store 1,474,560 bytes of data. MB in this context means 1,000×1,024 bytes.
95.
Gigabyte
–
The gigabyte is a multiple of the unit byte for digital information. The giga means 109 in the International System of Units, therefore one gigabyte is 1000000000bytes. The symbol for the gigabyte is GB. However, the term is also used in some fields of information technology to denote 1073741824 bytes, particularly for sizes of RAM. The use of gigabyte is thus ambiguous. For semiconductor RAM, the gigabyte denotes 1073741824bytes. To address this ambiguity, the binary prefixes are standardized in the International System of each binary prefix denoting an integer power of 1024. With these prefixes, a module, labeled as having the size 1GB is designated as 1GiB. The gigabyte is commonly used to mean either 10003 bytes or 10243 bytes. As 1024 is roughly corresponding to SI multiples, it was used for binary multiples as well. In 1998 the International Electrotechnical Commission published standards for binary prefixes, requiring that the gigabyte strictly gibibyte denote 10243 bytes. This is the recommended definition by the International Electrotechnical Commission. The Mac OS X manager from version 10.6 and higher is a notable example of this usage in software. Since Snow Leopard, file sizes are reported in decimal units. 1 GiB = 1073741824 bytes.
Gigabyte
–
This 2.5" hard drive can hold 500 GB of data.
96.
Binary prefix
–
In citations of main capacity, gigabyte customarily means 7009107374182400000 ♠ 1073741824 bytes. As 1024 is a power of two, this usage is referred to as a binary prefix. For example, a 1 Gbit/s Ethernet connection transfers data at 7009100000000000000 ♠ 1000000000 bit/s. In contrast with the binary usage, this use is described as a decimal prefix, as 1000 is a power of 10. The use of the same unit prefixes with two different meanings has caused confusion. In 2008, the IEC prefixes were incorporated into the ISO/IEC 80000 standard. Early computers used one of two addressing methods to access the system memory; binary or decimal. By the mid-1960s, main memory sizes were most commonly powers of two. Early computer documentation would specify the memory size with an exact number such as 4096, 8192, or 16384 words of storage. These furthermore are small multiples of 210, or 1024. As storage capacities increased, different methods were developed to abbreviate these quantities. The prefixes kilo - and mega -, meaning 1000 and 7006100000000000000 1000000 respectively, were commonly used in the electronics industry before World War II. This usage is not consistent with the SI. Compliance with the SI requires that the prefixes can not be used as placeholders for other numbers, like 1024. Gene Amdahl's seminal 1964 article on IBM System/360 used "1K" to mean 1024.
Binary prefix
–
The 7008536870912000000♠ 536 870 912 byte (512×2 20) capacity of these RAM modules is stated as "512 MB" on the label.
Binary prefix
–
Linear-log graph of percentage of the difference between decimal and binary interpretations of the unit prefixes versus the storage size.
97.
Rod calculus
–
The basic equipment for carrying out rod calculus is a bundle of a counting board. A counting board could be a wooden board with or without grid, on the floor or on sand. In 1975 a bundle of bamboo counting rods was unearthed. Rod Numerals is the only numeric system that uses different combination of a single symbol to convey any number or fraction in the Decimal System. For numbers in the units place, every vertical rod represent 1. Two vertical rods represent 2, so on, until 5 vertical rods, which represents 5. For number between 9, a biquinary system is used, in which a horizontal bar on top of the vertical bars represent 5. The second row is the same numbers in horizontal form. For numbers larger than 9, a decimal system is used. Rods placed one place to the left of the units place represent that number. For the hundreds place, another set of rods is placed to the left which represents 100 times of that number, so on. When doing calculation, usually there was no grid on the surface. In Rod Numerals, zeroes are represented by a space, which serves both as a place holder value. Unlike in Arabic Numerals, there is no specific symbol to represent zero. In the adjacent image, the zero is merely represented with a space.
Rod calculus
–
Japanese counting board with grids
Rod calculus
–
Rod calculus facsimile from the Yongle encyclopedia
Rod calculus
Rod calculus
–
representation of the number 231
98.
Tsinghua Bamboo Slips
–
On January 2014 the journal Nature announced that some Tsinghua Bamboo Slips represent "the world's oldest example" of a decimal multiplication table. The Tsinghua Bamboo Slips were donated by an alumnus of the university. The precise date of the illicit excavation that yielded the slips remain unknown. The director of the conservation and research project, has stated that the wishes of the alumnus to maintain his identity secret will be respected. The style of ornament on the accompanying box are in keeping with this conclusion. By the time they reached the university, the slips were badly affected by mold. A Center for Excavated Texts Research and Preservation was established at Tsinghua on April 25, 2009. There are 2388 slips altogether in the collection, including a number of fragments. A series of articles discussing the TBS, intended for an non-specialist Chinese audience, appeared in the Guangming Daily during late 2008 and 2009. The first volume of texts was published in 2010. A 2013 article in The New York Times reported to understanding the Chinese classics. "The classics are all political", Li Xueqin commented: "It would be like finding the Bible the'original' classics. It enables us to look at the classics before they were turned into'classics.' The questions now include, how they became what they are?" In some cases, a TBS text can be found with only variations in wording, title or orthography.
Tsinghua Bamboo Slips
–
The world's earliest artifacts of decimal multiplication table
Tsinghua Bamboo Slips
–
A diagram of the Warring States decimal multiplication table
99.
Abacus
–
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 ἄβαξ abax which means something without base, improperly, any piece of rectangular board or plank. Alternatively, "drawing-board covered with dust". Greek ἄβαξ itself is probably a borrowing of a Northwest Semitic, perhaps word akin to Hebrew ʾābāq, "dust". The preferred plural of abacus is a subject of disagreement, with both abaci in use. The user of an abacus is called an abacist. 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 been 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 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, metal for mathematical calculations. This Greek abacus saw use in Achaemenid Persia, the Etruscan civilization, until the French Revolution, the Western Christian world. A tablet found in 1846 AD, dates back to 300 BC, making it the oldest counting board discovered so far.
Abacus
–
A Chinese abacus
Abacus
–
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
–
Copy of a Roman abacus
Abacus
–
Japanese soroban
100.
Computer
–
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, make computers very useful. Such computers are used as control systems for a very wide variety of industrial and consumer devices. It connects millions of other computers. Since ancient times, 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 analog calculations in the early 20th century. The first electronic calculating machines were developed during World War II. The speed, versatility of computers has increased continuously and dramatically since then. Conventionally, a modern computer consists of at least one processing element, some form of memory. A sequencing and control unit can change the order of operations in response to stored information. Peripheral devices include input devices, input/output devices that perform both functions. They enable the result of operations to be saved and retrieved. This usage of the term referred to a person who carried out computations. The word continued until the middle of the 20th century.
Computer
–
Computer
Computer
–
Suanpan (the number represented on this abacus is 6,302,715,408)
Computer
Computer
101.
Binary numeral system
–
The base-2 system is a positional notation with a radix of 2. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used internally by almost all modern computers and computer-based devices. 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 l'Arithmétique Binaire. Systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, India. 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, which dates to around 1650 BC. 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. It is based on taoistic duality of yin and yang. The Indian scholar Pingala developed a binary system for describing prosody. He used binary numbers in the form of short and long syllables, making it similar to Morse code. Pingala's Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter.
Binary numeral system
–
Numeral systems
Binary numeral system
–
Gottfried Leibniz
Binary numeral system
–
George Boole
102.
ENIAC
–
ENIAC was amongst the earliest electronic general-purpose computers made. It was Turing-complete, could solve "a large class of numerical problems" through reprogramming. ENIAC was heralded as a "Giant Brain" by the press. ENIAC's 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, which equates to $6,816,000 in 2016. ENIAC was designed by J. Presper Eckert of the University of Pennsylvania, U.S.. In 1946, the researchers formed the Eckert-Mauchly Computer Corporation. ENIAC was a modular computer, composed of individual panels to perform different functions. Twenty of these modules were accumulators which hold a ten-digit decimal number in memory. Numbers were passed between these units across general-purpose buses. Key to its versatility was the ability to branch; it could trigger different operations, depending on the sign of a computed result. It weighed more than 30 short tons, consumed 150 kW of electricity. This requirement led to the rumor that whenever the computer was switched on, lights in Philadelphia dimmed. An IBM card punch was used for output. These cards could be used to produce printed offline using an IBM accounting machine, such as the IBM 405.
ENIAC
–
ENIAC
ENIAC
–
Glen Beck (background) and Betty Snyder (foreground) program ENIAC in BRL building 328. (U.S. Army photo)
ENIAC
–
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
–
A function table from ENIAC on display at Aberdeen Proving Ground museum.
