1.
Numeral system
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A numeral system is a writing system for expressing numbers, that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. It can be seen as the context that allows the symbols 11 to be interpreted as the symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases. The number the numeral represents is called its value, ideally, a numeral system will, Represent a useful set of numbers Give every number represented a unique representation Reflect the algebraic and arithmetic structure of the numbers. For example, the decimal representation of whole numbers gives every nonzero whole number a unique representation as a finite sequence of digits. Etc. all of which have the same meaning except for some scientific, such systems are, however, not the topic of this article. The most commonly used system of numerals is the Hindu–Arabic numeral system, two Indian mathematicians are credited with developing it. Aryabhata of Kusumapura developed the notation in the 5th century. The numeral system and the concept, developed by the Hindus in India, slowly spread to other surrounding countries due to their commercial. The Arabs adopted and modified it, even today, the Arabs call the numerals which they use Rakam Al-Hind or the Hindu numeral system. The Arabs translated Hindu texts on numerology and spread them to the world due to their trade links with them. The Western world modified them and called them the Arabic numerals, hence the current western numeral system is the modified version of the Hindu numeral system developed in India. It also exhibits a great similarity to the Sanskrit–Devanagari notation, which is used in India. The simplest numeral system is the numeral system, in which every natural number is represented by a corresponding number of symbols. If the symbol / is chosen, for example, then the seven would be represented by ///////. Tally marks represent one such system still in common use, the unary system is only useful for small numbers, although it plays an important role in theoretical computer science. Elias gamma coding, which is used in data compression. The unary notation can be abbreviated by introducing different symbols for new values. The ancient Egyptian numeral system was of type, and the Roman numeral system was a modification of this idea
Numeral system
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Numeral systems
2.
Arabic numerals
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In this numeral system, a sequence of digits such as 975 is read as a single number, using the position of the digit in the sequence to interpret its value. The symbol for zero is the key to the effectiveness of the system, the system was adopted by Arab mathematicians in Baghdad and passed on to the Arabs farther west. There is some evidence to suggest that the numerals in their current form developed from Arabic letters in the Maghreb, the current form of the numerals developed in North Africa, distinct in form from the Indian and eastern Arabic numerals. The use of Arabic numerals spread around the world through European trade, books, the term Arabic numerals is ambiguous. It most commonly refers to the widely used in Europe. Arabic numerals is also the name for the entire family of related numerals of Arabic. It may also be intended to mean the numerals used by Arabs and it would be more appropriate to refer to the Arabic numeral system, where the value of a digit in a number depends on its position. The decimal Hindu–Arabic numeral system was developed in India by AD700, the development was gradual, spanning several centuries, but the decisive step was probably provided by Brahmaguptas formulation of zero as a number in AD628. The system was revolutionary by including zero in positional notation, thereby limiting the number of digits to ten. It is considered an important milestone in the development of mathematics, one may distinguish between this positional system, which is identical throughout the family, and the precise glyphs used to write the numerals, which varied regionally. The glyphs most commonly used in conjunction with the Latin script since early modern times are 0123456789. The first universally accepted inscription containing the use of the 0 glyph in India is first recorded in the 9th century, in an inscription at Gwalior in Central India dated to 870. Numerous Indian documents on copper plates exist, with the symbol for zero in them, dated back as far as the 6th century AD. Inscriptions in Indonesia and Cambodia dating to AD683 have also been found and their work was principally responsible for the diffusion of the Indian system of numeration in the Middle East and the West. In the 10th century, Middle-Eastern mathematicians extended the decimal system to include fractions. The decimal point notation was introduced by Sind ibn Ali, who wrote the earliest treatise on Arabic numerals. Ghubar numerals themselves are probably of Roman origin, some popular myths have argued that the original forms of these symbols indicated their numeric value through the number of angles they contained, but no evidence exists of any such origin. In 825 Al-Khwārizmī wrote a treatise in Arabic, On the Calculation with Hindu Numerals, Algoritmi, the translators rendition of the authors name, gave rise to the word algorithm
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, that is sometimes discouraged as it can lead to confusion with Indian numerals, used in Brahmic scripts of India. Each numeral in the Persian variant has a different Unicode point even if it looks identical to the Eastern Arabic numeral counterpart, however the variants used with Urdu, Sindhi and other South Asian languages are not encoded separately from the Persian variants. See U+0660 through U+0669 and U+06F0 through U+06F9, written numerals are arranged with their lowest-value digit to the right, with higher value positions added to the left. That is identical to the arrangement used by Western texts using Hindu-Arabic numerals even though Arabic script is read from right to left. There is no conflict unless numerical layout is necessary, as is the case for arithmetic problems and lists of numbers, Eastern Arabic numerals remain strongly predominant vis-à-vis Western Arabic numerals in many countries to the East of the Arab world, particularly in Iran and Afghanistan. In Pakistan, Western Arabic numerals are more used as a considerable majority of the population is anglophone. Eastern numerals still continue to see use in Urdu publications and newspapers, in North Africa, only Western Arabic numerals are now commonly used. In medieval times, these used a slightly different set
Eastern Arabic numerals
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Numeral systems
Eastern Arabic numerals
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Arabic style Eastern Arabic numerals on a clock in the
Cairo Metro
Eastern Arabic numerals
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Ottoman clocks tended to use Eastern Arabic numerals styled to look like
Roman
4.
Indian numerals
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Indian numerals are the symbols representing numbers in India. These numerals are used in the context of the decimal Hindu–Arabic numeral system. Below is a list of the Indian numerals in their modern Devanagari form, the corresponding Hindu-Arabic equivalents, their Hindi and Sanskrit pronunciation, since Sanskrit is an Indo-European language, it is obvious that the words for numerals closely resemble those of Greek and Latin. The word Shunya for zero was translated into Arabic as صفر sifr, meaning nothing which became the zero in many European languages from Medieval Latin. The five Indian languages that have adapted the Devanagari script to their use also naturally employ the numeral symbols above, of course, for numerals in Tamil language see Tamil numerals. For numerals in Telugu language see Telugu numerals, Tamil and Malayalam scripts also have distinct forms for 10,100,1000 numbers, ௰, ௱, ௲and ൰, ൱, ൲ respectively in tamil and scripts. A decimal place system has been traced back to ca.500 in India, before that epoch, the Brahmi numeral system was in use, that system did not encompass the concept of the place-value of numbers. Instead, Brahmi numerals included additional symbols for the tens, as well as symbols for hundred. The Indian place-system numerals spread to neighboring Persia, where they were picked up by the conquering Arabs, in 662, Severus Sebokht - a Nestorian bishop living in Syria wrote, I will omit all discussion of the science of the Indians. Of their subtle discoveries in astronomy — discoveries that are more ingenious than those of the Greeks, I wish only to say that this computation is done by means of nine signs. But it is in Khmer numerals of modern Cambodia where the first extant material evidence of zero as a numerical figure, as it was from the Arabs that the Europeans learned this system, the Europeans called them Arabic numerals, the Arabs refer to their numerals as Indian numerals. In academic circles they are called the Hindu–Arabic or Indo–Arabic numerals, but what was the net achievement in the field of reckoning, the earliest art practiced by man. An inflexible numeration so crude as to progress well nigh impossible. Man used these devices for thousands of years without contributing an important idea to the system. Even when compared with the growth of ideas during the Dark Ages. When viewed in light, the achievements of the unknown Hindu. Sanskrit Siddham Numbers Karl Menninger, Number Words and Number Symbols - A Cultural History of Numbers ISBN 0-486-27096-3 David Eugene Smith and Louis Charles Karpinski, The Hindu-Arabic Numerals
Indian numerals
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Numeral systems
5.
Sinhala numerals
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Sinhalese belongs to the Indo-European language family with its roots deeply associated with Indo-Aryan sub family to which the languages such as Persian and Hindi belong. It is also surmised that Sinhala had evolved from an ancient variant of Apabramsa which is known as ‘Elu’, when tracing history of Elu, it was preceded by Hela or Pali Sihala. The Sinhala script had evolved from Southern Brahmi script from which almost all the Southern Indic Scripts such as Telugu, later Sinhala was influenced by Grantha writing of Southern India. Since 1250 AD, the Sinhala script had remained the same with few changes, although some scholars are of the view that the Brahmi Script arrived with the Buddhism, Mahavamsa speaks of written language even right after the arrival of Vijaya. Archeologists had found pottery fragments in Anuradhapura Sri Lanka with older Brahmi script inscriptions, the earliest Brahmi Script found in India had been dated to 6th Century BC in Tamil Nadu though most of Brahmi writing found in India had been attributed to emperor Ashoka in the 3rd century BC. Sinhala letters are round-shaped and are written left to right. The evolution of the script to the present shapes may have taken place due to writing on Ola leaves, unlike chiseling on a rock, writing on palm leaves has to be more round-shaped to avoid the stylus ripping the Palm leaf while writing on it. When drawing vertical or horizontal lines on Ola leaf, the leaves would have been ripped. Instead a stylistic stop which was known as ‘Kundaliya’ is used, period and commas were later introduced into Sinhala script after the introduction of paper due to the influence of Western languages. In modern Sinhala, Arabic numerals, which were introduced by Portuguese, Dutch and English, is used for writing numbers and it is accepted that Arabic numerals had evolved from Brahmi numerals. This article will touch upon Brahmi numerals, which were found in Sri Lanka. It had been found five different types of numerations were used in the Sinhala language at the time of the invasion of the Kandyan kingdom by the British. Out of the five types of numerations, two sets of numerations were in use in the century mainly for astrological calculations and to express traditional year. The five types or sets of numerals or numerations are listed below, according to Mr. Gunesekera, these numerals were used for ordinary calculations and to express simple numbers. These numerals had separate Symbols for 10,40,50,100,1000 and these numerals were also regarded as Lith Lakunu or ephemeris numbers by W. A. De Silva in his “Catalogue of Palm leaf manuscripts in the library of Colombo Museum”. This set of numerals was known as Sinhala illakkam or Sinhala archaic numerals, Arabic Figures are now universally used. For the benefit of the student, the old numerals are given in the plate opposite,11 clauses had been numbered in Arabic numerals in the English part of the agreement and in parallel Sinhala clauses were numbered in Sinhala archaic numerals. Numbers of lith illakkam look Sinhala letters and vowel modifiers, the number six is known as ‘akma’ in the Lith Illakkam
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.
6.
Balinese numerals
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The Balinese language has an elaborate decimal numeral system. The numerals 1–10 have basic, combining, and independent forms, the combining forms are used to form higher numbers. In some cases there is more than one word for a numeral, reflecting the Balinese register system, final orthographic -a is a schwa. * A less productive combining form of a-1 is sa- and it, ulung-, and sangang- are from Javanese. Dasa 10 is from Sankrit désa, like English, Balinese has compound forms for the teens and tens, however, it also has a series of compound tweens, 21–29. The teens are based on a root *-welas, the tweens on -likur, hyphens are not used in the orthography, but have been added to the table below to clarify their derivation. The high-register combining forms kalih-2 and tigang-3 are used with -likur, -dasa, and higher numerals, the teens are from Javanese, where the -olas forms are regular, apart from pele-kutus 18, which is suppletive. Sa-laé25, and se-ket 50 are also suppletive, and cognate with Javanese səlawé25, there are additional numerals pasasur ~ sasur 35 and se-timahan ~ se-timan 45, and a compound telung-benang for 75. The unit combining forms are combined with atus 100, atak 200, amas 400, tali 1000, laksa 10,000, keti 100,000, in addition, there is karobelah 150, lebak 175, and sepa for 1600. At least karobelah has a cognate in Javanese, ro-bəlah, where ro- is the form for two
Balinese numerals
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Numeral systems
7.
Burmese numerals
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Burmese numerals are a set of numerals traditionally used in the Burmese language, although the Arabic numerals are also used. Burmese numerals follow the Hindu-Arabic numeral system used in the rest of the world. 1 Burmese for zero comes from Sanskrit śūnya.2 Can be abbreviated to IPA, in list contexts, spoken Burmese has innate pronunciation rules that govern numbers when they are combined with another word, be it a numerical place or a measure word. Other suffixes such as ထောင်, သောင်း, သိန်း, and သန်း all shift to, for six and eight, no pronunciation shift occurs. These pronunciation shifts are exclusively confined to spoken Burmese and are not spelt any differently,1 Shifts to voiced consonant following three, four, five, and nine. Ten to nineteen are almost always expressed without including တစ်, another pronunciation rule shifts numerical place name from the low tone to the creaky tone. Number places from 10 up to 107 has increment of 101, beyond those Number places, larger number places have increment of 107. 1014 up to 10140 has increment of 107, numbers in the hundreds place, shift from ရာ to ရာ့, except for numbers divisible by 100. Numbers in the place, shift from ထောင် to ထောင့်. Hence, a number like 301 is pronounced, while 300 is pronounced, the digits of a number are expressed in order of decreasing digits place. When a number is used as an adjective, the word order is. However, for numbers, the word order is flipped to. The exception to rule is the number 10, which follows the standard word order. Ordinal numbers, from first to tenth, are Burmese pronunciations of their Pali equivalents and they are prefixed to the noun. Beyond that, cardinal numbers can be raised to the ordinal by suffixing the particle မြောက် to the number in the order, number + measure word + မြောက်. Colloquially, decimal numbers are formed by saying ဒသမ where the separator is located. For example,10.1 is ဆယ် ဒသမ တစ်, half is expressed primarily by တစ်ဝက်, although ထက်ဝက်, အခွဲ and အခြမ်း are also used. Quarter is expressed with အစိတ် or တစ်စိတ်, other fractional numbers are verbally expressed as follows, denominator + ပုံ + numerator + ပုံ
Burmese numerals
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Numeral systems
Burmese numerals
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Burmese numerals in various script styles
8.
Javanese numerals
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The Javanese language has a decimal numeral system with distinct words for the tweens from 21 to 29, called likuran. The basic numerals 1–10 have independent and combining forms, the latter derived via a suffix -ng, the combining forms are used to form the tens, hundreds, thousands, and millions. The numerals 1–5 and 10 have distinct high-register and low register forms, the halus forms are listed below in italics. Like English, Javanese has compound forms for the teens, however, it also has a series of compound tweens, the teens are based on a root -las, the tweens on -likur, and the tens are formed by the combining forms. Hyphens are not used in the orthography, but have added to the table below to clarify their derivation. Final orthographic -a tends to in many dialects, as does any preceding a, parallel to the tens are the hundreds, the thousands, and the millions, except that the compounds of five and six are formed with limang- and nem-. The names of the Old Javanese numerals were derived from their names in the Sanskrit language, balinese numerals, a related but yet more complex numeral system
Javanese numerals
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Numeral systems
9.
Lao alphabet
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Lao script, or Akson Lao, is the primary script used to write the Lao language and other minority languages in Laos. It was also used to write the Isan language, but was replaced by the Thai script and it has 27 consonants,7 consonantal ligatures,33 vowels, and 4 tone marks. Akson Lao is a system to the Thai script, with which it shares many similarities. However, Lao has fewer characters and is formed in a curvilinear fashion than Thai. Lao is traditionally written from left to right, Lao is considered an abugida, in which certain implied vowels are unwritten. However, due to spelling reforms by the communist Lao Peoples Revolutionary Party, despite this, most Lao outside of Laos, and many inside Laos, continue to write according to former spelling standards, so vernacular Lao functions as a pure abugida. For example, the old spelling of ສເລີມ to hold a ceremony, vowels can be written above, below, in front of, or behind consonants, with some vowel combinations written before, over and after. Spaces for separating words and punctuation were traditionally not used, but a space is used, the letters have no majuscule or minuscule differentiation. The Lao script was standardized in the Mekong River valley after the various Tai principalities of the region were merged under Lan Xang in the 14th century. This script, sometimes known as Tai Noi, has changed little since its inception and continued use in the Lao-speaking regions of modern-day Laos, conversely, the Thai alphabet continued to evolve, but the scripts still share similarities. This script was derived locally from the Khmer script of Angkor with additional influence from Mon, traditionally, only secular literature were written with the Lao alphabet. Religious literature was written in Tua Tham, a Mon-based script that is still used for the Tai Khün, Tai Lue. Mystical, magical, and some literature was written in a modified version of the Khmer alphabet. Essentially Thai and Lao are almost typographic variants of other just as in the Javanese and Balinese scripts. The Lao and Thai alphabets share the same roots, but Lao has fewer characters and is written in a curvilinear fashion than Thai. However this is apparent today due to the communist party simplifying the spelling to be phonetic. There is speculation that the Lao and Thai script both derive from a common script due to the similarities between the scripts. When examining older forms of Thai scripts, many letters are almost identical to the Lao alphabet, some minority languages use separate writing systems, The Hmong have adopted the Roman Alphabet
Lao alphabet
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Numeral systems
Lao alphabet
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Lao
10.
Thai numerals
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The Thai language lacks grammatical number. A count is expressed in the form of an uninflected noun followed by a number. In Thai, counting is kannap, the classifier, laksananam Variations to this pattern do occur, a partial list of Thai words that also classify nouns can be found in Wiktionary category, Thai classifiers. Thai sūn is written as oval 0 when using Arabic numerals, but a small circle ๐ when using traditional numerals and it is from Sanskrit śūnya, as are the alternate names for numbers one to four given below, but not the counting 1. Thai names for N +1 and the regular digits 2 through 9 as shown in the table, below, resemble those in Chinese varieties as spoken in Southern China, Thai and Lao words for numerals are almost identical, however, the numerical digits vary somewhat in shape. Shown below is a comparison between three languages using Cantonese and Minnan characters and pronunciations, the Thai transliteration uses the Royal Thai General System of Transcription. Sanskrit lakh designates the place value of a digit, which are named for the powers of ten, the place is lak nuai, tens place, lak sip, hundreds place, lak roi. The number one following any multiple of sip becomes et, the number ten is the same as Minnan 十. Numbers from twenty to twenty nine begin with yi sip, names of the lak sip for 30 to 90, and for the lak of 100,1000,10,000,100,000 and million, are almost identical to those of the like Khmer numerals. For the numbers twenty-one through twenty-nine, the part signifying twenty, yi sip, see the alternate numbers section below. The hundreds are formed by combining roi with the tens and ones values, for example, two hundred and thirty-two is song roi sam sip song. The words roi, phan, muen, and saen should occur with a preceding numeral, nueng never precedes sip, so song roi nueng sip is incorrect. Native speakers will sometimes use roi nueng with different tones on nueng to distinguish one hundred from one hundred, however, such distinction is often not made, and ambiguity may follow. To resolve this problem, if the number 101 is intended, numbers above a million are constructed by prefixing lan with a multiplier. For example, ten million is sip lan, and a trillion is lan lan, colloquially, decimal numbers are formed by saying chut where the decimal separator is located. For example,1.01 is nueng chut sun nueng, fractional numbers are formed by placing nai between the numerator and denominator or using x suan y to clearly indicate. For example, ⅓ is nueng nai sam or nueng suan sam, the word set can be omitted. The word khrueng is used for half and it precedes the measure word if used alone, but it follows the measure word when used with another number
Thai numerals
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Numeral systems
11.
Japanese numerals
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The system of Japanese numerals is the system of number names used in the Japanese language. The Japanese numerals in writing are based on the Chinese numerals. Two sets of pronunciations for the numerals exist in Japanese, one is based on Sino-Japanese readings of the Chinese characters, there are two ways of writing the numbers in Japanese, in Hindu-Arabic numerals or in Chinese numerals. The Hindu-Arabic numerals are often used in horizontal writing. Numerals with multiple On readings use the Go-on and Kan-on variants respectively, * The special reading 〇 maru is also found. It may be used when reading individual digits of a number one after another. A popular example is the famous 109 store in Shibuya, Tokyo which is read as ichi-maru-kyū and this usage of maru for numerical 0 is similar to reading numeral 0 in English as oh. However, as a number, it is written as 0 or rei. Additionally, two and five are pronounced with a vowel in phone numbers Starting at 万, numbers begin with 一 if no digit would otherwise precede. That is,100 is just 百 hyaku, and 1000 is just 千 sen and this differs from Chinese as numbers begin with 一 if no digit would otherwise precede starting at 百. And, if 千 sen directly precedes the name of powers of myriad, 一 ichi is normally attached before 千 sen and that is,10,000,000 is normally read as 一千万 issenman. But if 千 sen does not directly precede the name of powers of myriad or if numbers are lower than 2,000 and that is,15,000,000 is read as 千五百万 sengohyakuman or 一千五百万 issengohyakuman, and 1,500 as 千五百 sengohyaku or 一千五百 issengohyaku. The numbers 4 and 9 are considered unlucky in Japanese,4, pronounced shi, is a homophone for death,9, the number 13 is sometimes considered unlucky, though this is a carryover from Western tradition. On the contrary, numbers 7 and sometimes 8 are considered lucky in Japanese, in modern Japanese, cardinal numbers are given the on readings except 4 and 7, which are called yon and nana respectively. Alternate readings are used in names, day-of-month names. For instance, the decimal fraction 4.79 is always read yon-ten nana kyū, though April, July, and September are called shi-gatsu, shichi-gatsu, the on readings are also used when shouting out headcounts. Intermediate numbers are made by combining elements, Tens from 20 to 90 are -jū as in 二十 to 九十. Hundreds from 200 to 900 are -hyaku, thousands from 2000 to 9000 are -sen
Japanese numerals
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Numeral systems
12.
Counting rods
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Counting rods are small bars, typically 3–14 cm long, that were used by mathematicians for calculation in ancient China, Japan, Korea, and Vietnam. They are placed horizontally or vertically to represent any integer or rational number. The written forms based on them are called rod numerals and they are a true positional numeral system with digits for 1–9 and a blank for 0, from the Warring states period to the 16th century. Counting rods were used by ancient Chinese for more two thousand years. In 1954, forty-odd counting rods of the Warring States period were found in Zuǒjiāgōngshān Chu Grave No.15 in Changsha, in 1973, archeologists unearthed a number of wood scripts from a Han dynasty tomb in Hubei. On one of the scripts was written, “当利二月定算”. This is one of the earliest examples of using counting rod numerals in writing, in 1976, a bundle of Western Han counting rods made of bones was unearthed from Qianyang County in Shaanxi. The use of counting rods must predate it, Laozi said a good calculator doesnt use counting rods, the Book of Han recorded, they calculate with bamboo, diameter one fen, length six cun, arranged into a hexagonal bundle of two hundred seventy one pieces. At first calculating rods were round in section, but by the time of the Sui dynasty triangular rods were used to represent positive numbers. After the abacus flourished, counting rods were abandoned except in Japan, counting rods represent digits by the number of rods, and the perpendicular rod represents five. To avoid confusion, vertical and horizontal forms are alternately used, generally, vertical rod numbers are used for the position for the units, hundreds, ten thousands, etc. while horizontal rod numbers are used for the tens, thousands, hundred thousands etc. It is written in Sunzi Suanjing that one is vertical, ten is horizontal, red rods represent positive numbers and black rods represent negative numbers. Ancient Chinese clearly understood negative numbers and zero, though they had no symbol for the latter, later, a go stone was sometimes used to represent zero. This alternation of vertical and horizontal rod numeral form is important to understanding written transcription of rod numerals on manuscripts correctly. In the same manuscript,405 was transcribed as, with a space in between for obvious reasons, and could in no way be interpreted as 45. In other words, transcribed rod numerals may not be positional, the value of a number depends on its physical position on the counting board. A9 at the rightmost position on the stands for 9. Moving the batch of rods representing 9 to the one position gives 9 or 90
Counting rods
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Numeral systems
Counting rods
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Yang Hui (Pascal's)
triangle, as depicted by
Zhu Shijie in 1303, using rod numerals.
Counting rods
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rod numeral place value from Yongle Encyclopedia: 71,824
Counting rods
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Japanese counting board with grids
13.
Abjad numerals
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The Abjad numerals are a decimal numeral system in which the 28 letters of the Arabic alphabet are assigned numerical values. They have been used in the Arabic-speaking world since before the century when Arabic numerals were adopted. In modern Arabic, the word abjadīyah means alphabet in general, in the Abjad system, the first letter of the Arabic alphabet, alif, is used to represent 1, the second letter, bāʾ, is used to represent 2, etc. Individual letters also represent 10s and 100s, yāʾ for 10, kāf for 20, qāf for 100, the word abjad itself derives from the first four letters in the Phoenician alphabet, Aramaic alphabet, Hebrew alphabet and other scripts for Semitic languages. These older alphabets contained only 22 letters, stopping at taw, the Arabic Abjad system continues at this point with letters not found in other alphabets, ṯāʾ=500, etc. The Abjad order of the Arabic alphabet has two different variants. Loss of samekh was compensated for by the split of shin ש into two independent Arabic letters, ش and ﺱ, which moved up to take the place of samekh. The most common Abjad sequence, read right to left, is, This is commonly vocalized as follows. Before the introduction of the Hindu–Arabic numeral system, the numbers were used for all mathematical purposes. In modern Arabic, they are used for numbering outlines, items in lists. In English, points of information are sometimes referred to as A, B, and C, the abjad numbers are also used to assign numerical values to Arabic words for purposes of numerology. The common Islamic phrase بسم الله الرحمن الرحيم bismillāh al-Raḥmān al-Raḥīm has a value of 786. The name Allāh الله by itself has the value 66, a few of the numerical values are different in the alternative Abjad order. For four Persian letters these values are used, The Abjad numerals are equivalent to the earlier Hebrew numerals up to 400, the Hebrew numeral system is known as Gematria and is used in Kabbalistic texts and numerology. Like the Abjad order, it is used in times for numbering outlines and points of information. The Greek numerals differ in a number of ways from the Abjad ones, the Greek language system of letters-as-numbers is called isopsephy
Abjad numerals
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Numeral systems
14.
Greek numerals
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Greek numerals are a system of writing numbers using the letters of the Greek alphabet. These alphabetic numerals are known as Ionic or Ionian numerals, Milesian numerals. In modern Greece, they are used for ordinal numbers. For ordinary cardinal numbers, however, Greece uses Arabic numerals, attic numerals, which were later adopted as the basis for Roman numerals, were the first alphabetic set. They were acrophonic, derived from the first letters of the names of the numbers represented and they ran =1, =5, =10, =100, =1000, and =10000. 50,500,5000, and 50000 were represented by the letter with minuscule powers of ten written in the top right corner, the same system was used outside of Attica, but the symbols varied with the local alphabets, in Boeotia, was 1000. The present system probably developed around Miletus in Ionia, 19th-century classicists placed its development in the 3rd century BC, the occasion of its first widespread use. The present system uses the 24 letters adopted by Euclid as well as three Phoenician and Ionic ones that were not carried over, digamma, koppa, and sampi. The position of characters within the numbering system imply that the first two were still in use while the third was not. Greek numerals are decimal, based on powers of 10, the units from 1 to 9 are assigned to the first nine letters of the old Ionic alphabet from alpha to theta. Each multiple of one hundred from 100 to 900 was then assigned its own separate letter as well and this alphabetic system operates on the additive principle in which the numeric values of the letters are added together to obtain the total. For example,241 was represented as, in ancient and medieval manuscripts, these numerals were eventually distinguished from letters using overbars, α, β, γ, etc. In medieval manuscripts of the Book of Revelation, the number of the Beast 666 is written as χξϛ, although the Greek alphabet began with only majuscule forms, surviving papyrus manuscripts from Egypt show that uncial and cursive minuscule forms began early. These new letter forms sometimes replaced the ones, especially in the case of the obscure numerals. The old Q-shaped koppa began to be broken up and simplified, the numeral for 6 changed several times. During antiquity, the letter form of digamma came to be avoided in favor of a special numerical one. By the Byzantine era, the letter was known as episemon and this eventually merged with the sigma-tau ligature stigma. In modern Greek, a number of changes have been made
Greek numerals
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Numeral systems
Greek numerals
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A
Constantinopolitan map of the British Isles from
Ptolemy 's
Geography (c. 1300), using Greek numerals for its
graticule: 52–63°N of the
equator and 6–33°E from Ptolemy's
Prime Meridian at the
Fortunate Isles.
15.
Hebrew numerals
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The system of Hebrew numerals is a quasi-decimal alphabetic numeral system using the letters of the Hebrew alphabet. The system was adapted from that of the Greek numerals in the late 2nd century BC, the current numeral system is also known as the Hebrew alphabetic numerals to contrast with earlier systems of writing numerals used in classical antiquity. The Greek system was adopted in Hellenistic Judaism and had been in use in Greece since about the 5th century BC, in this system, there is no notation for zero, and the numeric values for individual letters are added together. Each unit is assigned a letter, each tens a separate letter. The later hundreds are represented by the sum of two or three letters representing the first four hundreds, to represent numbers from 1,000 to 999,999, the same letters are reused to serve as thousands, tens of thousands, and hundreds of thousands. In Israel today, the system of Arabic numerals is used in almost all cases. The Hebrew numerals are used only in cases, such as when using the Hebrew calendar, or numbering a list. Numbers in Hebrew from zero to one million, Hebrew alphabet are used to a limited extent to represent numbers, widely used on calendars. In other situations Arabic numerals are used, cardinal and ordinal numbers must agree in gender with the noun they are describing. If there is no such noun, the form is used. For ordinal numbers greater than ten the cardinal is used and numbers above the value 20 have no gender, note, For ordinal numbers greater than 10, cardinal numbers are used instead. Note, For numbers greater than 20, gender does not apply, cardinal and ordinal numbers must agree in gender with the noun they are describing. If there is no such noun, the form is used. Ordinal numbers must also agree in number and definite status like other adjectives, the cardinal number precedes the noun, except for the number one which succeeds it. The number two is special - shnayim and shtayim become shney and shtey when followed by the noun they count, for ordinal numbers greater than ten the cardinal is used. The Hebrew numeric system operates on the principle in which the numeric values of the letters are added together to form the total. For example,177 is represented as קעז which corresponds to 100 +70 +7 =177, mathematically, this type of system requires 27 letters. In practice the last letter, tav is used in combination with itself and/or other letters from kof onwards, to numbers from 500
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).
16.
Roman numerals
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The numeric system represented by Roman numerals originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet, Roman numerals, as used today, are based on seven symbols, The use of Roman numerals continued long after the decline of the Roman Empire. The numbers 1 to 10 are usually expressed in Roman numerals as follows, I, II, III, IV, V, VI, VII, VIII, IX, Numbers are formed by combining symbols and adding the values, so II is two and XIII is thirteen. Symbols are placed left to right in order of value. Named after the year of its release,2014 as MMXIV, the year of the games of the XXII Olympic Winter Games The standard forms described above reflect typical modern usage rather than a universally accepted convention. Usage in ancient Rome varied greatly and remained inconsistent in medieval, Roman inscriptions, especially in official contexts, seem to show a preference for additive forms such as IIII and VIIII instead of subtractive forms such as IV and IX. Both methods appear in documents from the Roman era, even within the same document, double subtractives also occur, such as XIIX or even IIXX instead of XVIII. Sometimes V and L are not used, with such as IIIIII. Such variation and inconsistency continued through the period and into modern times. Clock faces that use Roman numerals normally show IIII for four o’clock but IX for nine o’clock, however, this is far from universal, for example, the clock on the Palace of Westminster in London uses IV. Similarly, at the beginning of the 20th century, different representations of 900 appeared in several inscribed dates. For instance,1910 is shown on Admiralty Arch, London, as MDCCCCX rather than MCMX, although Roman numerals came to be written with letters of the Roman alphabet, they were originally independent symbols. The Etruscans, for example, used
Roman numerals
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Entrance to section LII (52) of the
Colosseum, with numerals still visible
Roman numerals
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Numeral systems
Roman numerals
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A typical
clock face with Roman numerals in
Bad Salzdetfurth, Germany
Roman numerals
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An inscription on
Admiralty Arch, London. The number is 1910, for which MCMX would be more usual
17.
Babylonian numerals
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Babylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record. The Babylonians, who were famous for their observations and calculations. Neither of the predecessors was a positional system and this system first appeared around 2000 BC, its structure reflects the decimal lexical numerals of Semitic languages rather than Sumerian lexical numbers. However, the use of a special Sumerian sign for 60 attests to a relation with the Sumerian system. The Babylonian system is credited as being the first known positional numeral system and this was an extremely important development, because non-place-value systems require unique symbols to represent each power of a base, which can make calculations more difficult. Only two symbols were used to notate the 59 non-zero digits and these symbols and their values were combined to form a digit in a sign-value notation quite similar to that of Roman numerals, for example, the combination represented the digit for 23. A space was left to indicate a place value, similar to the modern-day zero. Babylonians later devised a sign to represent this empty place and they lacked a symbol to serve the function of radix point, so the place of the units had to be inferred from context, could have represented 23 or 23×60 or 23×60×60 or 23/60, etc. A common theory is that 60, a highly composite number, was chosen due to its prime factorization, 2×2×3×5, which makes it divisible by 1,2,3,4,5,6,10,12,15,20. Integers and fractions were represented identically — a radix point was not written, the Babylonians did not technically have a digit for, nor a concept of, the number zero. Although they understood the idea of nothingness, it was not seen as a number—merely the lack of a number, what the Babylonians had instead was a space to mark the nonexistence of a digit in a certain place value. Babylon Babylonia History of zero Numeral system Menninger, Karl W. Number Words and Number Symbols, Number, From Ancient Civilisations to the Computer. CESCNC - a handy and easy-to use numeral converter
Babylonian numerals
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Numeral systems
Babylonian numerals
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Babylonian numerals
18.
Egyptian numerals
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The system of ancient Egyptian numerals was used in Ancient Egypt around 3000 BC until the early first millennium AD. It was a system of numeration based on the scale of ten, often rounded off to the power, written in hieroglyphs. The hieratic form of numerals stressed an exact finite series notation, the Ancient Egyptian system used bases of ten. The following hieroglyphics were used to denote powers of ten, Multiples of these values were expressed by repeating the symbol as many times as needed, for instance, a stone carving from Karnak shows the number 4622 as Egyptian hieroglyphs could be written in both directions. The symbol nfr, meaning beautiful, was used to indicate the base level in drawings of tombs and pyramids. Rational numbers could also be expressed, but only as sums of fractions, i. e. sums of reciprocals of positive integers, except for 2⁄3. The hieroglyph indicating a fraction looked like a mouth, which meant part, Fractions were written with this fractional solidus, i. e. the numerator 1, and the positive denominator below. As with most modern day languages, the ancient Egyptian language could also write out numerals as words phonetically, just like one can write thirty instead of 30 in English. The word, for instance, was written as while the numeral was This was, however, uncommon for most numbers other than one, instances of numerals written in hieratic can be found as far back as the Early Dynastic Period. The Old Kingdom Abusir Papyri are an important corpus of texts that utilize hieratic numerals. A large number like 9999 could thus be written only four signs—combining the signs for 9000,900,90. Boyer saw the new hieratic numerals as ciphered, mapping one number onto one Egyptian letter for the first time in human history, greeks adopted the new system, mapping their counting numbers onto two of their alphabets, the Doric and Ionian. In the oldest hieratic texts the individual numerals were written in a ciphered relationship to the Egyptian alphabet. But during the Old Kingdom a series of standardized writings had developed for sign-groups containing more than one numeral, however, repetition of the same numeral for each place-value was not allowed in the hieratic script. As the hieratic writing system developed over time, these sign-groups were further simplified for quick writing, two famous mathematical papyri using hieratic script are the Moscow Mathematical Papyrus and the Rhind Mathematical Papyrus. The majuscule letter A in some reconstructed forms means that the quality of that remains uncertain, Ancient Egypt Egyptian language Egyptian mathematics Allen. Middle Egyptian, An Introduction to the Language and Culture of Hieroglyphs, Egyptian Grammar, Being an Introduction to the Study of Hieroglyphs. Hieratische Paläographie, Die aegyptische Buchschrift in ihrer Entwicklung von der Fünften Dynastie bis zur römischen Kaiserzeit, Introduction Egyptian numerals Numbers and dates http, //egyptianmath. blogspot. com
Egyptian numerals
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Numeral systems
19.
Maya numerals
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The Maya numeral system is a vigesimal positional notation used in the Maya civilization to represent numbers. The numerals are made up of three symbols, zero, one and five, for example, thirteen is written as three dots in a horizontal row above two horizontal lines stacked above each other. Numbers after 19 were written vertically in powers of twenty, for example, thirty-three would be written as one dot above three dots, which are in turn atop two lines. The first dot represents one twenty or 1×20, which is added to three dots and two bars, or thirteen, upon reaching 202 or 400, another row is started. The number 429 would be written as one dot above one dot above four dots, the powers of twenty are numerals, just as the Hindu-Arabic numeral system uses powers of tens. Other than the bar and dot notation, Maya numerals can be illustrated by face type glyphs or pictures, the face glyph for a number represents the deity associated with the number. These face number glyphs were used, and are mostly seen on some of the most elaborate monumental carving. Addition and subtraction, Adding and subtracting numbers below 20 using Maya numerals is very simple, addition is performed by combining the numeric symbols at each level, If five or more dots result from the combination, five dots are removed and replaced by a bar. If four or more bars result, four bars are removed, similarly with subtraction, remove the elements of the subtrahend symbol from the minuend symbol, If there are not enough dots in a minuend position, a bar is replaced by five dots. If there are not enough bars, a dot is removed from the next higher minuend symbol in the column, the Maya/Mesoamerican Long Count calendar required the use of zero as a place-holder within its vigesimal positional numeral system. A shell glyph – – was used as a symbol for these Long Count dates. However, since the eight earliest Long Count dates appear outside the Maya homeland, it is assumed that the use of zero predated the Maya, indeed, many of the earliest Long Count dates were found within the Olmec heartland. However, the Olmec civilization had come to an end by the 4th century BC, in the Long Count portion of the Maya calendar, a variation on the strictly vigesimal numbering is used. The Long Count changes in the place value, it is not 20×20 =400, as would otherwise be expected. This is supposed to be because 360 is roughly the number of days in a year, subsequent place values return to base-twenty. In fact, every known example of large numbers uses this modified vigesimal system and it is reasonable to assume, but not proven by any evidence, that the normal system in use was a pure base-20 system. Maya Mathematics - online converter from decimal numeration to Maya numeral notation, anthropomorphic Maya numbers - online story of number representations
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.
20.
Binary number
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The base-2 system is a positional notation with a radix of 2. Because of its implementation in digital electronic circuitry using logic gates. Each digit is referred to as a bit, the modern binary number system was devised by Gottfried Leibniz in 1679 and appears in his article Explication de lArithmétique Binaire. Systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, Leibniz was specifically inspired by the Chinese I Ching. The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions, the method used for ancient Egyptian multiplication is also closely related to binary numbers. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus, the I Ching dates from the 9th century BC in China. The binary notation in the I Ching is used to interpret its quaternary divination technique and it is based on taoistic duality of yin and yang. Eight trigrams and a set of 64 hexagrams, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China. The Song Dynasty scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, the Indian scholar Pingala developed a binary system for describing prosody. He used binary numbers in the form of short and long syllables, Pingalas Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter. The binary representations in Pingalas system increases towards the right, the residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Africa, sets of binary combinations similar to the I Ching have also been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy. The base-2 system utilized in geomancy had long been applied in sub-Saharan Africa. Leibnizs system uses 0 and 1, like the modern binary numeral system, Leibniz was first introduced to the I Ching through his contact with the French Jesuit Joachim Bouvet, who visited China in 1685 as a missionary. Leibniz saw the I Ching hexagrams as an affirmation of the universality of his own beliefs as a Christian. Binary numerals were central to Leibnizs theology and he believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing. Is not easy to impart to the pagans, is the ex nihilo through Gods almighty power. In 1854, British mathematician George Boole published a paper detailing an algebraic system of logic that would become known as Boolean algebra
Binary number
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Numeral systems
Binary number
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Arithmetic values represented by parts of the Eye of Horus
Binary number
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Gottfried Leibniz
Binary number
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George Boole
21.