103.
IBM 650
–
The IBM 650 Magnetic Drum Data-Processing Machine is one of IBM's early computers, the world’s first mass-produced computer. It was announced in 1956 enhanced as the IBM 650 RAMAC with the addition of up to four disk storage units. Almost 2,000 systems were produced, the last in 1962. Support for its component units was withdrawn in 1969. The 650 was a two-address, bi-quinary coded computer, with memory on a rotating magnetic drum. Character support was provided by the input/output units converting special characters to/from a two-digit decimal code. The IBM 7070, announced 1958, was expected to be a "common successor to the 705". The IBM 1620, introduced in 1959, addressed the lower end of the market. The UNIVAC Solid State was announced by Sperry Rand in December 1958 to the 650. None of these had a 650 compatible set. The basic 650 system consisted of three units: IBM 650 Console Unit housed the magnetic drum storage, arithmetical device and the operator's console. Words on the drums were organized for the respective models. 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 second address in each instruction was the address of the next instruction. Instructions could then be interleaved, placing many at addresses that would be immediately accessible when execution of the previous instruction was completed.
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
–
An IBM 650 at Texas A&M University. The IBM 533 Card Read Punch unit is on the right.
IBM 650
–
IBM 650 console panel, showing bi-quinary indicators. (At House for the History of IBM Data Processing(closed), Sindelfingen)
IBM 650
–
Close-up of bi-quinary indicators
104.
Binary-coded decimal
–
Special bit patterns are sometimes used for other indications. The 4-bit encoding may vary however, for technical reasons, see Excess-3 for instance. The ten states representing a BCD decimal digit are sometimes called tetrades with those don't care-states unused named pseudo-tetrads or pseudo-decimal digit). BCD's principal drawbacks are a small increase in the complexity of the circuits needed to implement a slightly less dense storage. BCD takes advantage of the fact that any one numeral can be represented by a four bit pattern. This is also called "8421" encoding. Other encodings are also used, including "7421" -- named after the weighting used for the bits -- and "excess-3". For example, 6,' 0110' b in 8421 notation, is' 1100' b in 4221,' 0110' b in 7421,' 1001' b in excess-3. Packed: two numerals are encoded into a single byte, with one numeral in the least significant nibble and the other numeral in the most significant nibble. To represent numbers larger than the range of contiguous bytes may be used. Masking operations are used to pack or unpack a packed BCD digit. Logical operations are used to convert a numeral to its equivalent bit pattern or reverse the process. By employing BCD, the manipulation of numerical data for display can be greatly simplified by treating each digit as a single sub-circuit. If the numeric quantity were manipulated as pure binary, interfacing to such a display would require complex circuitry. Therefore, in cases where the calculations are relatively simple, working throughout with BCD can lead than converting to and from binary.
Binary-coded decimal
–
A binary clock might use LEDs to express binary values. In this clock, each column of LEDs shows a binary-coded decimal numeral of the traditional sexagesimal time.
105.
IEEE 754
–
The IEEE Standard for Floating-Point Arithmetic is a technical standard for floating-point computation established in 1985 by the Institute of Electrical and Electronics Engineers. The standard addressed many problems found in the floating point implementations that made them difficult to use reliably and portably. Many hardware floating point units now use the IEEE 754 standard. The international standard ISO/IEC/IEEE 60559:2011 has been published. The binary formats in the original standard are included in the new standard along with three basic formats. To conform to the current standard, an implementation must implement at least one of the basic formats as both an interchange format. As of September 2015, the standard is being revised to incorporate errata. An IEEE 754 format is a "set of representations of numerical symbols". A format may also include how the set is encoded. A format comprises: Finite numbers, which may be base 10. Each finite number is described by three integers: s = c = a significand, q = an exponent. The numerical value of a finite number is s × c × bq where b is the base, also called radix. Two infinities: +∞ and −∞. Two kinds of NaN: a quiet NaN and a signaling NaN. A NaN may carry a payload, intended for diagnostic information indicating the source of the NaN.
IEEE 754
–
Precision of binary32 and binary64 in the range 10 −12 to 10 12.
106.
Warring States
–
The "Warring States Period" derives its name from the Record of the Warring States, a work compiled early in the Han dynasty. This geographical position offered protection from the states of the Central Plains, but limited its initial influence. 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, controlling the eastern approaches to Qin. Wei, middle. Zhao, the northernmost of the three. Qi: located in the east of China, centred on the Shandong Peninsula, described as east of Mount Tai but whose territory extended far beyond. 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. 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. Zhongshan: Between the states of Zhao and Yan was the state of Zhongshan, eventually annexed by Zhao in 296 BC. 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.
Warring States
–
History of China
Warring States
–
Warring States about 350 BC
Warring States
–
Tomb Guardian held at Birmingham Museum of Art
Warring States
–
A jade -carved dragon garment ornament from the Warring States period
107.
Egyptian hieroglyphs
–
Egyptian hieroglyphs were the formal writing system used in Ancient Egypt. It combined alphabetic elements, with a total of some 1,000 distinct characters. Cursive hieroglyphs were used for religious literature on papyrus and wood. Egyptian scripts are derived from hieroglyphic writing; Meroitic was a late derivation from Demotic. The system continued to be used throughout the Late Period, well as the Persian and Ptolemaic periods. Late survivals of hieroglyphic use are found well into the Roman period, extending into the 4th century AD. The decipherment of hieroglyphs would only be solved with the help of the Rosetta Stone. The hieroglyph comes from a compound of ἱερός and γλύφω, supposedly a calque of an Egyptian phrase mdw · w-nṯr "god's words". The glyphs themselves were called τὰ ἱερογλυφικὰ γράμματα "the sacred engraved letters". The hieroglyph has become a noun in English, standing for an hieroglyphic character. As used in the previous sentence, the word hieroglyphic is an adjective, but hieroglyphic has also become a noun in English, at least in non-academic usage. Hieroglyphs emerged from the preliterate artistic traditions of Egypt. For example, symbols on Gerzean pottery from c. 4000 BC have been argued to resemble hieroglyphic writing. There are around 800 hieroglyphs dating back to New Kingdom Eras. By the Greco-Roman period, there are more than 5,000.
Egyptian hieroglyphs
–
A section of the Papyrus of Ani showing cursive hieroglyphs.
Egyptian hieroglyphs
–
Hieroglyphs on a funerary stela in Manchester Museum
Egyptian hieroglyphs
–
The Rosetta Stone in the British Museum
Egyptian hieroglyphs
–
Hieroglyphs typical of the Graeco-Roman period
108.
Cretan hieroglyphs
–
Cretan hieroglyphs are undeciphered hieroglyphs found on artefacts of early Bronze Age Crete, during the Minoan era. The two writing systems continued to be used in parallel for most of their history. The 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, 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 also found in the northeastern Aegean. It has been suggested that there was an evolution of the hieroglyphs into the linear scripts. Also, some relations to Anatolian hieroglyphs have been suggested. Symbol inventories have been compiled by Meijer, Olivier/Godart. The known corpus has been edited 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, two types of punctuation. Many symbols have apparent Linear A counterparts, so that it is tempting to insert Linear B sound values.
Cretan hieroglyphs
–
A green jasper seal with Cretan hieroglyphs. 1800 BC
109.
Minoans
–
It belongs to a period of Greek history preceding both Ancient Greece. It was rediscovered through the work of British archaeologist Arthur Evans. The term "Minoan" was originally given as a description to the pottery of this period. Minos was associated in Greek myth with the Minotaur, which Evans identified with the site at Knossos, the largest Minoan site. The Homer recorded a tradition that Crete once had 90 cities. The Minoan period saw significant contacts between Crete, the Mediterranean, particularly the Near East. Some of its best art is preserved on the island of Santorini, destroyed during the Thera eruption. The term "Minoan" refers to the mythic "king" Minos of Knossos. Who first coined the term is debated. It is commonly attributed to the archeologist Arthur Evans. Minos was associated with the labyrinth, which Evans identified with the site at Knossos. Likely, Arthur Evans read the book, continuing the use of the term in his own findings. Evans claims to have applied it, but not to have invented it. Hoeck had in mind the Crete of mythology. He had no idea that the archaeological Crete had existed.
Minoans
–
Minoan civilization
Minoans
–
Minoan copper ingot.
Minoans
–
Fresco showing three women who were possibly queens. [citation needed]
110.