Ternary numeral system
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The ternary numeral system has three as its base. Analogous to a bit, a digit is a trit. One trit is equivalent to bits of information. Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary, for example, decimal 365 corresponds to binary 101101101 and to ternary 111112. However, they are far less compact than the corresponding representations in bases such as decimal – see below for a compact way to codify ternary using nonary. The value of a number with n bits that are all 1 is 2n −1. Then N = M, N = /, and N = bd −1, for a three-digit ternary number, N =33 −1 =26 =2 ×32 +2 ×31 +2 ×30 =18 +6 +2. Nonary or septemvigesimal can be used for representation of ternary. A base-three system is used in Islam to keep track of counting Tasbih to 99 or to 100 on a hand for counting prayers. In certain analog logic, the state of the circuit is often expressed ternary and this is most commonly seen in Transistor–transistor logic using 7406 open collector logic. The output is said to either be low, high, or open, in this configuration the output of the circuit is actually not connected to any voltage reference at all. Where the signal is usually grounded to a reference, or at a certain voltage level. Thus, the voltage level is sometimes unpredictable. A rare ternary point is used to denote fractional parts of an inning in baseball, since each inning consists of three outs, each out is considered one third of an inning and is denoted as.1. For example, if a player pitched all of the 4th, 5th and 6th innings, plus 2 outs of the 7th inning, his Innings pitched column for that game would be listed as 3.2, meaning 3⅔. In this usage, only the part of the number is written in ternary form. Ternary numbers can be used to convey self-similar structures like the Sierpinski triangle or the Cantor set conveniently, additionally, it turns out that the ternary representation is useful for defining the Cantor set and related point sets, because of the way the Cantor set is constructed. The Cantor set consists of the points from 0 to 1 that have an expression that does not contain any instance of the digit 1
Ternary numeral system
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Numeral systems
22.
Duodecimal
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The duodecimal system is a positional notation numeral system using twelve as its base. In this system, the number ten may be written by a rotated 2 and this notation was introduced by Sir Isaac Pitman. These digit forms are available as Unicode characters on computerized systems since June 2015 as ↊ and ↋, other notations use A, T, or X for ten and B or E for eleven. The number twelve is written as 10 in duodecimal, whereas the digit string 12 means 1 dozen and 2 units. Similarly, in duodecimal 100 means 1 gross,1000 means 1 great gross, the number twelve, a superior highly composite number, is the smallest number with four non-trivial factors, and the smallest to include as factors all four numbers within the subitizing range. As a result, duodecimal has been described as the number system. Of its factors,2 and 3 are prime, which means the reciprocals of all 3-smooth numbers have a representation in duodecimal. In particular, the five most elementary fractions all have a terminating representation in duodecimal. This all makes it a convenient number system for computing fractions than most other number systems in common use, such as the decimal, vigesimal, binary. Although the trigesimal and sexagesimal systems do even better in respect, this is at the cost of unwieldy multiplication tables. In this section, numerals are based on decimal places, for example,10 means ten,12 means twelve. Languages using duodecimal number systems are uncommon, germanic languages have special words for 11 and 12, such as eleven and twelve in English. However, they are considered to come from Proto-Germanic *ainlif and *twalif, historically, units of time in many civilizations are duodecimal. There are twelve signs of the zodiac, twelve months in a year, traditional Chinese calendars, clocks, and compasses are based on the twelve Earthly Branches. There are 12 inches in a foot,12 troy ounces in a troy pound,12 old British pence in a shilling,24 hours in a day. The Romans used a system based on 12, including the uncia which became both the English words ounce and inch. The importance of 12 has been attributed to the number of cycles in a year. It is possible to count to 12 with the acting as a pointer
Duodecimal
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Numeral systems
Duodecimal
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A duodecimal multiplication table
23.
Hexadecimal
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In mathematics and computing, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, Hexadecimal numerals are widely used by computer system designers and programmers. As each hexadecimal digit represents four binary digits, it allows a more human-friendly representation of binary-coded values, one hexadecimal digit represents a nibble, which is half of an octet or byte. For example, a byte can have values ranging from 00000000 to 11111111 in binary form. In a non-programming context, a subscript is typically used to give the radix, several notations are used to support hexadecimal representation of constants in programming languages, usually involving a prefix or suffix. The prefix 0x is used in C and related languages, where this value might be denoted as 0x2AF3, in contexts where the base is not clear, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously, a numerical subscript can give the base explicitly,15910 is decimal 159,15916 is hexadecimal 159, which is equal to 34510. Some authors prefer a text subscript, such as 159decimal and 159hex, or 159d and 159h. example. com/name%20with%20spaces where %20 is the space character, thus ’, represents the right single quotation mark, Unicode code point number 2019 in hex,8217. In the Unicode standard, a value is represented with U+ followed by the hex value. Color references in HTML, CSS and X Window can be expressed with six hexadecimal digits prefixed with #, white, CSS allows 3-hexdigit abbreviations with one hexdigit per component, #FA3 abbreviates #FFAA33. *nix shells, AT&T assembly language and likewise the C programming language, to output an integer as hexadecimal with the printf function family, the format conversion code %X or %x is used. In Intel-derived assembly languages and Modula-2, hexadecimal is denoted with a suffixed H or h, some assembly languages use the notation HABCD. Ada and VHDL enclose hexadecimal numerals in based numeric quotes, 16#5A3#, for bit vector constants VHDL uses the notation x5A3. Verilog represents hexadecimal constants in the form 8hFF, where 8 is the number of bits in the value, the Smalltalk language uses the prefix 16r, 16r5A3 PostScript and the Bourne shell and its derivatives denote hex with prefix 16#, 16#5A3. For PostScript, binary data can be expressed as unprefixed consecutive hexadecimal pairs, in early systems when a Macintosh crashed, one or two lines of hexadecimal code would be displayed under the Sad Mac to tell the user what went wrong. Common Lisp uses the prefixes #x and #16r, setting the variables *read-base* and *print-base* to 16 can also used to switch the reader and printer of a Common Lisp system to Hexadecimal number representation for reading and printing numbers. Thus Hexadecimal numbers can be represented without the #x or #16r prefix code, MSX BASIC, QuickBASIC, FreeBASIC and Visual Basic prefix hexadecimal numbers with &H, &H5A3 BBC BASIC and Locomotive BASIC use & for hex. TI-89 and 92 series uses a 0h prefix, 0h5A3 ALGOL68 uses the prefix 16r to denote hexadecimal numbers, binary, quaternary and octal numbers can be specified similarly
Hexadecimal
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Numeral systems
Hexadecimal
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Bruce Alan Martin's hexadecimal notation proposal
Hexadecimal
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Hexadecimal finger-counting scheme.
24.
Non-standard positional numeral systems
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Each numeral represents one of the values 0,1,2, etc. up to b −1, but the value also depends on the position of the digit in a number. The value of a string like pqrs in base b is given by the polynomial form p × b 3 + q × b 2 + r × b + s. The numbers written in superscript represent the powers of the base used, and a minus sign −, all real numbers can be represented. This article summarizes facts on some non-standard positional numeral systems, in most cases, the polynomial form in the description of standard systems still applies. Some historical numeral systems may be described as non-standard positional numeral systems, however, most of the non-standard systems listed below have never been intended for general use, but are deviced by mathematicians or engineers for special academic or technical use. A bijective numeral system with base b uses b different numerals to represent all non-negative integers, however, the numerals have values 1,2,3, etc. up to and including b, whereas zero is represented by an empty digit string. For example, it is possible to have decimal without a zero, unary is the bijective numeral system with base b =1. In unary, one numeral is used to represent all positive integers, the value of the digit string pqrs given by the polynomial form can be simplified into p + q + r + s since bn =1 for all n. Non-standard features of this include, The value of a digit does not depend on its position. Thus, one can argue that unary is not a positional system at all. Introducing a radix point in this system will not enable representation of non-integer values, the single numeral represents the value 1, not the value 0 = b −1. The value 0 cannot be represented, in some systems, while the base is a positive integer, negative digits are allowed. Non-adjacent form is a system where the base is b =2. In the balanced system, the base is b =3. The reflected binary code, also known as the Gray code, is related to binary numbers. A few positional systems have been suggested in which the base b is not a positive integer, negative-base systems include negabinary, negaternary and negadecimal, in base −b the number of different numerals used is b. All integers, positive and negative, can be represented without a sign, in purely imaginary base bi the b2 numbers from 0 to b2 −1 are used as digits. It can be generalized to other bases, Complex-base system
Non-standard positional numeral systems
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Numeral systems
25.
Bijective numeration
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Bijective numeration is any numeral system in which every non-negative integer can be represented in exactly one way using a finite string of digits. The name derives from this bijection between the set of integers and the set of finite strings using a finite set of symbols. Most ordinary numeral systems, such as the decimal system, are not bijective because more than one string of digits can represent the same positive integer. In particular, adding leading zeroes does not change the value represented, even though only the first is usual, the fact that the others are possible means that decimal is not bijective. However, unary, with one digit, is bijective. A bijective base-k numeration is a positional notation. It uses a string of digits from the set to encode each positive integer, the base-k bijective numeration system uses the digit-set to uniquely represent every non-negative integer, as follows, The integer zero is represented by the empty string. The integer represented by the nonempty digit-string anan−1, a1a0 is an kn + an−1 kn−1 +. The digit-string representing the integer m >0 is anan−1, for a given base k ≥1, there are exactly kn bijective base-k numerals of length n ≥0. Thus, using 0 to denote the empty string, the base 1,2,3,8,10,12, 119A = 1×103 + 1×102 + 9×101 + 10×1 =1200. The bijective base-10 system is a base ten positional system that does not use a digit to represent zero. It instead has a digit to represent ten, such as A, as with conventional decimal, each digit position represents a power of ten, so for example 123 is one hundred, plus two tens, plus three units. All positive integers that are represented solely with non-zero digits in conventional decimal have the same representation in decimal without a zero. Addition and multiplication in decimal without a zero are essentially the same as with conventional decimal, except that carries occur when a position exceeds ten, rather than when it exceeds nine. So to calculate 643 +759, there are twelve units, ten tens, thirteen hundreds, in the bijective base-26 system one may use the Latin alphabet letters A to Z to represent the 26 digit values one to twenty-six. With this choice of notation, the sequence begins A, B, C. Each digit position represents a power of twenty-six, so for example, many spreadsheets including Microsoft Excel use this system to assign labels to the columns of a spreadsheet, starting A, B, C. For instance, in Excel 2013, there can be up to 16384 columns, a variant of this system is used to name variable stars
Bijective numeration
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Numeral systems
26.
Unary numeral system
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The unary numeral system is the bijective base-1 numeral system. It is the simplest numeral system to represent natural numbers, in order to represent a number N, for examples, the numbers 1,2,3,4,5. Would be represented in this system as 1,11,111,1111,11111 and these numbers should be distinguished from repunits, which are also written as sequences of ones but have their usual decimal numerical interpretation. This system is used in tallying, for example, using the tally mark |, the number 3 is represented as |||. In East Asian cultures, the three is represented as “三”, a character that is drawn with three strokes. Addition and subtraction are particularly simple in the system, as they involve little more than string concatenation. The Hamming weight or population count operation that counts the number of bits in a sequence of binary values may also be interpreted as a conversion from unary to binary numbers. However, multiplication is more cumbersome and has often used as a test case for the design of Turing machines. Compared to standard positional numeral systems, the system is inconvenient. It occurs in some decision problem descriptions in theoretical computer science, therefore, while the run-time and space requirement in unary looks better as function of the input size, it does not represent a more efficient solution. In computational complexity theory, unary numbering is used to distinguish strongly NP-complete problems from problems that are NP-complete, for such a problem, there exist hard instances for which all parameter values are at most polynomially large. Unary is used as part of data compression algorithms such as Golomb coding. It also forms the basis for the Peano axioms for formalizing arithmetic within mathematical logic, a form of unary notation called Church encoding is used to represent numbers within lambda calculus. Sloanes A000042, Unary representation of natural numbers, the On-Line Encyclopedia of Integer Sequences
Unary numeral system
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Numeral systems
27.
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, by converting a number less than n. General mixed radix systems were studied by Georg Cantor, the term factorial number system is used by Knuth, while the French equivalent numération factorielle was first used in 1888. The term factoradic, which is a portmanteau of factorial and mixed radix, appears to be of more recent date. The factorial number system is a mixed radix numeral system, the i-th digit from the right has base i, which means that the digit must be less than i. From this it follows that the rightmost digit is always 0, the second can be 0 or 1, the third 0,1 or 2, the factorial number system is sometimes defined with the 0. Place omitted because it is always zero, in this article, a factorial number representation will be flagged by a subscript. Stands for 354413021100, whose value is = 3×5, general properties of mixed radix number systems also apply to the factorial number system. Reading the remainders backward gives 341010, in principle, this system may be extended to represent fractional numbers, though rather than the natural extension of place values. Etc. which are undefined, the choice of radix values n =0,1,2,3,4. Again, the 0 and 1 places may be omitted as these are always zero, the corresponding place values are therefore 1/1, 1/1, 1/2, 1/6, 1/24. The following sortable table shows the 24 permutations of four elements with different inversion related vectors, the left and right inversion counts l and r are particularly eligible to be interpreted as factorial numbers. L gives the position in reverse colexicographic order, and the latter the position in lexicographic order. Sorting by a column that has the omissible 0 on the right makes the numbers in that column correspond to the index numbers in the immovable column on the left. The small columns are reflections of the next to them. The rightmost column shows the digit sums of the factorial numbers, for another example, the greatest number that could be represented with six digits would be 543210. Which equals 719 in decimal, 5×5, clearly the next factorial number representation after 543210. is 1000000. =72010, the value for the radix-7 digit
Factorial number system
–
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.
28.
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 sign often results in a negative-base number being one digit longer than its positive-base equivalent. Negative numerical bases were first considered by Vittorio Grünwald in his work Giornale di Matematiche di Battaglini, Grünwald gave algorithms for performing addition, subtraction, multiplication, division, root extraction, divisibility tests, and radix conversion. Negative bases were later rediscovered by A. J. Kempner in 1936 and Zdzisław Pawlak. Negabinary was implemented in the early Polish computer BINEG, built 1957–59, based on ideas by Z. Pawlak, implementations since then have been rare. The base −r expansion of a is given by the string dndn-1…d1d0. Negative-base systems may thus be compared to signed-digit representations, such as balanced ternary, some numbers have the same representation in base −r as in base r. For example, the numbers from 100 to 109 have the same representations in decimal and negadecimal, similarly,17 =24 +20 =4 +0 and is represented by 10001 in binary and 10001 in negabinary. The base −r expansion of a number can be found by repeated division by −r, recording the non-negative remainders of 0,1, …, r −1, and concatenating those remainders, starting with the last. Note that if a / b = c, remainder d, then bc + d = a, to arrive at the correct conversion, the value for c must be chosen such that d is non-negative and minimal. This is exemplified in the line of the following example wherein –5 ÷ –3 must be chosen to equal 2 remainder 1 instead of 1 remainder –2. Note that in most programming languages, the result of dividing a number by a negative number is rounded towards 0. In such a case we have a = c + d = c + d − r + r = +, because |d| < r, is the positive remainder. The conversion from integer to some negative base, Visual Basic implementation, The conversion to negabinary allows a remarkable shortcut, the bitwise XOR portion is originally due to Schroeppel. Adding negabinary numbers proceeds bitwise, starting from the least significant bits, while adding two negabinary numbers, every time a carry is generated an extra carry should be propagated to next bit. Unary negation, −x, can be computed as binary subtraction from zero,0 − x, shifting to the left multiplies by −2, shifting to the right divides by −2. To multiply, multiply like normal decimal or binary numbers, but using the rules for adding the carry. It is possible to compare negabinary numbers by slightly adjusting a normal unsigned binary comparator, when comparing the numbers A and B, invert each odd positioned bit of both numbers
Negative base
–
Numeral systems
29.
Quater-imaginary base
–
The quater-imaginary numeral system was first proposed by Donald Knuth in 1960. It is a positional numeral system which uses the imaginary number 2i as its base. It is able to represent every complex number using only the digits 0,1,2. The real and imaginary parts of complex number are thus readily expressed in base −4 as … d 4 d 2 d 0. D −2 … and 2 ⋅ respectively, to convert a digit string from the quater-imaginary system to the decimal system, the standard formula for positional number systems can be used. Additionally, for a given string d in the form d w −1, d w −2, every complex number has a quater-imaginary representation. Most numbers have a unique quater-imaginary representation, but just as 1 has the two representations 1 =0.999. in decimal notation, so 1/5 has the two quater-imaginary representations 1. …2i =0. …2i. For example, the representation of 6i is calculated by multiplying 6i • 2i = –12, which is expressed as 3002i. Finding the quater-imaginary representation of an arbitrary real number can be done manually by solving a system of simultaneous equations. But there are methods for both, real and imaginary, integers, as shown in section Negative base#To Negaquaternary. As an example of a number we can try to find the quater-imaginary counterpart of the decimal number 7. Since it is hard to exactly how long the digit string will be for a given decimal number. In this case, a string of six digits can be chosen, when an initial guess at the size of the string eventually turns out to be insufficient, a larger string can be used. Now the value of the coefficients d0, d2 and d4, because d0 −4 d2 +16 d4 =7 and because—by the nature of the quater-imaginary system—the coefficients can only be 0,1,2 or 3 the value of the coefficients can be found. A possible configuration could be, d0 =3, d2 =3 and this configuration gives the resulting digit string for 710. 710 =0103032 i =103032 i, finding a quater-imaginary representation of a purely imaginary integer number ∈ iZ is analogous to the method described above for a real number. For example, to find the representation of 6i, it is possible to use the general formula, then all coefficients of the real part have to be zero and the complex part should make 6. However, for 6i it is seen by looking at the formula that if d1 =3 and all other coefficients are zero
Quater-imaginary base
–
Numeral systems
30.
List of numeral systems
–
This is a list of numeral systems, that is, writing systems for expressing numbers. Numeral systems are classified here as to whether they use positional notation, the common names are derived somewhat arbitrarily from a mix of Latin and Greek, in some cases including roots from both languages within a single name. In this Youtube video, Matt Parker jokingly invented a base-1082 system and this turns out to be 1925. Radix Radix economy Table of bases List of numbers in various languages Numeral prefix
List of numeral systems
–
Numeral systems
List of numeral systems
31.
Positional numeral system
–
Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the symbol for the different orders of magnitude. This greatly simplified arithmetic, leading to the spread of the notation across the world. With the use of a point, the notation can be extended to include fractions. The Hindu–Arabic numeral system, base-10, is the most commonly used system in the world today for most calculations, today, the base-10 system, which is likely motivated by counting with the ten fingers, is ubiquitous. Other bases have been used in the past however, and some continue to be used today, for example, the Babylonian numeral system, credited as the first positional numeral system, was base-60, but it lacked a real 0 value. Zero was indicated by a space between sexagesimal numerals, by 300 BC, a punctuation symbol was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish, the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges, the Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number, thus numbers like 2 and 120,3 and 180,4 and 240, looked the same because the larger numbers lacked a final sexagesimal placeholder. Counting rods and most abacuses have been used to represent numbers in a numeral system. This approach required no memorization of tables and could produce practical results quickly, for four centuries there was strong disagreement between those who believed in adopting the positional system in writing numbers and those who wanted to stay with the additive-system-plus-abacus. Although electronic calculators have largely replaced the abacus, the continues to be used in Japan. After the French Revolution, the new French government promoted the extension of the decimal system, some of those pro-decimal efforts—such as decimal time and the decimal calendar—were unsuccessful. Other French pro-decimal efforts—currency decimalisation and the metrication of weights and measures—spread widely out of France to almost the whole world. According to Joseph Needham and Lam Lay Yong, decimal fractions were first developed and used by the Chinese in the 1st century BC, the written Chinese decimal fractions were non-positional. However, counting rod fractions were positional, the Jewish mathematician Immanuel Bonfils used decimal fractions around 1350, anticipating Simon Stevin, but did not develop any notation to represent them. A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century. A key argument against the system was its susceptibility to easy fraud by simply putting a number at the beginning or end of a quantity, thereby changing 100 into 5100
Positional numeral system
–
Numeral systems
32.
Integer
–
An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
Integer
–
Algebraic structure → Group theory
Group theory
33.
Number
–
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1,2,3, a notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are used for labels, for ordering. In common usage, number may refer to a symbol, a word, calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation. Their study or usage is called arithmetic, the same term may also refer to number theory, the study of the properties of numbers. Besides their practical uses, numbers have cultural significance throughout the world, for example, in Western society the number 13 is regarded as unlucky, and a million may signify a lot. Though it is now regarded as pseudoscience, numerology, the belief in a significance of numbers, permeated ancient. Numerology heavily influenced the development of Greek mathematics, stimulating the investigation of problems in number theory which are still of interest today. During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers, among the first were the hypercomplex numbers, which consist of various extensions or modifications of the complex number system. Numbers should be distinguished from numerals, the used to represent numbers. Boyer showed that Egyptians created the first ciphered numeral system, Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets. The number five can be represented by digit 5 or by the Roman numeral Ⅴ, notations used to represent numbers are discussed in the article numeral systems. The Roman numerals require extra symbols for larger numbers, different types of numbers have many different uses. Numbers can be classified into sets, called number systems, such as the natural numbers, the same number can be written in many different ways. For different methods of expressing numbers with symbols, such as the Roman numerals, each of these number systems may be considered as a proper subset of the next one. This is expressed, symbolically, by writing N ⊂ Z ⊂ Q ⊂ R ⊂ C, the most familiar numbers are the natural numbers,1,2,3, and so on. Traditionally, the sequence of numbers started with 1 However, in the 19th century, set theorists. Today, different mathematicians use the term to both sets, including 0 or not
Number
–
The number 605 in
Khmer numerals, from an inscription from 683 AD. An early use of zero as a decimal figure.
Number
–
Subsets of the
complex numbers.
34.
Nonnegative integer
–
In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers, the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1,10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory
Nonnegative integer
–
The
Ishango bone (on exhibition at the
Royal Belgian Institute of Natural Sciences) is believed to have been used 20,000 years ago for natural number arithmetic.
Nonnegative integer
–
Natural numbers can be used for counting (one
apple, two apples, three apples, …)
35.
Rational number
–
In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number. The set of all numbers, often referred to as the rationals, is usually denoted by a boldface Q, it was thus denoted in 1895 by Giuseppe Peano after quoziente. The decimal expansion of a rational number always either terminates after a number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number and these statements hold true not just for base 10, but also for any other integer base. A real number that is not rational is called irrational, irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating, since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. Rational numbers can be defined as equivalence classes of pairs of integers such that q ≠0, for the equivalence relation defined by ~ if. In abstract algebra, the numbers together with certain operations of addition and multiplication form the archetypical field of characteristic zero. As such, it is characterized as having no proper subfield or, alternatively, finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the numbers form a dense subset of the real numbers. The real numbers can be constructed from the numbers by completion, using Cauchy sequences, Dedekind cuts. The term rational in reference to the set Q refers to the fact that a number represents a ratio of two integers. In mathematics, rational is often used as a noun abbreviating rational number, the adjective rational sometimes means that the coefficients are rational numbers. However, a curve is not a curve defined over the rationals. Any integer n can be expressed as the rational number n/1, a b = c d if and only if a d = b c. Where both denominators are positive, a b < c d if and only if a d < b c. If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations, − a − b = a b, two fractions are added as follows, a b + c d = a d + b c b d
Rational number
–
A diagram showing a representation of the equivalent classes of pairs of integers
36.
Negative number
–
In mathematics, a negative number is a real number that is less than zero. If positive represents movement to the right, negative represents movement to the left, if positive represents above sea level, then negative represents below level. If positive represents a deposit, negative represents a withdrawal and they are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset, if a quantity may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage, negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common idea of an opposite is reflected in arithmetic. For example, − −3 =3 because the opposite of an opposite is the original thing, negative numbers are usually written with a minus sign in front. For example, −3 represents a quantity with a magnitude of three, and is pronounced minus three or negative three. To help tell the difference between a subtraction operation and a number, occasionally the negative sign is placed slightly higher than the minus sign. Conversely, a number that is greater than zero is called positive, the positivity of a number may be emphasized by placing a plus sign before it, e. g. +3. In general, the negativity or positivity of a number is referred to as its sign, every real number other than zero is either positive or negative. The positive whole numbers are referred to as natural numbers, while the positive and negative numbers are referred to as integers. In bookkeeping, amounts owed are often represented by red numbers, or a number in parentheses, Liu Hui established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers, islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Western mathematicians accepted the idea of numbers by the 17th century. Prior to the concept of numbers, mathematicians such as Diophantus considered negative solutions to problems false. Negative numbers can be thought of as resulting from the subtraction of a number from a smaller. For example, negative three is the result of subtracting three from zero,0 −3 = −3, in general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers
Negative number
–
This thermometer is indicating a negative
Fahrenheit temperature (−4°F).
37.
Minus sign
–
The plus and minus signs are mathematical symbols used to represent the notions of positive and negative as well as the operations of addition and subtraction. Their use has extended to many other meanings, more or less analogous. Plus and minus are Latin terms meaning more and less, respectively, though the signs now seem as familiar as the alphabet or the Hindu-Arabic numerals, they are not of great antiquity. In Europe in the early 15th century the letters P and M were generally used, the symbols appeared for the first time in Luca Pacioli’s mathematics compendium, Summa de arithmetica, geometria, proportioni et proportionalità, first printed and published in Venice in 1494. The + is a simplification of the Latin et, the − may be derived from a tilde written over m when used to indicate subtraction, or it may come from a shorthand version of the letter m itself. In his 1489 treatise Johannes Widmann referred to the symbols − and + as minus and mer, was − ist, das ist minus, und das + ist das mer. They werent used for addition and subtraction here, but to indicate surplus and deficit, the plus sign is a binary operator that indicates addition, as in 2 +3 =5. It can also serve as an operator that leaves its operand unchanged. This notation may be used when it is desired to emphasize the positiveness of a number, the plus sign can also indicate many other operations, depending on the mathematical system under consideration. Many algebraic structures have some operation which is called, or is equivalent to and it is conventional to use the plus sign to only denote commutative operations. Subtraction is the inverse of addition, directly in front of a number and when it is not a subtraction operator it means a negative number. For instance −5 is negative 5, a unary operator that acts as an instruction to replace the operand by its additive inverse. For example, if x is 3, then −x is −3, similarly, − is equal to 2. The above is a case of this. All three uses can be referred to as minus in everyday speech, further, some textbooks in the United States encourage −x to be read as the opposite of x or the additive inverse of x to avoid giving the impression that −x is necessarily negative. However, in programming languages and Microsoft Excel in particular, unary operators bind strongest, so in those cases −5^2 is 25. Some elementary teachers use raised plus and minus signs before numbers to show they are positive or negative numbers. For example, subtracting −5 from 3 might be read as positive three take away negative 5 and be shown as 3 − −5 becomes 3 +5 =8, in grading systems, the plus sign indicates a grade one level higher and the minus sign a grade lower
Minus sign
–
Plus, minus, and hyphen-minus.
38.
Europe
–
Europe is a continent that comprises the westernmost part of Eurasia. Europe is bordered by the Arctic Ocean to the north, the Atlantic Ocean to the west, yet the non-oceanic borders of Europe—a concept dating back to classical antiquity—are arbitrary. Europe covers about 10,180,000 square kilometres, or 2% of the Earths surface, politically, Europe is divided into about fifty sovereign states of which the Russian Federation is the largest and most populous, spanning 39% of the continent and comprising 15% of its population. Europe had a population of about 740 million as of 2015. Further from the sea, seasonal differences are more noticeable than close to the coast, Europe, in particular ancient Greece, was the birthplace of Western civilization. The fall of the Western Roman Empire, during the period, marked the end of ancient history. Renaissance humanism, exploration, art, and science led to the modern era, from the Age of Discovery onwards, Europe played a predominant role in global affairs. Between the 16th and 20th centuries, European powers controlled at times the Americas, most of Africa, Oceania. The Industrial Revolution, which began in Great Britain at the end of the 18th century, gave rise to economic, cultural, and social change in Western Europe. During the Cold War, Europe was divided along the Iron Curtain between NATO in the west and the Warsaw Pact in the east, until the revolutions of 1989 and fall of the Berlin Wall. In 1955, the Council of Europe was formed following a speech by Sir Winston Churchill and it includes all states except for Belarus, Kazakhstan and Vatican City. Further European integration by some states led to the formation of the European Union, the EU originated in Western Europe but has been expanding eastward since the fall of the Soviet Union in 1991. The European Anthem is Ode to Joy and states celebrate peace, in classical Greek mythology, Europa is the name of either a Phoenician princess or of a queen of Crete. The name contains the elements εὐρύς, wide, broad and ὤψ eye, broad has been an epithet of Earth herself in the reconstructed Proto-Indo-European religion and the poetry devoted to it. For the second part also the divine attributes of grey-eyed Athena or ox-eyed Hera. The same naming motive according to cartographic convention appears in Greek Ανατολή, Martin Litchfield West stated that phonologically, the match between Europas name and any form of the Semitic word is very poor. Next to these there is also a Proto-Indo-European root *h1regʷos, meaning darkness. Most major world languages use words derived from Eurṓpē or Europa to refer to the continent, in some Turkic languages the originally Persian name Frangistan is used casually in referring to much of Europe, besides official names such as Avrupa or Evropa
Europe
–
Reconstruction of
Herodotus ' world map
Europe
Europe
–
A medieval
T and O map from 1472 showing the three continents as domains of the sons of
Noah — Asia to Sem (
Shem), Europe to Iafeth (
Japheth), and Africa to Cham (
Ham)
Europe
–
Early modern depiction of
Europa regina ('Queen Europe') and the mythical
Europa of the 8th century BC.
39.
Science
–
Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe. The formal sciences are often excluded as they do not depend on empirical observations, disciplines which use science, like engineering and medicine, may also be considered to be applied sciences. However, during the Islamic Golden Age foundations for the method were laid by Ibn al-Haytham in his Book of Optics. In the 17th and 18th centuries, scientists increasingly sought to formulate knowledge in terms of physical laws, over the course of the 19th century, the word science became increasingly associated with the scientific method itself as a disciplined way to study the natural world. It was during this time that scientific disciplines such as biology, chemistry, Science in a broad sense existed before the modern era and in many historical civilizations. Modern science is distinct in its approach and successful in its results, Science in its original sense was a word for a type of knowledge rather than a specialized word for the pursuit of such knowledge. In particular, it was the type of knowledge which people can communicate to each other, for example, knowledge about the working of natural things was gathered long before recorded history and led to the development of complex abstract thought. This is shown by the construction of calendars, techniques for making poisonous plants edible. For this reason, it is claimed these men were the first philosophers in the strict sense and they were mainly speculators or theorists, particularly interested in astronomy. In contrast, trying to use knowledge of nature to imitate nature was seen by scientists as a more appropriate interest for lower class artisans. A clear-cut distinction between formal and empirical science was made by the pre-Socratic philosopher Parmenides, although his work Peri Physeos is a poem, it may be viewed as an epistemological essay on method in natural science. Parmenides ἐὸν may refer to a system or calculus which can describe nature more precisely than natural languages. Physis may be identical to ἐὸν and he criticized the older type of study of physics as too purely speculative and lacking in self-criticism. He was particularly concerned that some of the early physicists treated nature as if it could be assumed that it had no intelligent order, explaining things merely in terms of motion and matter. The study of things had been the realm of mythology and tradition, however. Aristotle later created a less controversial systematic programme of Socratic philosophy which was teleological and he rejected many of the conclusions of earlier scientists. For example, in his physics, the sun goes around the earth, each thing has a formal cause and final cause and a role in the rational cosmic order. Motion and change is described as the actualization of potentials already in things, while the Socratics insisted that philosophy should be used to consider the practical question of the best way to live for a human being, they did not argue for any other types of applied science
Science
–
Maize, known in some English-speaking countries as corn, is a large
grain plant domesticated by
indigenous peoples in
Mesoamerica in
prehistoric times.
Science
–
The scale of the universe mapped to the branches of science and the hierarchy of science.
Science
–
Aristotle, 384 BC – 322 BC, - one of the early figures in the development of the
scientific method.
Science
–
Galen (129—c.216) noted the optic chiasm is X-shaped. (Engraving from
Vesalius, 1543)
40.
Engineering
–
The term Engineering is derived from the Latin ingenium, meaning cleverness and ingeniare, meaning to contrive, devise. Engineering has existed since ancient times as humans devised fundamental inventions such as the wedge, lever, wheel, each of these inventions is essentially consistent with the modern definition of engineering. The term engineering is derived from the engineer, which itself dates back to 1390 when an engineer originally referred to a constructor of military engines. In this context, now obsolete, a referred to a military machine. Notable examples of the obsolete usage which have survived to the present day are military engineering corps, the word engine itself is of even older origin, ultimately deriving from the Latin ingenium, meaning innate quality, especially mental power, hence a clever invention. The earliest civil engineer known by name is Imhotep, as one of the officials of the Pharaoh, Djosèr, he probably designed and supervised the construction of the Pyramid of Djoser at Saqqara in Egypt around 2630–2611 BC. Ancient Greece developed machines in both civilian and military domains, the Antikythera mechanism, the first known mechanical computer, and the mechanical inventions of Archimedes are examples of early mechanical engineering. In the Middle Ages, the trebuchet was developed, the first steam engine was built in 1698 by Thomas Savery. The development of this gave rise to the Industrial Revolution in the coming decades. With the rise of engineering as a profession in the 18th century, similarly, in addition to military and civil engineering, the fields then known as the mechanic arts became incorporated into engineering. The inventions of Thomas Newcomen and the Scottish engineer James Watt gave rise to mechanical engineering. The development of specialized machines and machine tools during the revolution led to the rapid growth of mechanical engineering both in its birthplace Britain and abroad. John Smeaton was the first self-proclaimed civil engineer and is regarded as the father of civil engineering. He was an English civil engineer responsible for the design of bridges, canals, harbours and he was also a capable mechanical engineer and an eminent physicist. Smeaton designed the third Eddystone Lighthouse where he pioneered the use of hydraulic lime and his lighthouse remained in use until 1877 and was dismantled and partially rebuilt at Plymouth Hoe where it is known as Smeatons Tower. The United States census of 1850 listed the occupation of engineer for the first time with a count of 2,000, there were fewer than 50 engineering graduates in the U. S. before 1865. In 1870 there were a dozen U. S. mechanical engineering graduates, in 1890 there were 6,000 engineers in civil, mining, mechanical and electrical. There was no chair of applied mechanism and applied mechanics established at Cambridge until 1875, the theoretical work of James Maxwell and Heinrich Hertz in the late 19th century gave rise to the field of electronics
Engineering
–
The
steam engine, a major driver in the
Industrial Revolution, underscores the importance of engineering in modern history. This
beam engine is on display in the
Technical University of Madrid.
Engineering
–
Relief map of the
Citadel of Lille, designed in 1668 by
Vauban, the foremost military engineer of his age.
Engineering
–
The Ancient Romans built
aqueducts to bring a steady supply of clean fresh water to cities and towns in the empire.
Engineering
–
The
International Space Station represents a modern engineering challenge from many disciplines.
41.
Prime factor
–
In number theory, the prime factors of a positive integer are the prime numbers that divide that integer exactly. The prime factorization of an integer is a list of the integers prime factors, together with their multiplicities. The fundamental theorem of arithmetic says that every integer has a single unique prime factorization. To shorten prime factorizations, factors are expressed in powers. For example,360 =2 ×2 ×2 ×3 ×3 ×5 =23 ×32 ×5, in which the factors 2,3 and 5 have multiplicities of 3,2 and 1, respectively. For a prime factor p of n, the multiplicity of p is the largest exponent a for which pa divides n exactly. For a positive n, the number of prime factors of n. Perfect square numbers can be recognized by the fact all of their prime factors have even multiplicities. For example, the number 144 has the prime factors 144 =2 ×2 ×2 ×2 ×3 ×3 =24 ×32. These can be rearranged to make the more visible,144 =2 ×2 ×2 ×2 ×3 ×3 = × =2 =2. Because every prime factor appears a number of times, the original number can be expressed as the square of some smaller number. In the same way, perfect cube numbers will have prime factors whose multiplicities are multiples of three, and so on, positive integers with no prime factors in common are said to be coprime. Two integers a and b can also be defined as if their greatest common divisor gcd =1. Euclids algorithm can be used to determine whether two integers are coprime without knowing their prime factors, the runs in a time that is polynomial in the number of digits involved. The integer 1 is coprime to every integer, including itself. This is because it has no prime factors, it is the empty product and this implies that gcd =1 for any b ≥1. The function, ω, represents the number of prime factors of n, while the function, Ω. If n = ∏ i =1 ω p i α i, for example,24 =23 ×31, so ω =2 and Ω =3 +1 =4
Prime factor
–
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
42.
Long division
–
In arithmetic, long division is a standard division algorithm suitable for dividing multidigit numbers that is simple enough to perform by hand. It breaks down a division problem into a series of easier steps, as in all division problems, one number, called the dividend, is divided by another, called the divisor, producing a result called the quotient. It enables computations involving arbitrarily large numbers to be performed by following a series of simple steps, the abbreviated form of long division is called short division, which is almost always used instead of long division when the divisor has only one digit. Chunking is a less-efficient form of long division which may be easier to understand, while related algorithms have existed since the 12th century AD, the specific algorithm in modern use was introduced by Henry Briggs c.1600 AD. In English-speaking countries, long division does not use the division slash ⟨∕⟩ or obelus ⟨÷⟩ signs, the divisor is separated from the dividend by a right parenthesis ⟨)⟩ or vertical bar ⟨|⟩, the dividend is separated from the quotient by a vinculum. The combination of two symbols is sometimes known as a long division symbol or division bracket. It developed in the 18th century from an earlier single-line notation separating the dividend from the quotient by a left parenthesis, the process is begun by dividing the left-most digit of the dividend by the divisor. The quotient becomes the first digit of the result, and the remainder is calculated and this remainder carries forward when the process is repeated on the following digit of the dividend. When all digits have been processed and no remainder is left, an example is shown below, representing the division of 500 by 4. The largest number that the divisor 4 can be multiplied by without exceeding 5 is 1, next the 4 under the 5 is subtracted from the 5 to get the remainder,1, which is placed under the 4 under the 5. This remainder 1 is necessarily smaller than the divisor 4, next the first as-yet unused digit in the dividend, in this case the first digit 0 after the 5, is copied directly underneath itself and next to the remainder 1, to form the number 10. This remainder 2 is necessarily smaller than the divisor 4, the next digit of the dividend is copied directly below itself and next to the remainder 2, to form 20. Then the largest number by which the divisor 4 can be multiplied without exceeding 20 is ascertained, then this new quotient digit 5 is multiplied by the divisor 4 to get 20, which is written at the bottom below the existing 20. Then 20 is subtracted from 20, yielding 0, which is written below the 20 and we know we are done now because two things are true, there are no more digits to bring down from the dividend, and the last subtraction result was 0. If the last remainder when we ran out of dividend digits had been something other than 0, there would have been two possible courses of action. Or, we could extend the dividend by writing it as, say,500.000. and continue the process, in order to get a decimal answer, as in the following example. This example also illustrates that, at the beginning of the process, since the first digit 1 is less than the divisor 4, the first step is instead performed on the first two digits 12. Similarly, if the divisor were 13, one would perform the first step on 127 rather than 12 or 1, find the location of all decimal points in the dividend n and divisor m
Long division
–
An example of long division performed without a calculator.
43.