Linear A
–
Linear A is one of two currently undeciphered writing systems used in ancient Greece. Linear A was the primary script used in palace and religious writings of the Minoan civilization. It was discovered by archaeologist Sir Arthur Evans. It is the origin of the Linear B script, later used by the Mycenaean civilization. In the 1950s, Linear B was largely found to encode an early form of Greek. Although the two systems share many symbols, this did not lead to a subsequent decipherment of Linear A. Using the values associated with Linear B in Linear A mainly produces unintelligible words. If it uses the similar syllabic values as Linear B, then its underlying language appears unrelated to any known language. This has been dubbed the Minoan language. Linear A has hundreds of signs. They are believed to represent syllabic, semantic values in a manner similar to Linear B. It primarily occasionally appears as a right-to-left or boustrophedon script. An interesting feature is that of how numbers are recorded in the script. There are special symbols to indicate fractions and weights. Linear A has been unearthed chiefly but also at other sites in Greece, as well as Turkey and Israel.
Linear A
–
Linear A incised on tablets found in Akrotiri, Santorini.
Linear A
–
Linear A
Linear A
–
Linear A tablet – Chania Archaeological Museum.
Linear A
–
Linear A incised on a vase, also found in Akrotiri.
111.
Linear B
–
Linear B is a syllabic script, 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. The succeeding period, known as the Dark Ages, provides no evidence of the use of writing. It is also the only one of the three "Linears" to be deciphered, by English architect and self-taught linguist Michael Ventris. Linear B consists of over 100 ideographic signs. "signifying" signs symbolize objects or commodities. They 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: 66 in Knossos. From this fact, it could be thought 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 semantic values. The representations and naming of these signs have been standardized starting with the first in Paris in 1956. Colloquia continue: the 13th occurred in Paris.
Linear B
–
Linear B
Linear B
–
Linear B tablet discovered by Arthur Evans
Linear B
–
Tablets
112.
Classical Greece
–
Classical Greece was a period of around 200 years in Greek culture. This Classical period saw the annexation of much of modern-day Greece by its subsequent independence. Classical Greece had a powerful influence on the foundations of western civilization. Much of modern Western politics, artistic thought, scientific thought, theatre, philosophy derives from this period of Greek history. The Classical period in this sense is in turn succeeded by the Hellenistic period. From the perspective of Athenian culture in Classical Greece, the period generally referred to as the 5th BC extends slightly into the 4th century BC. The Persians were defeated in 490 BC. The Delian League then formed, as Athens' instrument. After both forces were spent, a brief peace came about; then the war resumed to Sparta's advantage. Internal Athenian agitations mark the end of the 5th century BC in Greece. Since its beginning, Sparta had been ruled by a diarchy. This meant that Sparta had two kings ruling concurrently throughout its entire history. The two kingships were both hereditary, vested in the Eurypontid dynasty. According to legend, the respective hereditary lines of these two dynasties sprang from twin descendants of Hercules. They were said to have conquered Sparta two generations after the Trojan War.
Classical Greece
–
The Parthenon, in Athens, a temple to Athena
Classical Greece
–
Statue of King Leonidas of Sparta
Classical Greece
–
Cities at the beginning of the Peloponnesian War
113.
Archimedes
–
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. He was also one of the first to apply mathematics to physical phenomena, founding statics, including an explanation of the principle of the lever. He is credited with designing innovative machines, such as his screw pump, defensive war machines to protect his native Syracuse from invasion. Archimedes died during the Siege of Syracuse when he was killed despite orders that he should not be harmed. Unlike his inventions, the mathematical writings of Archimedes were little known in antiquity. The date of birth is based by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years. In The Sand Reckoner, Archimedes gives his father's name as Phidias, an astronomer about whom nothing is known. Plutarch wrote in his Parallel Lives that Archimedes was related to the ruler of Syracuse. This work has been lost, leaving the details of his life obscure. It is unknown, for instance, whether he ever had children. During his youth, Archimedes may have studied in Alexandria, Egypt, where Conon of Samos and Eratosthenes of Cyrene were contemporaries. He referred as his friend while two of his works have introductions addressed to Eratosthenes. According to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. He declined, saying that he had to finish working on the problem.
Archimedes
–
Archimedes Thoughtful by Fetti (1620)
Archimedes
–
Cicero Discovering the Tomb of Archimedes by Benjamin West (1805)
Archimedes
–
Artistic interpretation of Archimedes' mirror used to burn Roman ships. Painting by Giulio Parigi.
Archimedes
–
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)
114.
Carl Friedrich Gauss
–
Carl Friedrich Gauss was born on 30 April 1777 in Brunswick, as the son of poor working-class parents. He was confirmed in a church near the school he attended as a child. Gauss was a prodigy. A contested story relates that, when he was eight, he figured out how to add up all the numbers from 1 to 100. He made his first ground-breaking mathematical discoveries while still a teenager. He completed his magnum opus, in 1798 at the age of 21, though it was not published until 1801. This work has shaped the field to the present day. While at university, Gauss independently rediscovered important theorems. Gauss was so pleased by this result that he requested that a regular heptadecagon be inscribed on his tombstone. The stonemason declined, stating that the difficult construction would essentially look like a circle. The 1796 was most productive for both Gauss and number theory. He discovered a construction of the heptadecagon on 30 March. He further advanced modular arithmetic, greatly simplifying manipulations in theory. On April he became the first to prove the quadratic reciprocity law. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic.
Carl Friedrich Gauss
–
Carl Friedrich Gauß (1777–1855), painted by Christian Albrecht Jensen
Carl Friedrich Gauss
–
Statue of Gauss at his birthplace, Brunswick
Carl Friedrich Gauss
–
Title page of Gauss's Disquisitiones Arithmeticae
Carl Friedrich Gauss
–
Gauss's portrait published in Astronomische Nachrichten 1828
115.
Hittites
–
The Hittites were an Ancient Anatolian people who established an empire centered on Hattusa in north-central Anatolia around 1600 BC. The Assyrians eventually annexed much of the Hittite empire, while the remainder was sacked by Phrygian newcomers to the region. They referred as Hatti. The conventional name "Hittites" is due to their initial identification with the Biblical Hittites in 19th archaeology. Before the discoveries, the only source of information about Hittites had been the Old Testament. French scholar Félix Marie Charles Texier did not identify them as Hittite. Some names in the tablets were neither Hattic nor Assyrian, but clearly Indo-European. In 1887, excavations at Tell El-Amarna in Egypt uncovered the diplomatic correspondence of his son Akhenaton. Others, such as Max Müller, proposed connecting it with Biblical Kittim, rather than with the "Children of Heth". He also proved that the ruins at Boğazköy were the remains of the capital of an empire that, at one point, controlled northern Syria. Under the direction of the German Archaeological Institute, excavations at Hattusa have been under way since 1907, during the world wars. Kültepe was successfully excavated by Professor Tahsin Özgüç from 1948 until his death in 2005. The Hittites used Mesopotamian Cuneiform script. The Museum of Anatolian Civilizations in Ankara, Turkey houses the richest collection of Hittite and Anatolian artifacts. The Hittite kingdom was centred on the lands surrounding Hattusa and Neša, known as "the land Hatti".
Hittites
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Bronze religious standard from a pre-Hittite tomb at Alacahöyük, dating to the third millennium B.C., from the Museum of Anatolian Civilizations, Ankara.
Hittites
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The Hittite Empire, ca. 1300 BC (shown in Blue).
Hittites
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Jewelry from Museum of Anatolian Civilizations.
Hittites
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Hittite chariot, from an Egyptian relief
116.
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 scriptures of Hinduism. 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, which are called smṛti. In Hindu Epic the Mahabharata, the creation of Vedas is credited to Brahma. The Vedic hymns themselves assert that they were skillfully created by Rishis, after inspired creativity, just as a carpenter builds a chariot. There are four Vedas: the Rigveda, the Yajurveda, the Atharvaveda. Each Veda has been subclassified into four major text types -- the Samhitas, the Aranyakas, the Upanishads. Some scholars add a fifth category – the Upasanas. Denominations have taken differing positions on the Vedas. Schools of Indian philosophy which cite the Vedas as their scriptural authority are classified as "orthodox". Despite their differences, just like the texts of the layers of texts in the Vedas discuss similar ideas and concepts. The Sanskrit véda "knowledge, wisdom" is derived from the root vid - "to know". This is reconstructed as meaning "see" or "know". The noun is from * u̯eidos, cognate to Greek εἶδος "aspect", "form".
Vedas
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Rigveda (padapatha) manuscript in Devanagari
117.