Algorithm
–
In mathematics and computer science, an algorithm is a self-contained sequence of actions to be performed. Algorithms can perform calculation, data processing and automated reasoning tasks, an algorithm is an effective method that can be expressed within a finite amount of space and time and in a well-defined formal language for calculating a function. The transition from one state to the next is not necessarily deterministic, some algorithms, known as randomized algorithms, 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 wasnt 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 and 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, the poem is a few hundred lines long and summarizes the art of calculating with the new style of Indian dice, or Talibus Indorum, or Hindu numerals. 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, but humans can do something equally useful, in the case of certain enumerably infinite sets, They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. 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 set of axioms. In logic, the time that an algorithm requires to complete cannot be measured, from such uncertainties, that characterize ongoing work, stems the unavailability of a definition of algorithm that suits both concrete and abstract usage of the term. Algorithms are essential to the way computers process data, thus, an algorithm can be considered to be any sequence of operations that can be simulated by a Turing-complete system. Although this may seem extreme, the arguments, in its favor are hard to refute. Gurevich. Turings informal argument in favor of his thesis justifies a stronger thesis, according to Savage, an algorithm is a computational process defined by a Turing machine. Typically, when an algorithm is associated with processing information, data can be read from a source, written to an output device. Stored data are regarded as part of the state of the entity performing the algorithm. In practice, the state is stored in one or more data structures, for some such computational process, the algorithm must be rigorously defined, specified in the way it applies in all possible circumstances that could arise. That is, any conditional steps must be dealt with, case-by-case
Algorithm
–
Alan Turing's statue at
Bletchley Park.
Algorithm
–
Logical NAND algorithm implemented electronically in
7400 chip
44.
Geometric series
–
In mathematics, a geometric series is a series with a constant ratio between successive terms. For example, the series 12 +14 +18 +116 + ⋯ is geometric, Geometric series are among the simplest examples of infinite series with finite sums, although not all of them have this property. Historically, geometric series played an important role in the development of calculus. Geometric series are used throughout mathematics, and they have important applications in physics, engineering, biology, economics, computer science, queueing theory, the terms of a geometric series form a geometric progression, meaning that the ratio of successive terms in the series is constant. This relationship allows for the representation of a series using only two terms, r and a. The term r is the ratio, and a is the first term of the series. In the case above, where r is one half, the series has the sum one, if r is greater than one or less than minus one the terms of the series become larger and larger in magnitude. The sum of the terms also gets larger and larger, if r is equal to one, all of the terms of the series are the same. If r is one the terms take two values alternately. The sum of the oscillates between two values. This is a different type of divergence and again the series has no sum, see for example Grandis series,1 −1 +1 −1 + ···. The sum can be computed using the self-similarity of the series, consider the sum of the following geometric series, s =1 +23 +49 +827 + ⋯. This series has common ratio 2/3, if we multiply through by this common ratio, then the initial 1 becomes a 2/3, the 2/3 becomes a 4/9, and so on,23 s =23 +49 +827 +1681 + ⋯. This new series is the same as the original, except that the first term is missing, subtracting the new series s from the original series s cancels every term in the original but the first, s −23 s =1, so s =3. A similar technique can be used to evaluate any self-similar expression, as n goes to infinity, the absolute value of r must be less than one for the series to converge. When a =1, this can be simplified to 1 + r + r 2 + r 3 + ⋯ =11 − r, the formula also holds for complex r, with the corresponding restriction, the modulus of r is strictly less than one. Since = 1−rn+1 and rn+1 →0 for | r | <1, convergence of geometric series can also be demonstrated by rewriting the series as an equivalent telescoping series. Consider the function, g = r K1 − r
Geometric series
–
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.
45.
Sign function
–
In mathematics, the sign function or signum function is an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign function is represented as sgn. The signum function of a number x is defined as follows. Any real number can be expressed as the product of its value and its sign function. The numbers cancel and all we are left with is the sign of x, D | x | d x = sgn for x ≠0. The signum function is differentiable with derivative 0 everywhere except at 0, using this identity, it is easy to derive the distributional derivative, d sgn d x =2 d H d x =2 δ. The signum can also be using the Iverson bracket notation. The signum can also be using the floor and the absolute value functions. For k ≫1, an approximation of the sign function is sgn ≈ tanh . Another approximation is sgn ≈ x x 2 + ε2, which gets sharper as ε →0, note that this is the derivative of √x2 + ε2. This is inspired from the fact that the above is equal for all nonzero x if ε =0. See Heaviside step function – Analytic approximations, the signum function can be generalized to complex numbers as, sgn = z | z | for any complex number z except z =0. The signum of a complex number z is the point on the unit circle of the complex plane that is nearest to z. Then, for z ≠0, sgn = e i arg z and we then have, csgn = z z 2 = z 2 z. At real values of x, it is possible to define a generalized function–version of the function, ε such that ε2 =1 everywhere. This generalized signum allows construction of the algebra of generalized functions, absolute value Heaviside function Negative number Rectangular function Sigmoid function Step function Three-way comparison Zero crossing Modulus function
Sign function
–
Signum function y = sgn(x)
46.
Common logarithm
–
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 ISO80000 specification recommends that log10 should be written lg, before the early 1970s, handheld electronic calculators were not available and mechanical calculators capable of multiplication were bulky, expensive and not widely available. Instead, tables of logarithms were used in science, engineering. Use of logarithms avoided laborious and error prone paper and pencil multiplications and divisions, because logarithms were so useful, tables of base-10 logarithms were given in appendices of many text books. Mathematical and navigation handbooks included tables of the logarithms of trigonometric functions as 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 number in a range. Such a range would cover all possible values of the mantissa, the integer part, called the characteristic, can be computed by simply counting how many places the decimal point must be moved so that it is just to the right of the first significant digit. For example, the logarithm of 120 is given by, log 10 120 = log 10 =2 + log 10 1.2 ≈2 +0.07918. The last number —the fractional part or the mantissa of the logarithm of 120—can be found in the table shown. The location of the point in 120 tells us that the integer part of the common logarithm of 120. Numbers greater than 0 and less than 1 have negative logarithms, when reading a number in bar notation out loud, the symbol n ¯ is read as bar n, so that 2 ¯.07918 is read as bar 2 point 07918. The following table shows how the same mantissa can be used for a range of numbers differing by powers of ten and this holds for any positive real number x because, log 10 = log 10 + log 10 = log 10 + i. Since i is always an integer the mantissa comes from log 10 which is constant for given x and this allows a table of logarithms to include only one entry for each mantissa. In the example of 5×10i,0.698970 will be listed once indexed by 5, or 0.5, common logarithms are sometimes also called Briggsian logarithms after Henry Briggs, a 17th-century British mathematician. In 1616 and 1617 Briggs visited John Napier, the inventor of what are now called natural logarithms at Edinburgh in order to suggest a change to Napiers logarithms. During these conferences the alteration proposed by Briggs was agreed upon, because base 10 logarithms were most useful for computations, engineers generally simply wrote log when they meant log10
Common logarithm
–
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
–
A graph of the common logarithm function for positive real numbers.
47.
0 (number)
–
0 is both a number and the numerical digit used to represent that number in numerals. The number 0 fulfills a role in mathematics as the additive identity of the integers, real numbers. As a digit,0 is used as a placeholder in place value systems, names for the number 0 in English include zero, nought or naught, nil, or—in contexts where at least one adjacent digit distinguishes it from the letter O—oh or o. Informal or slang terms for zero include zilch and zip, ought and aught, as well as cipher, have also been used historically. The word zero came into the English language via French zéro from Italian zero, in pre-Islamic time the word ṣifr had the meaning empty. Sifr evolved to mean zero when it was used to translate śūnya from India, the first known English use of zero was in 1598. The Italian mathematician Fibonacci, who grew up in North Africa and is credited with introducing the system to Europe. This became zefiro in Italian, and was contracted to zero in Venetian. The Italian word zefiro was already in existence and may have influenced the spelling when transcribing Arabic ṣifr, modern usage There are different words used for the number or concept of zero depending on the context. For the simple notion of lacking, the words nothing and none are often used, sometimes the words nought, naught and aught are used. Several sports have specific words for zero, such as nil in football, love in tennis and it is often called oh in the context of telephone numbers. Slang words for zero include zip, zilch, nada, duck egg and goose egg are also slang for zero. Ancient Egyptian numerals were base 10 and they used hieroglyphs for the digits and were not positional. By 1740 BC, the Egyptians had a symbol for zero in accounting texts. The symbol nfr, meaning beautiful, was used to indicate the base level in drawings of tombs and pyramids. By the middle of the 2nd millennium BC, the Babylonian mathematics had a sophisticated sexagesimal positional numeral system, the lack of a positional value 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 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
0 (number)
–
Example of the early Greek symbol for zero (lower right corner) from a 2nd-century papyrus
0 (number)
0 (number)
–
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)
–
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.
48.
Irrational number
–
In mathematics, the irrational numbers are all the real numbers, which are not rational numbers, the latter being the numbers constructed from ratios of integers. Irrational numbers may also be dealt with via non-terminating continued fractions, for example, the decimal representation of the number π starts with 3.14159265358979, but no finite number of digits can represent π exactly, nor does it repeat. Mathematicians do not generally take terminating or repeating to be the definition of the concept of rational number, as a consequence of Cantors proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational. The first proof of the existence of numbers is usually attributed to a Pythagorean. The then-current Pythagorean method would have claimed that there must be sufficiently small. However, Hippasus, in the 5th century BC, was able to deduce that there was in no common unit of measure. His reasoning is as follows, Start with a right triangle with side lengths of integers a, b. The ratio of the hypotenuse to a leg is represented by c, b, assume a, b, and 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, and therefore even. Since c2 is even, c must be even, since c is even, dividing c by 2 yields an integer. Squaring both sides of c = 2y yields c2 =2, or c2 = 4y2, substituting 4y2 for c2 in the first equation gives us 4y2= 2b2. Dividing by 2 yields 2y2 = b2, since y is an integer, and 2y2 = b2, b2 is divisible by 2, and therefore even. Since b2 is even, b must be even and we have just show that both b and c must be even. Hence they have a factor of 2. However this contradicts the assumption that they have no common factors and this contradiction proves that c and b cannot both be integers, and thus the existence of a number that cannot be expressed as a ratio of two integers. Greek mathematicians termed this ratio of incommensurable magnitudes alogos, or inexpressible. ”Another legend states that Hippasus was merely exiled for this revelation, the discovery of incommensurable ratios was indicative of another problem facing the Greeks, the relation of the discrete to the continuous. Brought into light by Zeno of Elea, who questioned the conception that quantities are discrete and composed of a number of units of a given size. ”However Zeno found that in fact “ in general are not discrete collections of units. That in fact, these divisions of quantity must necessarily be infinite, for example, consider a line segment, this segment can be split in half, that half split in half, the half of the half in half, and so on
Irrational number
–
The number is irrational.
49.
Golden mean 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 base, phi-base, or, colloquially, any non-negative real number can be represented as a base-φ numeral using only the digits 0 and 1, and avoiding the digit sequence 11 – this is called a standard form. A base-φ numeral that includes the digit sequence 11 can always be rewritten in standard form, despite using an irrational number base, when using standard form, all non-negative integers have a unique representation as a terminating base-φ expansion. Other numbers have standard representations in base-φ, with rational numbers having recurring representations and these representations are unique, except that numbers with a terminating expansion also have a non-terminating expansion, as they do in base-10, for example,1 =0. 99999…. In the following example the notation 1 is used to represent −1. 211. 01φ is not a standard base-φ numeral, since it contains a 11 and a 2, which isnt a 0 or 1, and contains a 1 = −1, which isnt a 0 or 1 either. To standardize a numeral, we can use the following substitutions, 011φ = 100φ, 0200φ = 1001φ, 010φ = 101φ and we can apply the substitutions in any order we like, as the result is the same. Below, the applied to the number on the previous line are on the right. Any positive number with a non-standard terminating base-φ representation can be standardized in this manner. If we get to a point where all digits are 0 or 1, except for the first digit being negative and this can be converted to the negative of a base-φ representation by negating every digit, standardizing the result, and then marking it as negative. For example, use a sign, or some other significance to denote negative numbers. If the arithmetic is being performed on a computer, a message may be returned. We can either consider our integer to be the digit of a nonstandard base-φ numeral, therefore, we can compute + =, − = and × =. So, using integer values only, we can add, subtract and multiply numbers of the form, > if and only if 2 − > × √5. If one side is negative, the positive, the comparison is trivial. Otherwise, square sides, to get an integer comparison, reversing the comparison direction if both sides were negative. On squaring both sides, the √5 is replaced with the integer 5, so, using integer values only, we can also compare numbers of the form. To convert an integer x to a number, note that x =
Golden mean base
–
Numeral systems
50.
Binary prefix
–
A binary prefix is a unit prefix for multiples of units in data processing, data transmission, and digital information, notably the bit and the byte, to indicate multiplication by a power of 2. The computer industry has used the units kilobyte, megabyte, and gigabyte, and the corresponding symbols KB, MB. In citations of main memory capacity, gigabyte customarily means 1073741824 bytes, as this is the third power of 1024, and 1024 is a power of two, this usage is referred to as a binary measurement. In most other contexts, the uses the multipliers kilo, mega, giga, etc. in a manner consistent with their meaning in the International System of Units. For example, a 500 gigabyte hard disk holds 500000000000 bytes, in contrast with the binary prefix 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/IEC80000 standard. Early computers used one of two addressing methods to access the memory, binary or decimal. For example, the IBM701 used binary and could address 2048 words of 36 bits each, while the IBM702 used decimal, by the mid-1960s, binary addressing had become the standard architecture in most computer designs, and main memory sizes were most commonly powers of two. Early computer system documentation would specify the size with an exact number such as 4096,8192. These are all powers of two, and furthermore are small multiples of 210, or 1024, as storage capacities increased, several different methods were developed to abbreviate these quantities. The method most commonly used today uses prefixes such as kilo, mega, giga, and corresponding symbols K, M, and G, the prefixes kilo- and mega-, meaning 1000 and 1000000 respectively, were commonly used in the electronics industry before World War II. Along with giga- or G-, meaning 1000000000, they are now known as SI prefixes after the International System of Units, introduced in 1960 to formalize aspects of the metric system. The International System of Units does not define units for digital information and this usage is not consistent with the SI. Compliance with the SI requires that the prefixes take their 1000-based meaning, the use of K in the binary sense as in a 32K core meaning 32 ×1024 words, i. e.32768 words, can be found as early as 1959. Gene Amdahls seminal 1964 article on IBM System/360 used 1K to mean 1024 and this style was used by other computer vendors, the CDC7600 System Description made extensive use of K as 1024. Thus the first binary prefix was born, the exact values 32768 words,65536 words and 131072 words would then be described as 32K, 65K and 131K. This style was used from about 1965 to 1975 and these two styles were used loosely around the same time, sometimes by the same company. In discussions of binary-addressed memories, the size was evident from context
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.
51.
Tsinghua Bamboo Slips
–
The Tsinghua Bamboo Slips are a collection of Chinese texts dating to the Warring States period and written in ink on strips of bamboo, that were acquired in 2008 by Tsinghua University, China. The texts were obtained by illegal excavation, probably of a tomb in the area of Hubei or Hunan province, the very large size of the collection and the significance of the texts for scholarship make it one of the most important discoveries of early Chinese texts to date. On 7 January 2014 the journal Nature announced that some Tsinghua Bamboo Slips represent the worlds oldest example of a multiplication table. The Tsinghua Bamboo Slips were donated to Tsinghua University in July 2008 by an alumnus of the university, the precise location and date of the illicit excavation that yielded the slips remain unknown. Li Xueqin, the director of the conservation and research project, has stated that the wishes of the alumnus to maintain his identity secret will be respected, a single radiocarbon date and the style of ornament on the accompanying box are in keeping with this conclusion. By the time reached the university, the slips were badly affected by mold. Conservation work on the slips was carried out, and a Center for Excavated Texts Research, there are 2388 slips altogether in the collection, including a number of fragments. A series of articles discussing the TBS, intended for an educated but non-specialist Chinese audience, the first volume of texts was published by the Tsinghua team in 2010. A2013 article in The New York Times reported on the TBSs importance to understanding the Chinese classics, the classics are all political, Li Xueqin commented, It would be like finding the original Bible or the original classics. It enables us to look at the classics before they were turned into classics, the questions now include, what were they in the beginning, and how they became what they are. In some cases, a TBS text can be found in the received Shang Shu, with variations in wording. Such examples include versions of the Jin Teng, Kang Gao, an annalistic history recording events from the beginning of the Western Zhou through to the early Warring States period is said to be similar in form and content to the received Bamboo Annals. Another text running across 14 slips recounts a celebratory gathering of the Zhou elite in the 8th year of the reign of King Wu of Zhou, prior to their conquest of the Shang dynasty. The gathering takes place in the temple of King Wen of Zhou, King Wus father, and consisted of beer drinking. Among the TBS texts in the style of the received Shang Shu, is one that has been titled The Admonition of Protection and this was the first text for which anything approaching a complete description and transcription was published. The text purports to be a record of a deathbed admonition by the Zhou king Wen Wang to his son and heir, Wu Wang. The content of the speech revolves around a concept of The Middle which seems to refer to an avoidance of extremes. Xinian 繫年, probably composed ca.370 BC, relates key events of Zhou history and it comprises 138 slips in a relatively well preserved condition
Tsinghua Bamboo Slips
–
The world's earliest artifacts of decimal multiplication table
Tsinghua Bamboo Slips
–
A diagram of the Warring States decimal multiplication table
52.
Abacus
–
The exact origin of the abacus is still unknown. Today, abaci are often constructed as a frame with beads sliding on wires. The use of the word abacus dates before 1387 AD, when a Middle English work borrowed the word from Latin to describe a sandboard abacus, the Latin word came from Greek ἄβαξ abax which means something without base, and improperly, any piece of rectangular board or plank. Alternatively, without reference to ancient texts on etymology, it has suggested that it means a square tablet strewn with dust. Whereas the table strewn with dust definition is popular, there are those that do not place credence in this at all, Greek ἄβαξ itself is probably a borrowing of a Northwest Semitic, perhaps Phoenician, word akin to Hebrew ʾābāq, dust. The preferred plural of abacus is a subject of disagreement, with both abacuses and abaci in use, the user of an abacus is called an abacist. The period 2700–2300 BC saw the first appearance of the Sumerian abacus, some scholars point to a character from the Babylonian cuneiform which may have been derived from a representation of the abacus. Archaeologists have found ancient disks of various sizes that are thought to have used as counters. However, wall depictions of this instrument have not been discovered, during the Achaemenid Empire, around 600 BC the Persians first began to use the abacus. The earliest archaeological evidence for the use of the Greek abacus dates to the 5th century BC, also Demosthenes talked of the need to use pebbles for calculations too difficult for your head. The Greek abacus was a table of wood or marble, pre-set with small counters in wood or metal for mathematical calculations and this Greek abacus saw use in Achaemenid Persia, the Etruscan civilization, Ancient Rome and, until the French Revolution, the Western Christian world. A tablet found on the Greek island Salamis in 1846 AD, dates back to 300 BC and it is a slab of white marble 149 cm long,75 cm wide, and 4.5 cm thick, on which are 5 groups of markings. Below these lines is a space with a horizontal crack dividing it. Also from this frame the Darius Vase was unearthed in 1851. It was covered with pictures including a holding a wax tablet in one hand while manipulating counters on a table with the other. The earliest known documentation of the Chinese abacus dates to the 2nd century BC. The Chinese abacus, known as the suanpan, is typically 20 cm tall and it usually has more than seven rods. There are two beads on each rod in the deck and five beads each in the bottom for both decimal and hexadecimal computation
Abacus
–
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
53.