Joseph Needham
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He was elected a fellow of the British Academy in 1971. Needham was the only child of a London family. His mother, Alicia Adelaïde, née Montgomery, was a music composer from Oldcastle, Co.. Meath, Ireland. He had switched to Biochemistry. His Terry Lecture of 1936 was published with Yale University Press under the title of Order and Life. Although his career as an academic was well established, his career developed in unanticipated directions during and after World War II. Three Chinese scientists came for graduate study in 1937: Lu Gwei-djen, Wang Ying-lai and Shen Shih-Chang. Daughter of a Nanjingese pharmacist, taught Needham Chinese, igniting his interest in China's ancient technological and scientific past. The study of Classical Chinese privately with Gustav Haloun. Under the Royal Society's direction, Needham was the director of the Sino-British Science Co-operation Office in Chongqing from 1942 to 1946. In 1944 he visited Yunnan in an attempt to reach the Burmese border. Everywhere he went he was given old historical and scientific books which he shipped back to Britain through diplomatic channels. They were to form the foundation of his later research. On his return to Europe, he was asked by Julian Huxley to become the first head of the Natural Sciences Section of UNESCO in Paris, France.
Joseph Needham
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Joseph Needham
118.
Mathematical Treatise in Nine Sections
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The Mathematical Treatise in Nine Sections is a mathematical text written by Chinese Southern Song dynasty mathematician Qin Jiushao in the year 1247. Each chapter contains a total of 81 problems. In 1971 Belgian sinologist Ulrich Libbrecht published Chinese Mathematics in the Thirteenth Century, which earned him a degree cum laude at Leiden University. Guo, Shuchun, "Qin Jiushao". Encyclopedia of 1st ed.
Mathematical Treatise in Nine Sections
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Mathematical Treatise in Nine Sections in The Siku Quanshu
Mathematical Treatise in Nine Sections
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1842 wood block printed Shu Shu Jiu Zhang
Mathematical Treatise in Nine Sections
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surveying a round city from afar.Shu Shu Jiu Zhang
Mathematical Treatise in Nine Sections
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Qin Jiushao solved third order equation with rod calculus
119.
Simon Stevin
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Simon Stevin, sometimes called Stevinus, was a Flemish mathematician, physicist and military engineer. Stevin was active in a many areas of engineering, both theoretical and practical. Very little is known with certainty about Stevin's life and what we know is mostly inferred from other recorded facts. The 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. Some documents regarding his father use the Stevijn. This is a normal spelling shift in 16th century Dutch. Stevin was born around Catelyne van der Poort. His father is believed to have been a son of a member of the schuttersgilde Sint-Barbara of Bruges. Many other Stevins were later mentioned in the Poorterboeken. Simon Stevin's mother Cathelijne was the daughter of a wealthy family from Ypres. Her father Hubert was a poorter of Bruges. Simon's Cathelijne later married Joost Sayon, involved in a member of the schuttersgilde Sint-Sebastiaan. Through her marriage Cathelijne became a member of a family of Calvinists and Simon was likely brought up in the Calvinist faith. It is believed that Stevin grew up in a relatively affluent environment and enjoyed a good education.
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
120.
Al Khwarizmi
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In the 12th century, Latin translations of his work on the Indian numerals introduced the positional system to the Western world. The Compendious Book on Calculation by Completion and Balancing presented the systematic solution of linear and quadratic equations in Arabic. He is often considered one of the fathers of algebra. He wrote on astrology. Some words reflect the importance of al-Khwārizmī's 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, both meaning digit. Few details of al-Khwārizmī's 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 al-Qutrubbulli could indicate he might instead have come from a viticulture district near Baghdad. Recently, G. J. Toomer... with naive confidence constructed an entire fantasy on the error which cannot be denied the merit of amusing the reader. Ibn al-Nadīm's 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.
Al Khwarizmi
Al Khwarizmi
Al Khwarizmi
121.
Indo-Aryan languages
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The Indo-Aryan or Indic languages are the dominant language family of the Indian subcontinent and South Asia, spoken largely by Indo-Aryan people. 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, more than half of all Indo-European languages recognized by Ethnologue. While the languages are primarily spoken in South Asia, pockets of Indo-Aryan languages are found to be spoken in Europe and the Middle East. Proto-Indo-Aryan, or sometimes Proto-Indic, is the reconstructed proto-language of the Indo-Aryan languages. 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, directly attested as Vedic and Mitanni-Aryan. Despite the great archaicity of Vedic, however, the other Indo-Aryan languages preserve a small number of archaic features lost in Vedic. Vedic has been used in the ancient preserved religious hymns, the foundational canon of Hinduism known as the Vedas. Mitanni-Aryan is of similar age to the language of the Rigveda, but the only evidence of it is a few proper names and specialized loanwords. Vedic Sanskrit is only marginally different from Proto-Indo-Aryan the proto-language of the Indo-Aryan languages. In about the 4th century BCE, an artificial language based on Vedic, called "Classical Sanskrit" by convention, was codified and standardized by the grammarian Panini. 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.
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.)
122.
Dravidian languages
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The Dravidian languages with the most speakers are Telugu, Tamil, Kannada and Malayalam. There are also small groups of Dravidian-speaking scheduled tribes, who live beyond the mainstream communities, such as the Kurukh and Gond tribes. Epigraphically the Dravidian languages have been attested since the 2nd century BCE. Only two Dravidian languages are exclusively spoken outside India: Brahui in Pakistan and Dhangar, a dialect of Kurukh, in Nepal. Caldwell coined the term "Dravidian" for this family of languages, based on the usage of the Sanskrit word drāviḍa in the work Tantravārttika by Kumārila Bhaṭṭa. In his own words, Caldwell says, The word I have chosen is'Dravidian', from Drāviḍa, the adjectival form of Draviḍa. I have, therefore, no doubt of the propriety of adopting it. The 1961 publication of the Dravidian etymological dictionary by T. Burrow and M. B. Emeneau proved a notable event in the study of Dravidian linguistics. As for the origin of the Sanskrit word drāviḍa itself, researchers have proposed various theories. Basically the theories deal with the direction of derivation between tamiẓ and drāviḍa. There is no definite philological and linguistic basis for asserting unilaterally that the name Dravida also forms the origin of the word Tamil. Sinhala BCE inscriptions cite dameḍa-, damela- denoting Tamil merchants. It appears that damiḷa- was older than draviḍa- which could be its Sanskritization.
Dravidian languages
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Language families in South Asia
Dravidian languages
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North
Dravidian languages
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Jambai Tamil Brahmi inscription dated to the early Sangam age
123.
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 also by Hungarian diaspora communities worldwide. Like Estonian, it belongs to the Uralic language family, its closest relatives being Mansi and Khanty. It is one of the European languages not part of the Indo-European languages. The Hungarian name for the language is magyar nyelv. The word "Magyar" is also used as an English word to refer as an ethnic group or its language. Hungarian is a member of the Uralic family. Current literature favors the hypothesis that it comes from the name of the Turkic tribe Onogur. There are numerous sound correspondences between Hungarian and the other Ugric languages. For example, Hungarian száz "hundred" vs. Khanty sot "hundred". The correspondences are also regular. During the later half of the 19th century, a competing hypothesis proposed a Turkic affinity of Hungarian. The Hungarians gradually changed their lifestyle from settled hunters probably as a result of early contacts with Iranian nomads. In Hungarian, Iranian loans probably span well over a millennium. Among these include tehén ‘cow’, tíz ‘ten’, tej ‘milk’, nád ‘reed’.
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
124.
Chinese language
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Chinese is a group of related, but in many cases mutually unintelligible, language varieties, forming a branch of the Sino-Tibetan language family. Chinese is spoken by the Han majority and many other ethnic groups in China. Nearly 1.2 billion people speak some form of Chinese as their first language. The internal diversity of Chinese has been likened to that of the Romance languages, but may be even more varied. There are between 13 regional groups of Chinese, of which the most spoken by far is Mandarin, followed by Wu, Yue and Min. Most of these groups are mutually unintelligible, although some, like certain Southwest Mandarin dialects, may share some degree of intelligibility. All varieties of Chinese are tonal and analytic. Standard Chinese is a standardized form of spoken Chinese based on the Beijing dialect of Mandarin. It is the official language of China and Taiwan, as well as one of four official languages of Singapore. It is one of the six official languages of the United Nations. The written form of the standard language, based on the logograms known as Chinese characters, is shared by literate speakers of otherwise unintelligible dialects. It is also influential in Guangdong province and much of Guangxi, is widely spoken among overseas communities. Hakka also has a sizeable diaspora in Taiwan and southeast Asia. Shanghainese and other Wu varieties are prominent in the lower Yangtze region of eastern China. Chinese can be traced back to a hypothetical Sino-Tibetan proto-language.
Chinese language
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The Tripitaka Koreana, a Korean collection of the Chinese Buddhist canon
Chinese language
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Hànyǔ (Chinese) written in traditional (left) and simplified (right) characters
Chinese language
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" Preface to the Poems Composed at the Orchid Pavilion " by Wang Xizhi, written in semi-cursive style
125.