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, such computers are used as control systems for a very wide variety of industrial and consumer devices. The Internet is run on computers and it millions of other computers. Since ancient times, simple manual devices like the abacus aided people in doing calculations, early in the Industrial Revolution, some mechanical devices were built to automate long tedious tasks, such as guiding patterns for looms. More sophisticated electrical machines did specialized analog calculations in the early 20th century, the first digital electronic calculating machines were developed during World War II. The speed, power, and versatility of computers has increased continuously and dramatically since then, conventionally, a modern computer consists of at least one processing element, typically a central processing unit, and some form of memory. The processing element carries out arithmetic and logical operations, and a sequencing, peripheral devices include input devices, output devices, and input/output devices that perform both functions. Peripheral devices allow information to be retrieved from an external source and this usage of the term referred to a person who carried out calculations or computations. The word continued with the same meaning until the middle of the 20th century, from the end of the 19th century the word began to take on its more familiar meaning, a machine that carries out computations. The Online Etymology Dictionary gives the first attested use of computer in the 1640s, one who calculates, the Online Etymology Dictionary states that the use of the term to mean calculating machine is from 1897. The Online Etymology Dictionary indicates that the use of the term. 1945 under this name, theoretical from 1937, as Turing machine, devices have been used to aid computation for thousands of years, mostly using one-to-one correspondence with fingers. The earliest counting device was probably a form of tally stick, later record keeping aids throughout the Fertile Crescent included calculi which represented counts of items, probably livestock or grains, sealed in hollow unbaked clay containers. The use of counting rods is one example, the abacus was initially used for arithmetic tasks. The Roman abacus was developed from used in Babylonia as early as 2400 BC. Since then, many forms of reckoning boards or tables have been invented. In a medieval European counting house, a checkered cloth would be placed on a table, the Antikythera mechanism is believed to be the earliest mechanical analog computer, according to Derek J. de Solla Price. It was designed to calculate astronomical positions and it was discovered in 1901 in the Antikythera wreck off the Greek island of Antikythera, between Kythera and Crete, and has been dated to circa 100 BC
Computer
–
Computer
Computer
–
Suanpan (the number represented on this abacus is 6,302,715,408)
Computer
Computer
54.
IBM 650
–
The IBM650 Magnetic Drum Data-Processing Machine is one of IBMs early computers, and the world’s first mass-produced computer. It was announced in 1953 and in 1956 enhanced as the IBM650 RAMAC with the addition of up to four disk storage units, almost 2,000 systems were produced, the last in 1962. Support for the 650 and its component units was withdrawn in 1969, the 650 was a two-address, bi-quinary coded decimal computer, with memory on a rotating magnetic drum. Character support was provided by the units converting punched card alphabetical and special character encodings to/from a two-digit decimal code. The 650 was marketed to business, scientific and engineering users as well as to users of punched card machines who were upgrading from calculating punches, such as the IBM604, to computers. The IBM7070, announced 1958, was expected to be a successor to at least the 650. The IBM1620, introduced in 1959, addressed the lower end of the market, the UNIVAC Solid State was announced by Sperry Rand in December 1958 as a response to the 650. None of these had a 650 compatible instruction set, the basic 650 system consisted of three units, IBM650 Console Unit housed the magnetic drum storage, arithmetical device and the operators console. IBM655 Power Unit IBM533 or IBM537 Card Read Punch Unit The IBM533 had separate feeds for reading and punching, the IBM537 had one feed, thus could read and then punch into the same card. Words on the drums were organized in bands around the drum, fifty words per band, a word could be accessed when its location on the drum surface passed under the read/write heads during rotation. Because of this timing, the address in each instruction was the address of the next instruction. Instructions could then be interleaved, placing many at addresses that would be accessible when execution of the previous instruction was completed. Instructions read from the drum went to a program register, Data read from the drum went through a 10-digit distributor. The 650 had a 20-digit accumulator, divided into 10-digit lower and upper accumulators with a common sign, arithmetic was performed by a one-digit adder. The console, distributor, lower and upper accumulators were all addressable,8000,8001,8002,8003 respectively. Three four-digit index registers at addresses 8005 to 8007, drum addresses were indexed by adding 2000,4000 or 6000 to them, the 4000-word systems required transistorized read/write circuitry for the drum memory and were available before 1963. Floating point – arithmetic instructions supported an eight-digit mantissa and two-digit characteristic – MMMMMMMMCC, the 650 instructions consisted of a two-digit operation code, a four-digit data address and the four-digit address of the next instruction. The sign was ignored on the machine, but was used on machines with optional features
IBM 650
–
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
55.
Egyptian hieroglyphs
–
Egyptian hieroglyphs were the formal writing system used in Ancient Egypt. It combined logographic, syllabic and alphabetic elements, with a total of some 1,000 distinct characters, cursive hieroglyphs were used for religious literature on papyrus and wood. The later hieratic and demotic Egyptian scripts are derived from hieroglyphic writing, the writing system continued to be used throughout the Late Period, as well as the Persian and Ptolemaic periods. Late survivals of hieroglyphic use are found well into the Roman period, with the closing of pagan temples in the 5th century, knowledge of hieroglyphic writing was lost, and the script remained undeciphered throughout the medieval and early modern period. The decipherment of hieroglyphs would only be solved in the 1820s by Jean-François Champollion, the word hieroglyph comes from the Greek adjective ἱερογλυφικός, a compound of ἱερός and γλύφω, supposedly a calque of an Egyptian phrase mdw·w-nṯr gods words. The glyphs themselves were called τὰ ἱερογλυφικὰ γράμματα the sacred engraved letters, the word hieroglyph has become a noun in English, standing for an individual hieroglyphic character. As used in the sentence, the word hieroglyphic is an adjective. Hieroglyphs emerged from the artistic traditions of Egypt. For example, symbols on Gerzean pottery from c.4000 BC have been argued to resemble hieroglyphic writing, proto-hieroglyphic symbol systems develop in the second half of the 4th millennium BC, such as the clay labels of a Predynastic ruler called Scorpion I recovered at Abydos in 1998. The first full sentence written in hieroglyphs so far discovered was found on a seal found in the tomb of Seth-Peribsen at Umm el-Qaab. There are around 800 hieroglyphs dating back to the Old Kingdom, Middle Kingdom, by the Greco-Roman period, there are more than 5,000. However, given the lack of evidence, no definitive determination has been made as to the origin of hieroglyphics in ancient Egypt. Since the 1990s, and discoveries such as the Abydos glyphs, as writing developed and became more widespread among the Egyptian people, simplified glyph forms developed, resulting in the hieratic and demotic scripts. These variants were more suited than hieroglyphs for use on papyrus. Hieroglyphic writing was not, however, eclipsed, but existed alongside the other forms, especially in monumental, the Rosetta Stone contains three parallel scripts – hieroglyphic, demotic, and Greek. Hieroglyphs continued to be used under Persian rule, and after Alexander the Greats conquest of Egypt, during the ensuing Ptolemaic and Roman periods. It appears that the quality of comments from Greek and Roman writers about hieroglyphs came about, at least in part. Some believed that hieroglyphs may have functioned as a way to distinguish true Egyptians from some of the foreign conquerors, another reason may be the refusal to tackle a foreign culture on its own terms, which characterized Greco-Roman approaches to Egyptian culture generally
Egyptian hieroglyphs
–
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
56.
Cretan hieroglyphs
–
Cretan hieroglyphs are undeciphered hieroglyphs found on artefacts of early Bronze Age Crete, during the Minoan era. It predates Linear A by about a century, but the two writing systems continued to be used in parallel for most of their history, the seals and sealings represent about 307 distinct sign-groups, consisting all together of ±832 signs. The other inscriptions represent about 274 distinct sign-groups, consisting all together of ±723 signs, more documents have been published since then, such as, for example, from the Petras deposit. Definitive edition was published in 2010, the relation of the last three items with the script of the main corpus is uncertain. Some Cretan Hieroglyphic inscriptions were found on the island of Samothrace in the northeastern Aegean. It has been suggested there was an evolution of the hieroglyphs into the linear scripts. Also, some relations to Anatolian hieroglyphs have been suggested, the overlaps between the Cretan script and other scripts, such as the hieroglyphic scripts of Cyprus and the Hittite lands of Anatolia, may suggest. That they all evolved from an ancestor, a now-lost script perhaps originating in Syria. Symbol inventories have been compiled by Evans, Meijer, Olivier/Godart, the known corpus has been edited in 1996 as CHIC, listing a total of 314 items. The glyph inventory as presented by CHIC includes 96 syllabograms, ten of which double as logograms, there are also 23 logograms representing four levels of numerals, numerical fractions, and two types of punctuation. Many symbols have apparent Linear A counterparts, so that it is tempting to insert Linear B sound values, Cretan Writing in the Second Millennium B. C. World Archaeology,17, 377–389, doi,10. 1080/00438243.1986.9979977 Yule, Paul, Early Cretan Seals, marburger Studien zur Vor und Frühgeschichte, ISBN 3-8053-0490-0 W. C. Brice, Notes on the Cretan Hieroglyphic Script, I, the Clay Bar from Malia, H20, Kadmos 29 1-10. Brice, Cretan Hieroglyphs & Linear A, Kadmos 29 171-2, brice, Notes on the Cretan Hieroglyphic Script, III. The Inscriptions from Mallia Quarteir Mu, the Clay Bar from Knossos, P116, Kadmos 30 93-104. Brice, Notes on the Cretan Hieroglyphic Script, Kadmos 31, M. Civitillo, LA SCRITTURA GEROGLIFICA MINOICA SUI SIGILLI. Il messaggio della glittica protopalaziale, Biblioteca di Pasiphae XII, Pisa-Roma 2016, G. M. Facchetti La questione della scrittura «geroglifica cretese» dopo la recente edizione del corpus dei testi. Pasiphae, Rivista di filologia e antichita egee, the Cretan hieroglyphic script of the second millennium BC, description, analysis, function and decipherment perspectives
Cretan hieroglyphs
–
A green
jasper seal with Cretan hieroglyphs. 1800 BC
57.
Vedas
–
The Vedas are a large body of knowledge texts originating in the ancient Indian subcontinent. Composed in Vedic Sanskrit, the texts constitute the oldest layer of Sanskrit literature, Hindus consider the Vedas to be apauruṣeya, which means not of a man, superhuman and impersonal, authorless. Vedas are also called śruti literature, distinguishing them from religious texts. The Veda, for orthodox Indian theologians, are considered revelations seen by ancient sages after intense meditation, in the Hindu Epic the Mahabharata, the creation of Vedas is credited to Brahma. The Vedic hymns themselves assert that they were created by Rishis, after inspired creativity. There are four Vedas, the Rigveda, the Yajurveda, the Samaveda, each Veda has been subclassified into four major text types – the Samhitas, the Aranyakas, the Brahmanas, and the Upanishads. Some scholars add a fifth category – the Upasanas, the various Indian philosophies and denominations have taken differing positions on the Vedas. Schools of Indian philosophy which cite the Vedas as their authority are classified as orthodox. Other śramaṇa traditions, such as Lokayata, Carvaka, Ajivika, Buddhism and Jainism, despite their differences, just like the texts of the śramaṇa traditions, the layers of texts in the Vedas discuss similar ideas and concepts. The Sanskrit word véda knowledge, wisdom is derived from the root vid- to know and this is reconstructed as being derived from the Proto-Indo-European root *u̯eid-, meaning see or know. The noun is from Proto-Indo-European *u̯eidos, cognate to Greek εἶδος aspect, not to be confused is the homonymous 1st and 3rd person singular perfect tense véda, cognate to Greek οἶδα oida I know. Root cognates are Greek ἰδέα, English wit, etc, the Sanskrit term veda as a common noun means knowledge. The term in some contexts, such as hymn 10.93.11 of the Rigveda, means obtaining or finding wealth, property, a related word Vedena appears in hymn 8.19.5 of the Rigveda. It was translated by Ralph T. H. Griffith as ritual lore, as studying the Veda by the 14th century Indian scholar Sayana, as bundle of grass by Max Müller, Vedas are called Maṛai or Vaymoli in parts of South India. Marai literally means hidden, a secret, mystery, in some south Indian communities such as Iyengars, the word Veda includes the Tamil writings of the Alvar saints, such as Divya Prabandham, for example Tiruvaymoli. The Vedas are among the oldest sacred texts, the Samhitas date to roughly 1700–1100 BC, and the circum-Vedic texts, as well as the redaction of the Samhitas, date to c. 1000-500 BC, resulting in a Vedic period, spanning the mid 2nd to mid 1st millennium BC, or the Late Bronze Age, Michael Witzel gives a time span of c.1500 to c. Witzel makes special reference to the Near Eastern Mitanni material of the 14th century BC the only record of Indo-Aryan contemporary to the Rigvedic period
Vedas
–
Rigveda (
padapatha) manuscript in
Devanagari
58.
Al Khwarizmi
–
Muḥammad ibn Mūsā al-Khwārizmī, formerly Latinized as Algoritmi, was a Persian mathematician, astronomer, and geographer during the Abbasid Caliphate, a scholar in the House of Wisdom in Baghdad. In the 12th century, Latin translations of his work on the Indian numerals introduced the decimal number system to the Western world. Al-Khwārizmīs The Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear and he is often considered one of the fathers of algebra. He revised Ptolemys Geography and wrote on astronomy and astrology, some words reflect the importance of al-Khwārizmīs contributions to mathematics. Algebra is derived from al-jabr, one of the two operations he used to solve quadratic equations, algorism and algorithm stem from Algoritmi, the Latin form of his name. His name is also the origin of guarismo and of algarismo, few details of al-Khwārizmīs life are known with certainty. He was born in a Persian family and Ibn al-Nadim gives his birthplace as Khwarezm in Greater Khorasan, muhammad ibn Jarir al-Tabari gives his name as Muḥammad ibn Musá al-Khwārizmiyy al-Majūsiyy al-Quṭrubbaliyy. The epithet al-Qutrubbulli could indicate he might instead have come from Qutrubbul and this would not be worth mentioning if a series of errors concerning the personality of al-Khwārizmī, occasionally even the origins of his knowledge, had not been made. Recently, G. J. Toomer. with naive confidence constructed an entire fantasy on the error which cannot be denied the merit of amusing the reader. Regarding al-Khwārizmīs religion, Toomer writes, Another epithet given to him by al-Ṭabarī, al-Majūsī, Ibn al-Nadīms Kitāb al-Fihrist includes a short biography on al-Khwārizmī together with a list of the books he wrote. Al-Khwārizmī accomplished most of his work in the period between 813 and 833, douglas Morton Dunlop suggests that it may have been possible that Muḥammad ibn Mūsā al-Khwārizmī was in fact the same person as Muḥammad ibn Mūsā ibn Shākir, the eldest of the three Banū Mūsā. Al-Khwārizmīs contributions to mathematics, geography, astronomy, and cartography established the basis for innovation in algebra, on the Calculation with Hindu Numerals written about 825, was principally responsible for spreading the Hindu–Arabic numeral system throughout the Middle East and Europe. It was translated into Latin as Algoritmi de numero Indorum, al-Khwārizmī, rendered as Algoritmi, led to the term algorithm. Some of his work was based on Persian and Babylonian astronomy, Indian numbers, al-Khwārizmī systematized and corrected Ptolemys data for Africa and the Middle East. Another major book was Kitab surat al-ard, presenting the coordinates of places based on those in the Geography of Ptolemy but with improved values for the Mediterranean Sea, Asia and he also wrote on mechanical devices like the astrolabe and sundial. He assisted a project to determine the circumference of the Earth and in making a map for al-Mamun. When, in the 12th century, his works spread to Europe through Latin translations, the Compendious Book on Calculation by Completion and Balancing is a mathematical book written approximately 830 CE. The term algebra is derived from the name of one of the operations with equations described in this book
Al Khwarizmi
Al Khwarizmi
Al Khwarizmi
59.
Indo-Aryan languages
–
The Indo-Aryan or Indic languages are the dominant language family of the Indian subcontinent. They constitute a branch of the Indo-Iranian languages, itself a branch of the Indo-European language family, Indo-Aryan speakers form about one half of all Indo-European speakers, and more than half of all Indo-European languages recognized by Ethnologue. While the languages are spoken in South Asia, pockets of Indo-Aryan languages are found to be spoken in Europe. The largest in terms of speakers are Hindustani, Bengali, Punjabi. Proto-Indo-Aryan, or sometimes Proto-Indic, is the reconstructed proto-language of the Indo-Aryan languages and it is intended to reconstruct the language of the Proto-Indo-Aryans. Proto-Indo-Aryan is meant to be the predecessor of Old Indo-Aryan which is attested as Vedic. Despite the great archaicity of Vedic, however, the other Indo-Aryan languages preserve a number of archaic features lost in Vedic. Vedic has been used in the ancient preserved religious hymns, the canon of Hinduism known as the Vedas. Mitanni-Aryan is of age to the language of the Rigveda. The language of the Vedas – commonly referred to as Vedic Sanskrit by modern scholars – is only marginally different from Proto-Indo-Aryan the proto-language of the Indo-Aryan languages. From the Vedic, Sanskrit developed as the language of culture, science and religion, as well as the court, theatre. Sanskrit is, by convention, referred to by scholars as Classical Sanskrit in contra-distinction to the so-called Vedic Sanskrit. Outside the learned sphere of Sanskrit, vernacular dialects continued to evolve, the oldest attested Prakrits are the Buddhist and Jain canonical languages Pali and Ardha Magadhi, respectively. By medieval times, the Prakrits had diversified into various Middle Indo-Aryan dialects, Apabhramsa is the conventional cover term for transitional dialects connecting late Middle Indo-Aryan with early Modern Indo-Aryan, spanning roughly the 6th to 13th centuries. Some of these dialects showed considerable literary production, the Sravakachar of Devasena is now considered to be the first Hindi book, the next major milestone occurred with the Muslim conquests on the Indian subcontinent in the 13th–16th centuries. Under the flourishing Turco-Mongol Mughal empire, Persian became very influential as the language of prestige of the Islamic courts due to adoptation of the language by the Mughal emperors. However, Persian was soon displaced by Hindustani and this Indo-Aryan language is a combination with Persian, Arabic, and Turkic elements in its vocabulary, with the grammar of the local dialects. The two largest languages that formed from Apabhramsa were Bengali and Hindustani, others include Sindhi, Gujarati, Odia, Marathi, the Indo-Aryan languages of Northern India and Pakistan form a dialect continuum
Indo-Aryan languages
–
Dogri–Kangri region
Indo-Aryan languages
–
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.)
60.
Dravidian languages
–
The Dravidian languages with the most speakers are Telugu, Tamil, Kannada and Malayalam. There are also groups of Dravidian-speaking scheduled tribes, who live beyond the mainstream communities, such as the Kurukh. 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, I have, therefore, no doubt of the propriety of adopting it. The 1961 publication of the Dravidian etymological dictionary by T, 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, Kamil Zvelebil cites the forms such as dramila damiḷa and then goes on to say, The forms damiḷa/damila almost certainly provide a connection of drviḍa and. Tamiḷ < tamiẓ. whereby the further development might have been *tamiẓ > *damiḷ > damiḷa- / damila- and further, with the intrusive, hypercorrect -r-, an analogical case of DED1033 Ta. kamuku, Tu. kangu areca nut, Skt. kramu. Sinhala BCE inscriptions cite dameḍa-, damela- denoting Tamil merchants and it appears that damiḷa- was older than draviḍa- which could be its Sanskritization. Based on what Krishnamurti states, the Sanskrit word draviḍa itself is later than damiḷa since the dates for the forms with -r- are centuries later than the dates for the forms without -r-. The Monier-Williams Sanskrit Dictionary lists for the Sanskrit word draviḍa a meaning of collective Name for 5 peoples, viz. the Āndhras, Karṇāṭakas, Gurjaras, Tailaṅgas, the Dravidian languages form a close-knit family. They are descended from the Proto-Dravidian language, there is reasonable agreement on how they are related to each other. Most scholars agree on four groups, North, Central, South-Central, earlier classifications grouped Central and South-Central Dravidian in a single branch. The classification below follows Krishnamurti in grouping South-Central and South Dravidian, languages recognized as official languages of India appear here in boldface. Some authors deny that North Dravidian forms a subgroup, splitting it into Northeast and Northwest. In some words, *c is retracted to /k/ and this development is however also found in several other Dravidian languages, including Kannada, Kodagu and Tulu. A few morphological parallels between Brahui and Kurukh-Malto are also known, but according to McAlpin they are analyzable as shared archaisms rather than shared innovations, approximately 29% of Indias population spoke Dravidian languages in 1981
Dravidian languages
–
Language families in South Asia
Dravidian languages
–
North
Dravidian languages
–
Jambai Tamil Brahmi inscription dated to the early Sangam age
61.
Japanese language
–
Japanese is an East Asian language spoken by about 125 million speakers, primarily in Japan, where it is the national language. It is a member of the Japonic language family, whose relation to language groups, particularly to Korean. Little is known of the prehistory, or when it first appeared in Japan. Chinese documents from the 3rd century recorded a few Japanese words, 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, the standard dialect moved from the Kansai region to the Edo region in the Early Modern Japanese period. Following the end in 1853 of Japans self-imposed isolation, the flow of loanwords from European languages increased significantly, English loanwords in particular have become frequent, and Japanese words from English roots have proliferated. Japanese is an agglutinative, mora-timed language with simple phonotactics, a vowel system, phonemic vowel and consonant length. Word order is normally subject–object–verb with particles marking the grammatical function of words, sentence-final particles are used to add emotional or emphatic impact, or make questions. Nouns have no number or gender, and 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 system of honorifics with verb forms and vocabulary to indicate the relative status of the speaker, the listener. Japanese has no relationship with Chinese, but it makes extensive use of Chinese characters, or kanji, in its writing system. Along with kanji, the Japanese writing system uses two syllabic scripts, hiragana and katakana. Latin script is used in a fashion, such as for imported acronyms. Very little is known about the Japanese of this period, Old Japanese is the oldest attested stage of the Japanese language. Through the spread of Buddhism, the Chinese writing system was imported to Japan, the earliest texts found in Japan are written in Classical Chinese, but they may have been meant to be read as Japanese by the kanbun method. Some of these Chinese texts show the influences of Japanese grammar, in these hybrid texts, Chinese characters are also occasionally used phonetically to represent Japanese particles. The earliest text, the Kojiki, dates to the early 8th century, the end of Old Japanese coincides with the end of the Nara period in 794
Japanese language
–
A page from
Nihon Shoki (The Chronicles of Japan), the second oldest book of classical
Japanese history.
Japanese language
–
Map of Japanese dialects and Japonic languages
Japanese language
–
Two pages from a 12th-century
emaki scroll of
The Tale of Genji from the 11th century.
Japanese language
–
Calligraphy
62.
Korean language
–
It is also one of the two official languages in the Yanbian Korean Autonomous Prefecture and Changbai Korean Autonomous County of the Peoples Republic of China. Approximately 80 million people worldwide speak Korean and this implies that Korean is not an isolate, but a member of a small family. There is still debate on whether Korean and Japanese are related with each other, 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, Chinese characters arrived in Korea together with Buddhism during the pre-Three Kingdoms period. Mainly privileged elites were educated to read and write in hanja, however, today, the hanja are largely unused in everyday life, but in South Korea they experience revivals on artistic works and are important in historic and/or linguistic studies of Korean. Since the Korean War, through 70 years of separation, North–South differences have developed in standard Korean, including variations in pronunciation, verb inflection, the Korean names for the language are based on the names for Korea used in North Korea and South Korea. In South Korea, the Korean language is referred to by names including hanguk-eo Korean language, hanguk-mal, Korean speech and uri-mal. In hanguk-eo and hanguk-mal, the first part of the word, hanguk, refers to the Korean nation while -eo and -mal mean language and speech, Korean is also simply referred to as guk-eo, literally national language. This name is based on the same Chinese characters meaning nation + language that are used in Taiwan and Japan to refer to their respective national languages. In North Korea and China, the language is most often called Chosŏn-mal, or more formally, the English word Korean is derived from Goryeo, which is thought to be the first dynasty known to Western countries. Korean people in the former USSR refer to themselves as Koryo-saram and Goryeo In, the majority of historical and modern linguists classify Korean as a language isolate. Such factors of typological divergence as Middle Mongolians exhibition of gender agreement can be used to argue that a relationship with Altaic is unlikely. Sergei Anatolyevich Starostin found about 25% of potential cognates in the Japanese–Korean 100-word Swadesh list, a good example might be Middle Korean sàm and Japanese asa, meaning hemp. Also, the doublet wo meaning hemp is attested in Western Old Japanese and it is thus plausible to assume a borrowed term. Among ancient languages, various relatives of Korean have been proposed. Some classify the language of Jeju Island as a distinct modern Koreanic language, Other famous theories are the Dravido-Korean languages theory and the mostly unknown southern-theory which suggest an Austronesian relation. Korean is spoken by the Korean people in North Korea and South Korea and by the Korean diaspora in countries including the Peoples Republic of China, the United States, Japan. Korean-speaking minorities exist in these states, but because of cultural assimilation into host countries, Korean is the official language of South Korea and North Korea
Korean language
–
Two names for Korean, Hangugeo and Chosŏnmal,
written vertically in
Hangul
Korean language
–
Street signs in Korean; Daegu, Korea.
Korean language
–
Korean writing systems
63.
Units of measurement
–
A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same quantity. Any other value of quantity can be expressed as a simple multiple of the unit of measurement. For example, length is a physical quantity, the metre is a unit of length that represents a definite predetermined length. When we say 10 metres, we actually mean 10 times the definite predetermined length called metre, the definition, agreement, and practical use of units of measurement have played a crucial role in human endeavour from early ages up to this day. Different systems of units used to be very common, now there is a global standard, the International System of Units, the modern form of the metric system. In trade, weights and measures is often a subject of regulation, to ensure fairness. The International Bureau of Weights and Measures is tasked with ensuring worldwide uniformity of measurements, metrology is the science for developing nationally and internationally accepted units of weights and measures. In physics and metrology, units are standards for measurement of quantities that need clear definitions to be useful. Reproducibility of experimental results is central to the scientific method, a standard system of units facilitates this. Scientific systems of units are a refinement of the concept of weights, science, medicine, and engineering often use larger and smaller units of measurement than those used in everyday life and indicate them more precisely. The judicious selection of the units of measurement can aid researchers in problem solving, in the social sciences, there are no standard units of measurement and the theory and practice of measurement is studied in psychometrics and the theory of conjoint measurement. A unit of measurement is a quantity of a physical property. Units of measurement were among the earliest tools invented by humans, primitive societies needed rudimentary measures for many tasks, constructing dwellings of an appropriate size and shape, fashioning clothing, or bartering food or raw materials. Weights and measures are mentioned in the Bible and it is a commandment to be honest and have fair measures. As of the 21st Century, multiple unit systems are used all over the world such as the United States Customary System, the British Customary System, however, the United States is the only industrialized country that has not yet completely converted to the Metric System. The systematic effort to develop an acceptable system of units dates back to 1790 when the French National Assembly charged the French Academy of Sciences to come up such a unit system. After this treaty was signed, a General Conference of Weights, the CGPM produced the current SI system which was adopted in 1954 at the 10th conference of weights and measures. Currently, the United States is a society which uses both the SI system and the US Customary system
Units of measurement
–
The former Weights and Measures office in
Seven Sisters, London
Units of measurement
–
Units of measurement,
Palazzo della Ragione,
Padua
Units of measurement
–
An example of
metrication in 1860 when Tuscany became part of modern Italy (ex. one "libbra" = 339,54 grams)
64.
Units of information
–
In computing and telecommunications, a unit of information is the capacity of some standard data storage system or communication channel, used to measure the capacities of other systems and channels. In information theory, units of information are used to measure the information contents or entropy of random variables. The most common units are the bit, the capacity of a system which can exist in two states, and the byte, which is equivalent to eight bits. Multiples of these units can be formed from these with the SI prefixes or the newer IEC binary prefixes, information capacity is a dimensionless quantity. In particular, if b is an integer, then the unit is the amount of information that can be stored in a system with N possible states. When b is 2, the unit is the shannon, equal to the content of one bit. A system with 8 possible states, for example, can store up to log28 =3 bits of information, other units that have been named include, Base b =3, the unit is called trit, and is equal to log23 bits. Base b =10, the unit is called decimal digit, hartley, ban, decit, or dit, Base b = e, the base of natural logarithms, the unit is called a nat, nit, or nepit, and is worth log2 e bits. Several conventional names are used for collections or groups of bits, a byte can represent 256 distinct values, such as the integers 0 to 255, or -128 to 127. The IEEE 1541-2002 standard specifies B as the symbol for byte, bytes, or 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 bytes or groups of bytes, a group of four bits, or half a byte, is sometimes called a nibble or nybble. This unit is most often used in the context of number representations. Computers usually manipulate bits in groups of a size, conventionally called words. The number of bits in a word is defined by the size of the registers in the computers CPU. Some machine instructions and computer number formats use two words, or four words, computer memory caches usually operate on blocks of memory that consist of several consecutive words. These units are customarily called cache blocks, or, in CPU caches, virtual memory systems partition the computers main storage into even larger units, traditionally called pages. Terms for large quantities of bits can be formed using the range of SI prefixes for powers of 10, e. g. kilo =103 =1000, mega- =106 =1000000. These prefixes are often used for multiples of bytes, as in kilobyte, megabyte
Units of information
–
Comparison of units of information:
bit,
trit,
nat,
ban. Quantity of information is the height of bars. Dark green level is the "Nat" unit.
65.
Quantum information
–
In physics and computer science, quantum information is information that is held in the state of a quantum system. Quantum information is the entity of study in quantum information theory. Quantum information differs strongly from classical information, epitomized by the bit, in many striking, among these are the following, A unit of quantum information is the qubit. Unlike classical 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 quantum information. The reason for this indivisibility is due to the Heisenberg uncertainty principle, despite the state being continuously-valued. A qubit cannot be converted into classical bits, that is, despite the awkwardly-named 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 can neither be copied, nor destroyed. This is the content of the no cloning theorem and the no-deleting theorem, although a single qubit can be transported from place to place, it cannot be delivered to multiple recipients, this is the no-broadcast theorem, and is essentially implied by the no-cloning theorem. Qubits can be changed, by applying linear transformations or quantum gates to them, Classical bits may be combined with and extracted from configurations of multiple qubits, through the use of quantum gates. That is, two or more qubits can be arranged in such a way as to convey classical bits, the simplest such configuration is the Bell state, which consists of two qubits and four classical bits. Quantum information can be moved about, in a quantum channel, Quantum messages have a finite size, measured in qubits, quantum channels have a finite channel capacity, measured in qubits per second. Multiple qubits can be used to carry classical bits, although n qubits can carry more than n classical bits of information, the greatest amount of classical information that can be retrieved is n. Quantum information, and changes in information, can be quantitatively measured by using an analogue of Shannon entropy. Given a statistical ensemble of mechanical systems with the density matrix ρ. Many of the entropy measures in classical information theory can also be generalized to the quantum case, such as Holevo entropy. Quantum algorithms have a different computational complexity than classical algorithms, the most famous example of this is Shors factoring algorithm, which is not known to have a polynomial time classical algorithm, but does have a polynomial time quantum algorithm. Other examples include Grovers search algorithm, where the algorithm gives a quadratic speed-up over the best possible classical algorithm. Quantum key distribution allows unconditionally secure transmission of information, unlike classical encryption
Quantum information
–
General
66.
Pre-Columbian
–
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 used is Pre-Hispanic. Other civilizations were contemporary with the period and were described in European historical accounts of the time. A few, such as the Maya civilization, had their own written records, because many Christian Europeans of the time viewed such texts as heretical, men like Diego de Landa destroyed many texts in pyres, even while seeking to preserve native histories. Only a few documents have survived in their original languages, while others were transcribed or dictated into Spanish, giving modern historians glimpses of ancient culture. Indigenous American cultures continue to evolve after the pre-Columbian era, many of these peoples and their descendants continue traditional practices, while evolving and adapting new cultural practices and technologies into their lives. Now, the study of pre-Columbian cultures is most often based on scientific. Asian nomads are thought to have entered the Americas via the Bering Land Bridge, now the Bering Strait, 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. The first is the short chronology theory with the first movement beyond Alaska into the New World occurring no earlier than 14, 000–17,000 years ago, followed by successive waves of immigrants. The second belief is the long chronology theory, which proposes that the first group of people entered the hemisphere at an earlier date, possibly 50. In that case, the Eskimo peoples would have arrived separately and at a later date, probably no more than 2,000 years ago. The North American climate was unstable as the ice age receded and it finally stabilized by about 10,000 years ago, climatic conditions were then very similar to todays. Within this timeframe, roughly pertaining to the Archaic Period, numerous archaeological cultures have been identified, the unstable climate led to widespread migration, with early Paleo-Indians soon spreading throughout the Americas, diversifying into many hundreds of culturally distinct tribes. The paleo-indians were hunter-gatherers, likely characterized by small, mobile bands consisting of approximately 20 to 50 members of an extended family and these groups moved from place to place as preferred resources were depleted and new supplies were sought. During much of the Paleo-Indian period, bands are thought to have subsisted primarily through hunting now-extinct giant land animals such as mastodon, Paleo-Indian groups carried a variety of tools
Pre-Columbian
–
Hopewell mounds from the
Mound City Group in Ohio
Pre-Columbian
–
hunter-gatherers
Pre-Columbian
–
Mississippian site in Arkansas,
Parkin Site, circa 1539. Illustration by Herb Roe.
Pre-Columbian
–
One of the pyramids in the upper level of
Yaxchilán
67.
Toe
–
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, humans, and other animals that walk on the soles of their feet, are described as being plantigrade, unguligrade animals are those that walk on hooves at the tips of their toes. There are five toes present on each human foot, each toe consists of three phalanx bones, the proximal, middle and distal, with the exception of the big toe. The hallux only contains two bones, the proximal and distal. The phalanx bones of the toe join to the bones of the foot at the interphalangeal joints. Outside the hallux bones is skin, and present on all five toes is a toenail, the toes are, from medial to lateral, The first toe, also known as the hallux, the innermost toe, the thumb toe. With the exception of the hallux, toe movement is governed by action of the flexor digitorum brevis. These attach to the sides of the bones, making it impossible to move individual toes independently, muscles between the toes on their top and bottom also help to abduct and adduct the toes. Additional flexion control is provided by the flexor hallucis brevis and it is extended by the abductor hallucis muscle and the adductor hallucis muscle. The little toe has a set of control muscles and tendon attachments. Numerous other foot muscles contribute to motor control of the foot. The connective tendons between the minor toes accounts for the inability to actuate individual toes, the toes receive blood from the digital branches of the plantar metatarsal arteries and drain blood into the dorsal venous arch of the foot. Sensation to the skin of the toes is provided by five nerves, in humans, the hallux is usually longer than the second toe, but in some individuals, it may not be the longest toe. There is a trait in humans, where the dominant gene causes a longer second toe while the homozygous recessive genotype presents with the more common trait. People with the genetic disease fibrodysplasia ossificans progressiva characteristically have a short hallux which appears to turn inward, or medially. Humans usually have five toes on each foot, when more than five toes are present, this is known as polydactyly. Other variants may include syndactyly or arachnodactyly, forefoot shape, including toe shape, exhibits significant variation among people, these differences can be measured and have been statistically correlated with ethnicity. Such deviations may affect comfort and fit for various shoe types, research conducted for the U. S. Army indicated that larger feet may still have smaller arches, toe length, and toe-breadth
Toe
–
Toes on the
foot. The innermost toe (bottom-left in image), which is normally called the big toe, is the
hallux.
Toe
–
An individual's toes that follow the common trend of the hallux outsizing the second toe.
Toe
–
View of the feet in
Michellangelo's David
68.
Papua New Guinea
–
Its capital, located along its southeastern coast, is Port Moresby. The western half of New Guinea forms the Indonesian provinces of Papua, Papua New Guinea is one of the most culturally diverse countries in the world. There are 852 known languages in the country, of which 12 have no known living speakers, most of the population of more than 7 million people live in customary communities, which are as diverse as the languages. It is also one of the most rural, as only 18 percent of its live in urban centres. The country is one of the worlds least explored, culturally and geographically and it is known to have numerous groups of uncontacted peoples, and researchers believe there are many undiscovered species of plants and animals in the interior. Papua New Guinea is classified as an economy by the International Monetary Fund. Strong growth in Papua New Guineas mining and resource sector led to the becoming the sixth fastest-growing economy in the world in 2011. Growth was expected to slow once major resource projects came on line in 2015, mining remains a major economic factor, however. Local and national governments are discussing the potential of resuming mining operations in Panguna mine in Bougainville Province, nearly 40 percent of the population lives a self-sustainable natural lifestyle with no access to global capital. Most of the still live in strong traditional social groups based on farming. Their social lives combine traditional religion with modern practices, including primary education, at the national level, after being ruled by three external powers since 1884, Papua New Guinea established its sovereignty in 1975. This followed nearly 60 years of Australian administration, which started during the Great War and it became an independent Commonwealth realm with Queen Elizabeth II as its head of state and became a member of the Commonwealth of Nations in its own right. Archaeological evidence indicates that humans first arrived in Papua New Guinea around 42,000 to 45,000 years ago and they were descendants of migrants out of Africa, in one of the early waves of human migration. Agriculture was independently developed in the New Guinea highlands around 7000 BC, a major migration of Austronesian-speaking peoples to coastal regions of New Guinea took place around 500 BC. This has been correlated with the introduction of pottery, pigs, in the 18th century, traders brought the sweet potato to New Guinea, where it was adopted and became part of the staples. Portuguese traders had obtained it from South America and introduced it to the Moluccas, the far higher crop yields from sweet potato gardens radically transformed traditional agriculture and societies. Sweet potato largely supplanted the previous staple, taro, and resulted in a significant increase in population in the highlands. In 1901, on Goaribari Island in the Gulf of Papua, missionary Harry Dauncey found 10,000 skulls in the islands Long Houses, traders from Southeast Asia had visited New Guinea beginning 5,000 years ago to collect bird of paradise plumes
Papua New Guinea
–
Kerepunu villagers, British New Guinea, 1885.
Papua New Guinea
–
Flag
Papua New Guinea
–
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
–
Australian forces attack Japanese positions during the
Battle of Buna–Gona. 7 January 1943.
69.
Pentadecimal
–
This is a list of numeral systems, that is, writing systems for expressing numbers. Numeral systems are classified here as to whether they use positional notation, the common names are derived somewhat arbitrarily from a mix of Latin and Greek, in some cases including roots from both languages within a single name. In this Youtube video, Matt Parker jokingly invented a base-1082 system and this turns out to be 1925. Radix Radix economy Table of bases List of numbers in various languages Numeral prefix
Pentadecimal
–
Numeral systems
70.
10 (number)
–
10 is an even natural number following 9 and preceding 11. Ten is the base of the numeral system, by far the most common system of denoting numbers in both spoken and written language. The reason for the choice of ten is assumed to be that humans have ten fingers, a collection of ten items is called a decade. The ordinal adjective is decimal, the adjective is denary. Increasing a quantity by one order of magnitude is most widely understood to mean multiplying the quantity by ten, to reduce something by one tenth is to decimate. A theoretical highest number in topics that require a rating, by contrast having 0 or 1 as the lowest number, Ten is a composite number, its proper divisors being 1,2 and 5. Ten is the smallest noncototient, a number that cannot be expressed as the difference between any integer and the number of coprimes below it. Ten is the discrete semiprime and the second member of the discrete semiprime family. Ten has an aliquot sum σ of 8 and is accordingly the first discrete semiprime to be in deficit, all subsequent discrete semiprimes are in deficit. The aliquot sequence for 10 comprises five members with this number being the second member of the 7-aliquot tree. Ten is the smallest semiprime that is the sum of all the prime numbers from its lower factor through its higher factor Only three other small semiprimes share this attribute. It is the sum of only one number the discrete semiprime 14. Ten is the sum of the first three numbers, of the four first numbers, of the square of the two first odd numbers and also of the first four factorials. Ten is the eighth Perrin number, preceded in the sequence by 5,5,7, a polygon with ten sides is a decagon, and 10 is a decagonal number. Because 10 is the product of a power of 2 with nothing but distinct Fermat primes, Ten is also a triangular number, a centered triangular number, and a tetrahedral number. Ten is the number of n queens problem solutions for n =5, Ten is the smallest number whose status as a possible friendly number is unknown. As is the case for any base in its system, ten is the first two-digit number in decimal, any integer written in the decimal system can be multiplied by ten by adding a zero to the end. The Roman numeral for ten is X, it is thought that the V for five is derived from an open hand, incidentally, the Chinese word numeral for ten, is also a cross, 十
10 (number)
–
10
playing cards of all four suits
10 (number)
–
The
tetractys
71.