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 native language of the Vietnamese people, well as a first or second language for the many ethnic minorities of Vietnam. Vietnamese has also been officially recognized as a language in the Czech Republic. It is part of the Austroasiatic family of which it has by far the most speakers. It formerly used a modified set of Chinese characters called chữ nôm given vernacular pronunciation. Today is a Latin alphabet with additional diacritics for tones and certain letters. As the national language, Vietnamese is spoken by ethnic Vietnamese and by Vietnam's many minorities. Vietnamese is also the native language of the Gin group in southern Guangxi Province in China. A significant number of native speakers also reside in neighboring Cambodia and Laos. In the United States, Vietnamese is the sixth most spoken language, over 1.5 million speakers, who are concentrated in a handful of states. It is the third most spoken language in Texas, fifth in California. Vietnamese is the seventh most spoken language in Australia. In France, it is the eighth most spoken immigrant language at home. Vietnamese is the sole official and national language of Vietnam. It is the first language of the majority of the Vietnamese population, well as a first or second language for country's ethnic minority groups.
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.
126.
Japanese language
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Japanese is an East Asian language spoken by about 125 million speakers, primarily in Japan, where it is the national language. Little is known of the language's prehistory, or when it first appeared in Japan. Chinese documents from the 3rd century recorded a few Japanese words, but substantial texts did not appear until the 8th century. During the Heian period, Chinese had considerable influence on the vocabulary and phonology of Old Japanese. Late Middle Japanese saw changes in features that brought it closer to the modern language, as well as the first appearance of European loanwords. The standard dialect moved in the Early Modern Japanese period. Following the end in 1853 of Japan's self-imposed isolation, the flow of loanwords from European languages increased significantly. English loanwords in particular have become frequent, Japanese words from English roots have proliferated. Japanese is an mora-timed language with simple phonotactics, a pure vowel system, a lexically significant pitch-accent. Structure is topic -- comment. Sentence-final particles are used to add emotional or emphatic impact, or make questions. Nouns have no grammatical number or gender, there are no articles. Verbs are conjugated, primarily for tense and voice, but not person. Japanese equivalents of adjectives are also conjugated. Japanese has a complex system of honorifics with verb forms and vocabulary to indicate the relative status of the speaker, the listener, persons mentioned.
Japanese language
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A page from Nihon Shoki (The Chronicles of Japan), the second oldest book of classical Japanese history.
Japanese language
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Map of Japanese dialects and Japonic languages
Japanese language
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Two pages from a 12th-century emaki scroll of The Tale of Genji from the 11th century.
Japanese language
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Calligraphy
127.
Korean language
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Approximately 80 million people worldwide speak Korean. Korean has a extinct relatives, which together with Korean itself form the Koreanic family. Despite this, historical and modern linguists classify Korean as a language isolate. The idea that Korean belongs to a putative Altaic language family or to the Dravido-Korean family has been generally discredited. There is still debate on whether Korean and Japanese are related languages. The Korean language is agglutinative in its morphology and SOV in its syntax. A relation of Korean with Japonic languages has been proposed by linguists like William George Aston and Samuel Martin. Others supported the inclusion of Koreanic and Japonic languages in the purported Altaic family, now not accepted by most specialists. Only privileged elites were educated to fluently read and write, however, most of the population was illiterate. Since the Korean War, through 70 years of separations North–South differences have developed in standard Korean, including variance in pronunciation, verb inflection, vocabulary chosen. The Korean names for the language are based on the names for Korea used in North and South Korea. In South Korea, the Korean language is referred by many names including hanguk-eo hanguk-mal, "Korean speech" and uri-mal, "our language". In "hanguk-eo" and "hanguk-mal", the first part of "hanguk", refers to the Korean while" - eo" and" - mal" mean "language" and "speech", respectively. Korean is also referred to as guk-eo, literally "national language". In North Korea and China, the language is most often called Chosŏn-mal, or more formally, Chosŏn-ŏ.
Korean language
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Two names for Korean, Hangugeo and Chosŏnmal, written vertically in Hangul
Korean language
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Street signs in Korean; Daegu, Korea.
Korean language
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Korean writing systems
128.
Thai language
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Thai is a member of the Tai group of the Tai -- Kadai family. Over half of the words in Thai are borrowed from Pali, Sanskrit and Old Khmer. It is a analytic language. Thai also has relational markers. Spoken Thai is mutually intelligible with Laotian. Thai is the official language of Thailand, spoken by over million people. Standard Thai is based on the register of the educated classes of Bangkok. In addition to Central Thai, Thailand is home to other related Tai languages. It is spoken by about million people. Thais from both inside and outside the Isan region simply call this variant "Lao" when speaking informally. Northern Thai, spoken by about 6 million in the formerly independent kingdom of Lanna. Shares strong similarities to the point that in the past the Siamese Thais referred to it as Lao. Southern Thai, spoken by Phu Thai, spoken by about half a million around Nakhon Phanom Province, 300,000 more in Laos and Vietnam. Phuan, spoken by 200,000 in central Thailand and Isan, 100,000 more in northern Laos. Shan, spoken by about 100,000 along the border with the Shan States of Burma, by 3.2 million in Burma.
Thai language
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Thai
129.
Quechua languages
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It is perhaps most widely known for being the main language of the Inca Empire, was disseminated by the colonizers throughout their reign. Quechua had already expanded across wide ranges of the central Andes long before the expansion of the Inca Empire. The Inca were one among many peoples in present-day Peru who already spoke forms of Quechua. In the Cusco region, Quechua was influenced by local languages such as Aymara. The Cuzco variety of Quechua developed as quite distinct. In similar ways, diverse dialects developed in different areas, related to existing local languages, when the Inca Empire ruled and imposed Quechua as the official language. Clergy of the Catholic Church adopted Quechua to use as the language of evangelization. Given its use by the Catholic missionaries, the range of Quechua continued to expand in some areas. The Crown banned even "loyal" pro-Catholic texts in Quechua, such as Garcilaso de la Vega's Comentarios Reales. Despite a brief revival of the language immediately after the Latin American nations achieved independence in the 19th century, the prestige of Quechua had decreased sharply. Gradually its use declined so that it was spoken mostly by indigenous people in the more isolated and conservative rural areas. But in the 21st century, those speaking Quechua language speakers number 8 to 10 million people across South America, the most speakers of any indigenous language. The oldest written records of the language are by missionary Domingo de Santo Tomás, who arrived in Peru in 1538 and learned the language from 1540. He published his Grammatica o arte de la lengua general de los indios de los reynos del Perú in 1560. In 1975 Peru became the first country to recognize Quechua as one of its official languages.
Quechua languages
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Quechua
130.
Aymara language
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Aymara /aɪməˈrɑː/ is an Aymaran language spoken by the Aymara people of the Andes. It is one of only a handful of American languages with over million speakers. Aymara, along with Quechua and Spanish, is an official language of Bolivia. It is also spoken around the Lake Titicaca region of southern Peru and, to a much lesser extent, by some communities in northern Chile. Some linguists have claimed that Aymara is related to its more widely spoken neighbor, Quechua. That claim, however, is disputed. Aymara is an agglutinating and, to a certain extent, a polysynthetic language. It has a subject–object–verb word order. In this document, he uses the term aymaraes to refer to the people. The language was then called Colla. However, Cerron-Palomino asserts that Colla were in Puquina speakers who were the rulers of Tiwanaku in the first and third centuries. This hypothesis suggests that the linguistically-diverse area ruled by the Puquina came to adopt Aymara languages in their southern region. It took over another century for this usage of "Aymara" in reference to the language spoken by the Aymara people to become general use. In the meantime the Aymara language was referred to as "the language of the Colla". The entire history of this term is thoroughly outlined in Lingüística Aimara.
Aymara language
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Geographic Distribution of the Aymara language
131.
Units of measurement
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Any other value of that 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". 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 the International System of the modern form of the metric system. In trade, measures is often a subject of governmental regulation, to ensure transparency. The International Bureau of Measures is tasked with ensuring worldwide uniformity of their traceability to the International System of Units. Metrology internationally accepted units of measures. In physics and metrology, units are standards for measurement of physical 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 measures developed ago for commercial purposes. Engineering often use larger and smaller units of measurement than those used in everyday life and indicate them more precisely.
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)
132.