Algorism
–
Algorism is the technique of performing basic arithmetic by writing numbers in place value form and applying a set of memorized rules and facts to the digits. One who practices algorism is known as an algorist and this title means Algoritmi on the numbers of the Indians, where Algoritmi was the translators Latinization of Al-Khwarizmis name. Al-Khwarizmi was the most widely read mathematician in Europe in the late Middle Ages, primarily through his other book, in late medieval Latin, algorismus, the corruption of his name, simply meant the decimal number system that is still the meaning of modern English algorism. In 17th century French the words form, but not its meaning, changed to algorithm, following the model of the word logarithm, this form alluding to the ancient Greek arithmos = number. English adopted the French very soon afterwards, but it wasnt until the late 19th century that took on the meaning that it has in modern English. In English, it was first used about 1230 and then by Chaucer in 1391, another early use of the word is from 1240, in a manual titled Carmen de Algorismo composed by Alexandre de Villedieu. It begins thus, Haec algorismus ars praesens dicitur, in qua / Talibus Indorum fruimur bis quinque figuris, which translates as, This present art, in which we use those twice five Indian figures, is called algorismus. The word algorithm also derives from algorism, a generalization of the meaning to any set of rules specifying a computational procedure, occasionally algorism is also used in this generalized meaning, especially in older texts. These included the concept of the decimal fractions as an extension of the notation and this system was popularized in Europe by Leonardo of Pisa, now known as Fibonacci. Algorithmic art Positional notation Hindu–Arabic numeral system History of the Hindu–Arabic numeral system
Algorism
–
Calculating-Table by Gregor Reisch: Margarita Philosophica, 1508
72.
Decimal computer
–
Some also had a variable wordlength, which enabled operations on numbers with a large number of digits. Early computers that were exclusively decimal include the ENIAC, IBM NORC, IBM650, IBM1620, IBM7070. In these machines the basic unit of data was the decimal digit, encoded in one of several schemes, including binary-coded decimal or BCD, bi-quinary, excess-3, except for the 1620, these machines used word addressing. When non-numeric characters were used in machines, they were encoded as two decimal digits. Other early computers were character oriented, providing instructions for performing arithmetic on character strings of decimal numerals, on these machines the basic data element was an alphanumeric character, typically encoded in six bits. UNIVAC I and UNIVAC II used word addressing, with 12-character words, IBM examples include IBM702, IBM705, the IBM1400 series, IBM7010, and the IBM7080. It used 8-bit characters and introduced EBCDIC encoding, though ASCII was also supported, several microprocessor families offer limited decimal support. For example, the 80x86 family of microprocessors provide instructions to convert one-byte BCD numbers to binary format before or after arithmetic operations and these operations were not extended to wider formats and hence are now slower than using 32-bit or wider BCD tricks to compute in BCD. The x87 FPU has instructions to convert 10-byte packed decimal data, Decimal arithmetic is now becoming more common, for instance, three decimal types with two binary encodings were added to the 2008 IEEE 754r standard, with 7-, 16-, and 34-digit decimal significands. The IBM Power6 processor and the IBM System z9 have implemented these types using the Densely Packed Decimal binary encoding, the first in hardware, binghamton, New York, USA, John Wiley & Sons, Inc. Malabar, Florida, USA, Robert E. Krieger Publishing Company, Iowa City, Iowa, USA, The University of Iowa, Department of Computer Science
Decimal computer
–
IBM 650 front panel with
bi-quinary coded decimal displays
73.
Decimal separator
–
A decimal mark is a symbol used to separate the integer part from the fractional part of a number written in decimal form. Different countries officially designate different symbols for the decimal mark, the choice of symbol for the decimal mark also affects the choice of symbol for the thousands separator used in digit grouping, so the latter is also treated in this article. In mathematics the decimal mark is a type of radix point, in the Middle Ages, before printing, a bar over the units digit was used to separate the integral part of a number from its fractional part, e. g.9995. His Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear, a similar notation remains in common use as an underbar to superscript digits, especially for monetary values without a decimal mark, e. g.9995. Later, a separatrix between the units and tenths position became the norm among Arab mathematicians, e. g. 99ˌ95, when this character was typeset, it was convenient to use the existing comma or full stop instead. The separatrix was also used in England as an L-shaped or vertical bar before the popularization of the period, gerbert of Aurillac marked triples of columns with an arc when using his Hindu–Arabic numeral-based abacus in the 10th century. Fibonacci followed this convention when writing numbers such as in his influential work Liber Abaci in the 13th century, in France, the full stop was already in use in printing to make Roman numerals more readable, so the comma was chosen. Many other countries, such as Italy, also chose to use the comma to mark the decimal units position and it has been made standard by the ISO for international blueprints. However, English-speaking countries took the comma to separate sequences of three digits, in some countries, a raised dot or dash may be used for grouping or decimal mark, this is particularly common in handwriting. In the United States, the stop or period was used as the standard decimal mark. g. However, as the mid dot was already in use in the mathematics world to indicate multiplication. In the event, the point was decided on by the Ministry of Technology in 1968, the three most spoken international auxiliary languages, Ido, Esperanto, and Interlingua, all use the comma as the decimal mark. Interlingua has used the comma as its decimal mark since the publication of the Interlingua Grammar in 1951, Esperanto also uses the comma as its official decimal mark, while thousands are separated by non-breaking spaces,12345678,9. Idos Kompleta Gramatiko Detaloza di la Linguo Internaciona Ido officially states that commas are used for the mark while full stops are used to separate thousands, millions. So the number 12,345,678.90123 for instance, the 1931 grammar of Volapük by Arie de Jong uses the comma as its decimal mark, and uses the middle dot as the thousands separator. In 1958, disputes between European and American delegates over the representation of the decimal mark nearly stalled the development of the ALGOL computer programming language. ALGOL ended up allowing different decimal marks, but most computer languages, the 22nd General Conference on Weights and Measures declared in 2003 that the symbol for the decimal marker shall be either the point on the line or the comma on the line. It further reaffirmed that numbers may be divided in groups of three in order to facilitate reading, neither dots nor commas are ever inserted in the spaces between groups
Decimal separator
–
Numeral systems
74.
Scientific notation
–
Scientific notation is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. It is commonly used by scientists, mathematicians and engineers, in part because it can simplify certain arithmetic operations, on scientific calculators it is known as SCI display mode. In scientific notation all numbers are written in the form m × 10n, where the exponent n is an integer, however, the term mantissa may cause confusion because it is the name of the fractional part of the common logarithm. If the number is then a minus sign precedes m. In normalized notation, the exponent is chosen so that the value of the coefficient is at least one. Decimal floating point is an arithmetic system closely related to scientific notation. Any given integer can be written in the form m×10^n in many ways, in normalized scientific notation, the exponent n is chosen so that the absolute value of m remains at least one but less than ten. Thus 350 is written as 3. 5×102 and this form allows easy comparison of numbers, as the exponent n gives the numbers order of magnitude. In normalized notation, the exponent n is negative for a number with absolute value between 0 and 1, the 10 and exponent are often omitted when the exponent is 0. Normalized scientific form is the form of expression of large numbers in many fields, unless an unnormalized form. Normalized scientific notation is often called exponential notation—although the latter term is general and also applies when m is not restricted to the range 1 to 10. Engineering notation differs from normalized scientific notation in that the exponent n is restricted to multiples of 3, consequently, the absolute value of m is in the range 1 ≤ |m| <1000, rather than 1 ≤ |m| <10. Though similar in concept, engineering notation is rarely called scientific notation, engineering notation allows the numbers to explicitly match their corresponding SI prefixes, which facilitates reading and oral communication. A significant figure is a digit in a number that adds to its precision and this includes all nonzero numbers, zeroes between significant digits, and zeroes indicated to be significant. Leading and trailing zeroes are not significant because they exist only to show the scale of the number. Therefore,1,230,400 usually has five significant figures,1,2,3,0, and 4, when a number is converted into normalized scientific notation, it is scaled down to a number between 1 and 10. All of the significant digits remain, but the place holding zeroes are no longer required, thus 1,230,400 would become 1.2304 ×106. However, there is also the possibility that the number may be known to six or more significant figures, thus, an additional advantage of scientific notation is that the number of significant figures is clearer
Scientific notation
–
A calculator display showing the
Avogadro constant in E notation
75.
SI prefix
–
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 use today are decadic, historically there have been a number of binary metric prefixes as well. Each prefix has a symbol that is prepended to the unit symbol. The prefix kilo-, for example, may be added to gram to indicate multiplication by one thousand, the prefix milli-, 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 system with six dating back to the systems introduction in the 1790s. Metric prefixes have even been prepended to non-metric units, the SI prefixes are standardized for use in the International System of Units by the International Bureau of Weights and Measures in resolutions dating from 1960 to 1991. Since 2009, they have formed part of the International System of Quantities, the BIPM specifies twenty prefixes for the International System of Units. Each prefix name has a symbol which is used in combination with the symbols for units of measure. For example, the symbol for kilo- is k, and is used to produce km, kg, and kW, which are the SI symbols for kilometre, kilogram, prefixes corresponding to an integer power of one thousand are generally preferred. Hence 100 m is preferred over 1 hm or 10 dam, the prefixes hecto, deca, deci, and centi are commonly used for everyday purposes, and the centimetre is especially common. However, some building codes require that the millimetre be used in preference to the centimetre, because use of centimetres leads to extensive usage of decimal points. Prefixes may not be used in combination and this also applies to mass, for which the SI base unit already contains a prefix. For example, milligram is used instead of microkilogram, in the arithmetic of measurements having units, the units are treated as multiplicative factors to values. If they have prefixes, all but one of the prefixes must be expanded to their numeric multiplier,1 km2 means one square kilometre, or the area of a square of 1000 m by 1000 m and not 1000 square metres. 2 Mm3 means two cubic megametres, or the volume of two cubes of 1000000 m by 1000000 m by 1000000 m or 2×1018 m3, and not 2000000 cubic metres, examples 5 cm = 5×10−2 m =5 ×0.01 m =0. The prefixes, including those introduced after 1960, are used with any metric unit, metric prefixes may also be used with non-metric units. The choice of prefixes with a unit is usually dictated by convenience of use. Unit prefixes for amounts that are larger or smaller than those actually encountered are seldom used
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.
76.
Encyclopedia of Mathematics
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The Encyclopedia of Mathematics is a large reference work in mathematics. It is available in form and on CD-ROM. The 2002 version contains more than 8,000 entries covering most areas of mathematics at a level. The encyclopedia is edited by Michiel Hazewinkel and was published by Kluwer Academic Publishers until 2003, the CD-ROM contains animations and three-dimensional objects. Until November 29,2011, a version of the encyclopedia could be browsed online free of charge online This URL now redirects to the new wiki incarnation of the EOM. A new dynamic version of the encyclopedia is now available as a public wiki online and this new wiki is a collaboration between Springer and the European Mathematical Society. This new version of the encyclopedia includes the entire contents of the online version. All entries will be monitored for content accuracy by members of a board selected by the European Mathematical Society. Vinogradov, I. M. Matematicheskaya entsiklopediya, Moscow, Sov, Hazewinkel, M. Encyclopaedia of Mathematics, Kluwer,1994. Hazewinkel, M. Encyclopaedia of Mathematics, Vol.1, Hazewinkel, M. Encyclopaedia of Mathematics, Vol.2, Kluwer,1988. Hazewinkel, M. Encyclopaedia of Mathematics, Vol.3, Hazewinkel, M. Encyclopaedia of Mathematics, Vol.4, Kluwer,1989. Hazewinkel, M. Encyclopaedia of Mathematics, Vol.5, Hazewinkel, M. Encyclopaedia of Mathematics, Vol.6, Kluwer,1990. Hazewinkel, M. Encyclopaedia of Mathematics, Vol.7, Hazewinkel, M. Encyclopaedia of Mathematics, Vol.8, Kluwer,1992. Hazewinkel, M. Encyclopaedia of Mathematics, Vol.9, Hazewinkel, M. Encyclopaedia of Mathematics, Vol.10, Kluwer,1994. Hazewinkel, M. Encyclopaedia of Mathematics, Supplement I, Kluwer,1997, Hazewinkel, M. Encyclopaedia of Mathematics, Supplement II, Kluwer,2000. Hazewinkel, M. Encyclopaedia of Mathematics, Supplement III, Kluwer,2002, Hazewinkel, M. Encyclopaedia of Mathematics on CD-ROM, Kluwer,1998. Encyclopedia of Mathematics, public wiki monitored by a board under the management of the European Mathematical Society. List of online encyclopedias Current page of M. Hazewinkel Online Encyclopedia of Mathematics
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.
77.
Hermann Schmid (computer scientist)
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Some also had a variable wordlength, which enabled operations on numbers with a large number of digits. Early computers that were exclusively decimal include the ENIAC, IBM NORC, IBM650, IBM1620, IBM7070. In these machines the basic unit of data was the decimal digit, encoded in one of several schemes, including binary-coded decimal or BCD, bi-quinary, excess-3, except for the 1620, these machines used word addressing. When non-numeric characters were used in machines, they were encoded as two decimal digits. Other early computers were character oriented, providing instructions for performing arithmetic on character strings of decimal numerals, on these machines the basic data element was an alphanumeric character, typically encoded in six bits. UNIVAC I and UNIVAC II used word addressing, with 12-character words, IBM examples include IBM702, IBM705, the IBM1400 series, IBM7010, and the IBM7080. It used 8-bit characters and introduced EBCDIC encoding, though ASCII was also supported, several microprocessor families offer limited decimal support. For example, the 80x86 family of microprocessors provide instructions to convert one-byte BCD numbers to binary format before or after arithmetic operations and these operations were not extended to wider formats and hence are now slower than using 32-bit or wider BCD tricks to compute in BCD. The x87 FPU has instructions to convert 10-byte packed decimal data, Decimal arithmetic is now becoming more common, for instance, three decimal types with two binary encodings were added to the 2008 IEEE 754r standard, with 7-, 16-, and 34-digit decimal significands. The IBM Power6 processor and the IBM System z9 have implemented these types using the Densely Packed Decimal binary encoding, the first in hardware, binghamton, New York, USA, John Wiley & Sons, Inc. Malabar, Florida, USA, Robert E. Krieger Publishing Company, Iowa City, Iowa, USA, The University of Iowa, Department of Computer Science
Hermann Schmid (computer scientist)
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IBM 650 front panel with
bi-quinary coded decimal displays
78.
Mike Cowlishaw
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Mike F. Cowlishaw is a Visiting Professor at the Department of Computer Science at the University of Warwick, and a Fellow of the Royal Academy of Engineering. He is a retired IBM Fellow, and was a Fellow of the Institute of Engineering and Technology, and he was educated at Monkton Combe School and The University of Birmingham. Cowlishaw joined IBM in 1974 as an engineer but is best known as a programmer and writer. He has contributed to and/or edited various computing standards, including ISO, BSI, ANSI, IETF, W3C, ECMA and he retired from IBM in March 2010. Cowlishaws decimal arithmetic specification formed the proposal for the parts of the IEEE754 standard, as well as being followed by many implementations, such as Python. His decNumber decimal package is available as open source under several licenses and is now part of GCC. They are integrated into the IBM Power6 and IBM System z10 processor cores, and in numerous IBM software products such as DB2, TPF, WebSphere MQ, operating systems, Cowlishaw wrote an emulator for the Acorn System 1, and collected related documentation. Outside computing, he caved in the UK, New England, Spain and he is a life member of the National Speleological Society, wrote articles in the 1970s and 1980s on battery technology and on the shock strength of caving ropes, and designed LED-based caving lamps. The NetRexx Language, Cowlishaw, M. F. ISBN 0-13-806332-X, Prentice-Hall,1997 The REXX Language, Cowlishaw, Proceedings 16th IEEE Symposium on Computer Arithmetic, ISBN 0-7695-1894-X, pp. 104–111, IEEE Comp. June 2003 Densely Packed Decimal Encoding, Cowlishaw, M. F. IEE Proceedings – Computers and Digital Techniques ISSN 1350-2387, Vol.149, june 2001 NetRexx – an alternative for writing Java classes, Cowlishaw, M. F. 4, Winter 1994, pp. 15–24 A large-scale computer conferencing system, Chess and Cowlishaw, IBM Systems Journal, Vol 26,1,1987, 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, IBM Systems Journal, Volume 23, No. 4,1984, IBM Reprint order number G321-5228 The Characteristics and Use of Lead-Acid Cap Lamps, Cowlishaw, British Cave Research Association, Vol 1, No
Mike Cowlishaw
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Michael F. Cowlishaw
79.
Joseph Needham
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Noel Joseph Terence Montgomery Needham CH FRS FBA was a British scientist, historian and sinologist known for his scientific research and writing on the history of Chinese science. He was elected a fellow of the Royal Society in 1941, in 1992, the Queen conferred on him the Companionship of Honour, and the Royal Society noted he was the only living person to hold these three titles. Needham was the child of a London family. His father was a doctor, and his mother, Alicia Adelaïde, née Montgomery, was a composer from Oldcastle. Needham was educated at Oundle School before attending Gonville and Caius College, Cambridge, where he received a BA in 1921, an Oxbridge MA in January 1925 and he had intended to study medicine but came under the influence of Frederick Hopkins, resulting in his switch to biochemistry. His three-volume work Chemical Embryology, published in 1931, includes a history of embryology from Egyptian times up to the early 19th century and his Terry Lecture of 1936 was published by Cambridge University Press in association with Yale University Press under the title of Order and Life. In 1939 he produced a work on morphogenesis that a Harvard reviewer claimed will go down in the history of science as Joseph Needhams magnum opus. Although his career as biochemist and an academic was well established, his career developed in unanticipated directions during, three Chinese scientists came to Cambridge for graduate study in 1937, Lu Gwei-djen, Wang Ying-lai and Shen Shih-Chang. Lu, daughter of a Nanjingese pharmacist, taught Needham Chinese, igniting his interest in Chinas ancient technological and he then pursued, and mastered, the study of Classical Chinese privately with Gustav Haloun. Under the Royal Societys direction, Needham was the director of the Sino-British Science Co-operation Office in Chongqing from 1942 to 1946, during this time he made several long journeys through war-torn China and many smaller ones, visiting scientific and educational establishments and obtaining for them much needed supplies. His longest trip in late 1943 ended in far west in Gansu at the caves in Dunhuang at the end of the Great Wall where the earliest dated printed book - a copy of the Diamond Sutra - was found. The other long trip reached Fuzhou on the east coast, returning across the Xiang River just two days before the Japanese blew up the bridge at Hengyang and cut off part of China. In 1944 he visited Yunnan in an attempt to reach the Burmese border, everywhere he went he purchased and 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. In fact it was Needham who insisted that science should be included in the organisations mandate at a planning meeting. He devoted his energy to the history of Chinese science until his retirement in 1990, in 1948, Needham proposed a project to the Cambridge University Press for a book on Science and Civilisation in China. Within weeks of being accepted, the project had grown to seven volumes and his initial collaborator was the historian Wang Ling, whom he had met in Lizhuang and obtained a position for at Trinity. The first years were devoted to compiling a list of every mechanical invention and abstract idea that had been made and conceived in China
Joseph Needham
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Joseph Needham