Units of information
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In theory, units of information are also used to measure the information contents or entropy of random variables. Multiples of these units can be formed from these with the newer IEC binary prefixes. Information capacity is a dimensionless quantity. When b is 2, the unit is the shannon, equal to the information content of one "bit". A system for example, can store up to log28 = 3 bits of information. Other units that have been named include: Base b = 3: the unit is equal to log2 3 bits. Base b = 10: the unit is equal to log2 10 bits. Base b = e, the base of natural logarithms: the unit is worth log2 e bits. Conventional names are used for collections or groups of bits. The name given to 2 bits. The name is a reference to the song, "a haircut". A byte can represent 256 distinct values, such as -128 to 127. The IEEE 1541-2002 standard specifies "B" as the symbol for byte. Multiples thereof, are almost always used to specify the sizes of computer files and the capacity of storage units. Most modern computers and peripheral devices are designed to manipulate data in whole groups of bytes, rather than individual bits.
Units of information
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Comparison of units of information: bit, trit, nat, ban. Quantity of information is the height of bars. Dark green level is the "Nat" unit.
133.
Base e
–
The natural logarithm of x is generally written as ln x, loge x, or sometimes, if the base e is implicit, simply log x. Parentheses are sometimes added for clarity, giving ln, loge or log. 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. For example, ln is 2.0149... because e2.0149... = 7.5. Ln, is 1, because e1 = e, while the natural logarithm of 1, ln, is 0, since e0 = 1. The natural logarithm can be defined as the area under the curve y = 1/x from 1 to a. The simplicity of this definition, matched in other formulas involving the natural logarithm, leads to the term "natural". Like all logarithms, the natural logarithm maps multiplication into addition: ln = + ln. Logarithms can be defined to any positive base other than 1, not only e. However, logarithms in other bases differ only by a constant multiplier from the natural logarithm, are usually defined in terms of the latter. For instance, the binary logarithm is the natural logarithm divided by 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 in exponential decay problems.
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.
134.
Trit (computing)
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The ternary numeral system has three as its base. Analogous to a ternary digit is a trit. One trit is equivalent to log23 bits of information. Representations of integer numbers in ternary do not get uncomfortably lengthy as as in binary. For example, decimal 365 corresponds to ternary 111112. The value of a binary number with n bits that are all 1 is 2n 1. Then N = M, N = bd − 1. Nonary or septemvigesimal can be used for compact representation of ternary, similar to how hexadecimal systems are used in place of binary. A base-three system is used in Islam to keep track of counting Tasbih to 100 on a single hand for counting prayers. In analog logic, the state of the circuit is often expressed ternary. This is most commonly seen in Transistor -- logic using 7406 open collector logic. The output is said to either be low, open. In this configuration the output of the circuit is actually not connected to any voltage reference at all. Thus, the actual level is sometimes unpredictable. A rare "ternary point" is used to denote fractional parts of an inning in baseball.
Trit (computing)
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Numeral systems
135.
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 quantum-mechanical system, such as the polarization of a single photon: here the two states are vertical polarization and horizontal polarization. In a classical system, a bit would have to be in the other. However, quantum mechanics allows the qubit to be at the same time, a property, fundamental to quantum computing. 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 as 1. A qubit is overall very different. There are two possible outcomes for the measurement of a qubit -- usually 1, like a bit. It is possible to fully encode one bit in one qubit. However, a qubit can hold even more information, e.g. up to two bits using superdense coding. The two states in which a qubit may be measured are known as basis states.
Qubit
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Bloch sphere representation of a qubit. The probability amplitudes in the text are given by and.
136.
Quantum information
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In physics and computer science, quantum information is information, held in the state of a quantum system. Quantum information can be manipulated using engineering techniques known as quantum information processing. Quantum information differs strongly from classical information, epitomized by the bit, in many unfamiliar ways. Among these are the following: A unit of information is the qubit. Unlike digital states, a qubit is continuous-valued, describable by a direction on the Bloch sphere. Despite being continuously valued in this way, a qubit is the smallest possible unit of information. The reason for this indivisibility is due to the Heisenberg principle: despite the qubit state being continuously-valued, it is impossible to measure the value precisely. A qubit cannot be converted into classical bits;, it cannot be "read". This is the no-teleportation theorem. Despite the no-teleportation theorem, qubits can be moved from one physical particle to another, by means of quantum teleportation. That is, qubits can be transported, independently of the underlying physical particle. An arbitrary qubit destroyed. This is the content of the the no-deleting theorem. Qubits can be changed, by applying linear transformations or quantum gates to them, to alter their state. Classical bits may be extracted from configurations of multiple qubits, through the use of quantum gates.
Quantum information
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General
137.
Pre-Columbian
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For this reason the alternative terms of Precontact Americas, Pre-Colonial Americas or Prehistoric Americas are also in use. In areas of Latin America the term usually used is Pre-Hispanic. Pre-Columbian civilizations established hallmarks which included permanent settlements, cities, complex societal hierarchies. Other civilizations were contemporary with the colonial period and were described in European historical accounts of the time. A few, such as the Maya civilization, had their own written records. Indigenous American cultures continue to evolve after the pre-Columbian era. Many of their descendants continue traditional practices, while adapting cultural technologies into their lives. Asian nomads are thought to have entered the Americas via the Bering Land Bridge, now the Bering Strait and possibly along the coast. Genetic evidence found in Amerindians' maternally inherited mitochondrial DNA supports the theory of multiple genetic populations migrating from Asia. Over the course of millennia, Paleo-Indians spread throughout North and South America. Exactly when the first group of people migrated into the Americas is the subject of much debate. One of the earliest identifiable cultures was the Clovis culture, with sites dating from some 13,000 years ago. However, older sites dating back to 20,000 years ago have been claimed. Some genetic studies estimate the colonization of the Americas dates from between 40,000 and 13,000 years ago. The chronology of migration models is currently divided into two general approaches.
Pre-Columbian
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Hopewell mounds from the Mound City Group in Ohio
Pre-Columbian
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hunter-gatherers
Pre-Columbian
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Mississippian site in Arkansas, Parkin Site, circa 1539. Illustration by Herb Roe.
Pre-Columbian
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One of the pyramids in the upper level of Yaxchilán
138.
Toe
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Toes are the digits of the foot of a tetrapod. Animal species such as cats that walk on their toes are described as being digitigrade. There are five toes present on each human foot. Each toe consists of the proximal, middle and distal, with the exception of the big toe. The hallux only contains two phalanx bones, distal. The phalanx bones of the toe join to the metatarsal bones of the foot at the interphalangeal joints. Present on all five toes is a toenail. With the exception of the hallux, movement is generally governed by action of the flexor digitorum brevis and extensor digitorum brevis muscles. These attach to the sides of the bones, making it impossible to move individual toes independently. Muscles between the toes on their bottom also help to abduct and adduct the toes. Additional control is provided by the flexor hallucis brevis. It is extended by the adductor hallucis muscle. The little toe has a separate set of control muscles and tendon attachments, abductor digiti minimi. Other foot muscles contribute to fine motor control of the foot. The connective tendons between the minor toes accounts for the inability to actuate individual toes.
Toe
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Toes on the foot. The innermost toe (bottom-left in image), which is normally called the big toe, is the hallux.
Toe
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An individual's toes that follow the common trend of the hallux outsizing the second toe.
Toe
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View of the feet in Michellangelo's David
139.
Yuki tribe
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The Yuki are an indigenous people of California, whose traditional territory is around Round Valley, Mendocino County. They are enrolled members of the Round Valley Indian Tribes of the Round Valley Reservation. Yuki tribes are thought to have settled as south as Hood Mountain in present-day Sonoma County. As European-American settlers began to Northern California in the early 1850s, they drove the Yuki from their lands. Captives were taken into slavery. In 1856, the US government established the Indian reservation of Nome Cult Farm at Round Valley. It forced thousands of other local tribes on to these lands, often without sufficient support for the transition. These tensions led to the Mendocino War, where US forces killed hundreds of Yuki and took others by force to Nome Cult Farm. The Yuki language is no longer spoken. Scholarly estimates have varied substantially in California as historians and anthropologists have tried to evaluate early documentation. Alfred L. Kroeber estimated the 1770 population of Coast Yuki as 2,000, 500, 500, respectively, or 3,000 in all. Sherburne F. Cook initially raised this total slightly to 3,500. Subsequently, he proposed a higher estimate of 9,730 Yuki. In the 2010 census, 569 people claimed Yuki ancestry. 255 of them were full-blooded.
Yuki tribe
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Yuki men at the Nome Cult Farm, ca. 1858
140.
California
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California is the most populous state in the United States and the third most extensive by area. The capital is Sacramento. Los Angeles is California's most populous city, largest after New York City. The state also has the nation's most populous county, its largest county by area, San Bernardino County. A major agricultural area, dominates the state's center. The Spanish Empire then claimed it in their New Spain colony. The western portion of Alta California then was admitted as the 31st state on September 9, 1850. If it were a country, California would be the 35th most populous. Fifty-eight percent of the state's economy is centered on finance, government, real estate services, professional, scientific and technical business services. Although it accounts for only 1.5 percent of the state's economy, California's industry has the highest output of any U.S. state. The kingdom of Queen Calafia, according to Montalvo, was said to be a remote land rich in gold. They were robust of body with great virtue. The island itself is one of the wildest in the world on account of the craggy rocks. This conventional wisdom that maps were drawn to reflect this way, lasted as late as the 1700's. Shortened forms of the state's name include CA, Cal. Calif. and US-CA.
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
141.
Chumashan languages
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Southern Baja languages are spoken in areas with long-established populations of a physical type. The population in the core Chumashan area has been stable for the past 10,000 years. However, the attested range of Chumashan is recent. All of the Chumashan languages are now extinct, although they are well documented in the unpublished fieldnotes of linguist John Peabody Harrington. Especially well documented are Barbareño, Ineseño, Ventureño. The last native speaker of a Chumashan language was Barbareño speaker Mary Yee, who died in 1965. Six Chumashan languages are attested, all now extinct. However, most of them are with language classes. Contemporary Chumash people now prefer to refer to their languages by native names rather than the older names based on the local missions. I. Northern Chumash 1. Obispeño Also known after the name of the major village near which the mission was founded. II. Southern Chumash a. Island Chumash 2.
Chumashan languages
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Pre-contact distribution of Chumashan languages
142.
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 and West Papua. Most of the population of over 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 people live in urban centres. Papua New Guinea is classified as a developing economy by the International Monetary Fund. Mining remains a economic factor, however, with talks of resuming mining operations in the previously closed-off Panguna ongoing with the local and national governments. Nearly 40 percent of the population lives a self-sustainable natural lifestyle with no access to global capital. These societies and clans are explicitly acknowledged within the nation's constitutional framework. Archaeological evidence indicates that humans first arrived in Papua New Guinea around 42,000 to 45,000 years ago. 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 certain fishing techniques. On Goaribari Island in the Gulf of Papua, a missionary, Harry Dauncey, found 10,000 skulls in the island's Long Houses. Traders from Southeast Asia had visited New Guinea beginning 5,000 years ago to collect bird of paradise plumes. The country's dual name results from its complex administrative history before independence. The papua is derived from an local term of uncertain origin.
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.
143.
Pentadecimal
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This is a list of numeral systems, writing systems for expressing numbers. Numeral systems are classified as to whether they use positional notation, further categorized by radix or base. The common names are derived 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. This turns out to be 1925. Radix Radix economy Table of bases List of numbers in various languages Numeral prefix
Pentadecimal
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Numeral systems
144.
Base 24
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Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the same symbol for the different orders of magnitude. This greatly simplified arithmetic, leading to the rapid spread of the notation across the world. With the use of a point, the notation can be extended to include fractions and the numeric expansions of real numbers. The Babylonian system, base-60, was the first positional system developed, is still used today to count time and angles. The Hindu–Arabic numeral system, base-10, is the most commonly used system in the world today for most calculations. The base-10 system, likely motivated by counting with the ten fingers, is ubiquitous. Some continue to be used today. For example, it lacked a real 0 value. Zero was indicated by a space between sexagesimal numerals. By 300 BC, a symbol was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish, the Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges. The Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number. Thus numbers like 120, 3 and 180, 4 and 240, looked the same because the larger numbers lacked a final sexagesimal placeholder.
Base 24
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Numeral systems
145.
Base-6
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The senary numeral system has six as its base. It has been adopted independently by a small number of cultures. As six is a superior highly composite number, many of the arguments made in favor of the duodecimal system also apply to this base. This is proved by contradiction. This property maximizes the probability that the result of an multiplication will end in zero, given that neither of its factors do. E.g. if three fingers are extended on the right, 34senary is represented. This is equivalent to 3 × 6 + 4, 22decimal. Flipping the ` sixes' hand around to its backside may help to further disambiguate which represents the units. More abstract finger counting systems, such as chisanbop or binary, allow counting to 99, 1,023, or even higher depending on the method. The Ndom language of Papua New Guinea is reported to have senary numerals. Mer means 6, mer an thef means 6 × 2 = 12, nif thef means 36 × 2 = 72. Another example from Papua New Guinea are the Morehead-Maro languages. In these languages, counting is connected to ritualized yam-counting. These languages count from a base six, employing words for the powers of six; running up for some of the languages. One example is Kómnzo with the following numerals: nimbo, féta, tarumba, wärämäkä, wi.
Base-6
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Numeral systems
Base-6
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34 senary = 22 decimal, in senary finger counting
Base-6
146.
Algorism
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One who practices algorism is known as an algorist. This title means "Algoritmi on the numbers of the Indians", where "Algoritmi" was the translator's Latinization of Al-Khwarizmi's name. Al-Khwarizmi was the most widely read mathematician in Europe in the late Middle Ages, primarily through the Algebra. In medieval Latin, algorismus, the corruption of his name, simply meant the "decimal number system", still the meaning of modern English algorism. In English, it was first used then by Chaucer in 1391. Another early use of the word is 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 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, which in turn led to the notion of the decimal point. 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
147.
Decimal computer
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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 7070. Except for the 1620, these machines used word addressing. When non-numeric characters were used in these machines, they were encoded as two decimal digits. Other early computers were character oriented, but provided instructions for performing arithmetic on strings of character that consisted of decimal numerals. On these machines the basic data element was an alphanumeric character, typically encoded in 6 bits. UNIVAC I and UNIVAC II used word addressing, with 12 character words. IBM examples include IBM 702, other members of the IBM 1400 series, including the IBM 7010. The IBM character machines nominally used decimal addressing, but some allowed the use of non-decimal characters in addresses to expand the available address space. Addresses referenced a single character encoded with two additional bits per a parity bit. The word mark enabled operations on variable length words. 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. 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.
Decimal computer
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IBM 650 front panel with bi-quinary coded decimal displays
148.
Dewey Decimal Classification
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The Dewey Decimal Classification, or Dewey Decimal System, is a proprietary library classification system first published in the United States by Melvil Dewey in 1876. It is also available in an abridged version suitable for smaller libraries. It is currently maintained by the Online Computer Library Center, a non-profit cooperative that serves libraries. OCLC licenses access to an online version for catalogers called WebDewey. Libraries previously had given books permanent shelf locations that were related to the order of acquisition rather than topic. The classification's notation makes use of three-digit Arabic numerals for main classes, with fractional decimals allowing expansion for further detail. Using Arabic numerals for symbols, it is flexible to the degree that numbers can be expanded in linear fashion to cover special aspects of general subjects. The number makes it possible to find any book and to return it to its proper place on the library shelves. The classification system is used in 200,000 libraries in at least 135 countries. The major competing classification system to the Dewey Decimal system is the Library of Congress Classification system created by the U.S. Library of Congress. Melvil Dewey was an American librarian and self-declared reformer. He developed the ideas for his library classification system in 1873 while working at Amherst College library. He applied the classification to the books in that library, until in 1876 he had a first version of the classification. He used the pamphlet, published in more than one version during the year, to solicit comments from other librarians.
Dewey Decimal Classification
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Melvil Dewey, the inventor of the Dewey Decimal classification
Dewey Decimal Classification
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A library book shelf in Hong Kong catalogued using the Dewey classification
149.
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. However, the term "mantissa" may cause confusion because it can also refer to the fractional part of the common logarithm. If the number is negative then a minus sign precedes m. In normalized notation, the exponent is chosen so that the absolute value of the coefficient is at least one but less than ten. Decimal floating point is a system closely related to scientific notation. Any given integer can be written in the form 10 ^ n in many ways: for example, 350 can be written as 7002350000000000000 ♠ 3.5 × 102 or 7002350000000000000 ♠ 35 × 101 or 7002350000000000000 ♠ 350 × 100. In scientific notation, the n is chosen so that the absolute value of m remains at least one but less than ten. Thus 350 is written as 7002350000000000000♠3.5×102. This form allows easy comparison of numbers, as the exponent n gives the number's order of magnitude. In normalized notation, the n is negative for a number with absolute value between 1. The 10 and exponent are often omitted when the exponent is 0. Scientific form is the typical form of expression of large numbers in many fields, unless an unnormalized form, such as notation, is desired. Engineering notation differs from scientific notation in that the n is restricted to multiples of 3.
Scientific notation
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A calculator display showing the Avogadro constant in E notation
150.
SI prefix
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A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or fraction of the unit. While all metric prefixes in common today are decadic, historically there have been a number of binary metric prefixes as well. Each prefix has a unique symbol, prepended to the symbol. The prefix -, for example, may be added to gram to indicate multiplication by one thousand: one kilogram is equal to one thousand grams. The prefix -, likewise, may be added to metre to indicate division by one thousand; one millimetre is equal to one thousandth of a metre. Decimal multiplicative prefixes have been a feature of all forms of the metric system with six dating back in the 1790s. Metric prefixes have even been prepended to non-metric units. Since 2009, they have formed part of the International System of Quantities. The BIPM specifies twenty prefixes for the International System of Units. Each name has a symbol, used in combination with the symbols for units of measure. For example, the symbol for kilo - is used to produce ` km', ` kg', ` kW', which are the SI symbols for kilometre, kilogram, kilowatt, respectively. Prefixes may not be used in combination. This also applies to mass, for which the SI unit already contains a prefix. For example, milligram is used instead of microkilogram. In arithmetic of measurements having prefixed units, the prefixes must be expanded except when adding or subtracting values with identical units.
SI prefix
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Distance marker on the Rhine: 36 (XXXVI) myriametres from Basel. Note that the stated distance is 360 km; comma is the decimal mark in Germany.
151.
Encyclopedia of Mathematics
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The Encyclopedia of Mathematics is a large reference work in mathematics. It is available on CD-ROM. The presentation is technical in nature. The encyclopedia was published by Kluwer Academic Publishers until 2003, when Kluwer became part of Springer. The CD-ROM contains three-dimensional objects. A dynamic version of the encyclopedia is now available as a public wiki online. This new wiki is a collaboration between the European Mathematical Society. All entries will be monitored by members of an editorial board selected by the European Mathematical Society. Vinogradov, I. M. Matematicheskaya entsiklopediya, Moscow, Sov. Entsiklopediya, 1977. Hazewinkel, M. Encyclopaedia of Mathematics, Kluwer, 1994. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 1, Kluwer, 1987. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 2, Kluwer, 1988. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 3, Kluwer, 1989. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 4, Kluwer, 1989.
Encyclopedia of Mathematics
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Encyclopedia of Mathematics snap shot
Encyclopedia of Mathematics
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A complete set of Encyclopedia of Mathematics at a university library.
152.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each variation of a book. For example, an e-book, a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned after 1 January 2007, 10 digits long if assigned before 2007. The method of assigning an ISBN varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated based upon the 9-digit Standard Book Numbering created in 1966. The 10-digit ISBN format was published in 1970 as international standard ISO 2108. The International Standard Serial Number, identifies periodical publications such as magazines; and the International Standard Music Number covers for musical scores. The ISBN configuration of recognition was generated in 1967 in the United Kingdom by Emery Koltay. The 10-digit ISBN format was published as international standard ISO 2108. The United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978. An SBN may be converted by prefixing the digit "0". This can be converted to ISBN 0-340-01381-8; the digit does not need to be re-calculated. Since 1 ISBNs have contained 13 digits, a format, compatible with "Bookland" European Article Number EAN-13s.
International Standard Book Number
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A 13-digit ISBN, 978-3-16-148410-0, as represented by an EAN-13 bar code
153.
Hermann Schmid (computer scientist)
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Some also had a variable wordlength, which enabled operations with a large number of digits. Early computers that were exclusively decimal include the ENIAC, IBM NORC, IBM 650, IBM 7070. Except for the 1620, these machines used word addressing. When non-numeric characters were used in these machines, they were encoded as two decimal digits. Early computers were character oriented, but provided instructions for performing arithmetic on strings of character that consisted of decimal numerals. On these machines the basic data element was an alphanumeric character, typically encoded in 6 bits. UNIVAC I and UNIVAC II used word addressing, with 12 character words. IBM examples include IBM 702, IBM 705, IBM 7080, other members of the IBM 1400 series, including the IBM 7010. Some allowed the use of non-decimal characters in addresses to expand the available address space. Addresses referenced a single character encoded with two additional bits per character, a word mark and a parity bit. The mark enabled operations on variable length words. It introduced EBCDIC encoding, though ASCII was also supported. Several microprocessor families offer decimal support. For example, the 80x86 family of microprocessors provide instructions to convert one-byte BCD numbers to binary format after arithmetic operations. These operations hence are now slower than using 32-bit or wider BCD ` tricks' to compute in BCD.
Hermann Schmid (computer scientist)
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IBM 650 front panel with bi-quinary coded decimal displays
154.
John Wiley & Sons
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Founded in 1807, Wiley is also known for publishing For Dummies. As of 2015, the company had a revenue of $ billion. Wiley was established in 1807 when Charles Wiley opened a print shop in Manhattan. Wiley later shifted its focus to scientific, technical, engineering subject areas, abandoning its literary interests. Charles Wiley's son John took over the business when his father died in 1826. The firm was successively named Wiley, Lane & Co. then Wiley & Putnam, then John Wiley. The company acquired its present name in 1876, when John's second son William H. Wiley joined his brother Charles in the business. Through the 20th century, the company expanded higher education. Wiley in December 2010 opened an office in Dubai, to build on its business in the Middle East more effectively. The company has had an office in Beijing, China, since 2001, China is now its sixth-largest market for STEM content. On April 2012, the company announced the establishment of Wiley Brasil Editora LTDA in Brazil, effective May 1, 2012. Wiley's scientific, technical, medical business was significantly expanded by the acquisition of Blackwell Publishing in February 2007. Through a backfile initiative completed in 2007, million pages of content have been made available online, a collection dating back to 1799. Major journals published include Angewandte Chemie, Advanced Materials, Hepatology, Liver Transplantation. Launched commercially in 1999, Wiley InterScience provided online access to Wiley journals, major reference works, books, including backfile content.
John Wiley & Sons
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The Hoboken, New Jersey headquarters
155.
Mike Cowlishaw
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He was a Fellow of the Institute of Engineering and Technology, the British Computer Society. He was educated at The University of Birmingham. Cowlishaw is best known as a programmer and writer. He has edited various computing standards, including ISO, BSI, ANSI, IETF, W3C, ECMA, IEEE. He retired in March 2010. Cowlishaw collected related documentation. Outside computing, he caved in the UK, New England, Mexico. Soc. June 2003 Densely Packed Decimal Encoding, Cowlishaw, M. F. IEE Proceedings – Computers and Digital Techniques ISSN 1350-2387, Vol. 149, No. 4, Winter 1994, pp. 15–24 A large-scale computer conferencing system, Chess and Cowlishaw, IBM Systems Journal, Vol 26, No. 1, 1987, IBM Reprint order number G321-5291 LEXX – A programmable structured editor, Cowlishaw, M. F. IBM Journal of Research and Development, Vol 31, No. 1, 1987, IBM Reprint order number G322-0151 Fundamental requirements for picture presentation, Cowlishaw, M. F. Proc. Society for Information Display, Volume 26, No. 2 The design of the REXX language, Cowlishaw, M. F. IBM Systems Journal, Volume 23, No. 4, 1984, IBM Reprint order number G321-5228 The Characteristics and Use of Lead-Acid Cap Lamps, Cowlishaw, M. F. Trans. British Cave Research Association, Vol 1, No. 4, pp. 199–214, December 1974
Mike Cowlishaw
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Michael F. Cowlishaw
156.
JSTOR
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JSTOR is a digital library founded in 1995. Originally containing back issues of academic journals, it now also includes books and primary sources, current issues of journals. It provides full-text searches of almost 2,000 journals. President of Princeton University from 1972 to 1988, founded JSTOR. Most libraries found it prohibitively expensive in terms of space to maintain a comprehensive collection of journals. By digitizing many journal titles, JSTOR allowed libraries to outsource the storage of journals with the confidence that they would remain available long-term. Full-text search ability improved access dramatically. Bowen initially considered using CD-ROMs for distribution. JSTOR originally encompassed ten economics and history journals. It became a fully searchable index accessible from any ordinary web browser. Special software was put in place to make graphs clear and readable. With the success of this limited project, then-president of JSTOR, wanted to expand the number of participating journals. The work of adding these volumes to JSTOR was completed by December 2000. The Andrew W. Mellon Foundation funded JSTOR initially. Until January 2009 JSTOR operated in New York City and in Ann Arbor, Michigan.
JSTOR
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The JSTOR front page
157.
Integrated Authority File
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The Integrated Authority File or GND is an international authority file for the organisation of personal names, subject headings and corporate bodies from catalogues. It is used mainly increasingly also by archives and museums. The GND is managed with various regional library networks in German-speaking Europe and other partners. The GND falls under the Creative Commons Zero license. The GND specification provides a hierarchy of high-level sub-classes, useful in library classification, an approach to unambiguous identification of single elements. It also comprises an ontology intended for knowledge representation in the semantic web, available in the RDF format.
Integrated Authority File
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GND screenshot