Kharosthi
The Kharosthi script spelled Kharoshthi or Kharoṣṭhī, was an ancient Indian script used in Gandhara to write Gandhari Prakrit and Sanskrit. It was popular in Central Asia as well. An abugida, it was introduced at least by the middle of the 3rd century BCE during the 4th century BCE, remained in use until it died out in its homeland around the 3rd century CE, it was in use in Bactria, the Kushan Empire and along the Silk Road, where there is some evidence it may have survived until the 7th century in the remote way stations of Khotan and Niya. Kharosthi is encoded in the Unicode range U+10A00–U+10A5F, from version 4.1. Kharosthi is written right to left, but some inscriptions show the left to right direction, to become universal for the South Asian scripts; each syllable includes the short /a/ sound by default, with other vowels being indicated by diacritic marks. Recent epigraphic evidence highlighted by Professor Richard Salomon of the University of Washington has shown that the order of letters in the Kharosthi script follows what has become known as the Arapacana alphabet.
As preserved in Sanskrit documents, the alphabet runs: a ra pa ca na la da ba ḍa ṣa va ta ya ṣṭa ka sa ma ga stha ja śva dha śa kha kṣa sta jñā rtha bha cha sma hva tsa gha ṭha ṇa pha ska ysa śca ṭa ḍhaSome variations in both the number and order of syllables occur in extant texts. Kharosthi includes only one standalone vowel, used for initial vowels in words. Other initial vowels use the a character modified by diacritics. Using epigraphic evidence, Salomon has established that the vowel order is /a e i o u/, rather than the usual vowel order for Indic scripts /a i u e o/; that is the same as the Semitic vowel order. There is no differentiation between long and short vowels in Kharosthi. Both are marked using the same vowel markers; the alphabet was used in Gandharan Buddhism as a mnemonic for remembering a series of verses on the nature of phenomena. In Tantric Buddhism, the list was incorporated into ritual practices and became enshrined in mantras. There are two special modified forms of these consonants: Various additional marks are used to modify vowels and consonants: Nine Kharosthi punctuation marks have been identified: Kharosthi included a set of numerals that are reminiscent of Roman numerals.
The system is based on an additive and a multiplicative principle, but does not have the subtractive feature used in the Roman number system. The numerals, like the letters, are written from right to left. There is no zero and no separate signs for the digits 5–9. Numbers in Kharosthi use an additive system. For example, the number 1996 would be written as 1000 4 4 1 100 20 20 20 20 10 4 2; the Kharosthi script was deciphered by James Prinsep using the bilingual coins of the Indo-Greek Kingdom. This in turn led to the reading of the Edicts of Ashoka, some of which, from the northwest of South Asia, were written in the Kharosthi script. Scholars are not in agreement as to whether the Kharosthi script evolved or was the deliberate work of a single inventor. An analysis of the script forms shows a clear dependency on the Aramaic alphabet but with extensive modifications to support the sounds found in Indic languages. One model is that the Aramaic script arrived with the Achaemenid Empire's conquest of the Indus River in 500 BCE and evolved over the next 200+ years, reaching its final form by the 3rd century BCE where it appears in some of the Edicts of Ashoka found in northwestern part of South Asia.
However, no intermediate forms have yet been found to confirm this evolutionary model, rock and coin inscriptions from the 3rd century BCE onward show a unified and standard form. An inscription in Aramaic dating back to the 4th century BCE was found in Sirkap, testifying to the presence of the Aramaic script in northwestern India at that period. According to Sir John Marshall, this seems to confirm that Kharoshthi was developed from Aramaic; the study of the Kharosthi script was invigorated by the discovery of the Gandhāran Buddhist texts, a set of birch bark manuscripts written in Kharosthi, discovered near the Afghan city of Hadda just west of the Khyber Pass in Pakistan. The manuscripts were donated to the British Library in 1994; the entire set of manuscripts are dated to the 1st century CE, making them the oldest Buddhist manuscripts yet discovered. Kharosthi was added to the Unicode Standard in March, 2005 with the release of version 4.1. The Unicode block for Kharosthi is U+10A00–U+10A5F: Brahmi History of Afghanistan History of Pakistan Pre-Islamic scripts in Afghanistan Kaschgar und die Kharoṣṭhī Dani, Ahmad Hassan.
Kharoshthi Primer, Lahore Museum Publication Series - 16, Lahore, 1979 Falk, Harry. Schrift im alten Indien: Ein Forschungsbericht mit Anmerkungen, Gunter Narr Verlag, 1993 Fussman's, Gérard. Les premiers systèmes d'écriture en Inde, in Annuaire du Collège de France 1988-1989 Hinüber, Oscar von. Der Beginn der Schrift und frühe Schriftlichkeit in Indien, Franz Steiner Verlag, 1990 Nasim Khan, M.. Ashokan Inscriptions: A Palaeographical Study. Atthariyyat, Vol. I, pp. 131–150. Peshawar Nasim Khan, M.. Two Dated Kharoshthi Inscriptions from Gandhara. Journal of Asian Civilizations, Vol. XXII, No.1, July 1999: 99-103. Nasim Khan, M.. An Inscribed Relic-Casket from Dir; the Journal of Humanities and Social Sciences, Vol. V, No. 1, March 1997: 21-33. Peshawar Nasim Khan, M.. Kharoshthi Inscription from Swabi - Gandhara; the Journal of Humanities and Soc
Hebrew numerals
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 BCE; the current numeral system is known as the Hebrew alphabetic numerals to contrast with earlier systems of writing numerals used in classical antiquity. These systems were inherited from usage in the Aramaic and Phoenician scripts, attested from c. 800 BC in the so-called Samaria ostraca and sometimes known as Hebrew-Aramaic numerals derived from the Egyptian Hieratic numerals. The Greek system was adopted in Hellenistic Judaism and had been in use in Greece since about the 5th century BC. In this system, there is no notation for zero, the numeric values for individual letters are added together; each unit is assigned a separate letter, each tens a separate letter, the first four hundreds a separate letter. The hundreds are represented by the sum of two or three letters representing the first four hundreds.
To represent numbers from 1,000 to 999,999, the same letters are reused to serve as thousands, tens of thousands, hundreds of thousands. Gematria uses these transformations extensively. In Israel today, the decimal system of Arabic numerals is used in all cases; the Hebrew numerals are used only in special cases, such as when using the Hebrew calendar, or numbering a list, much as Roman numerals are used in the West. The Hebrew language has names for common numbers. Letters of the Hebrew alphabet are used to represent numbers in a few traditional contexts, for example in calendars. In other situations Arabic numerals are used. Cardinal and ordinal numbers must agree in gender with the noun. If there is no such noun, the feminine 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. Numbers greater than million were represented by the long scale.
Cardinal and ordinal numbers must agree in gender with the noun. If there is no such noun, the feminine form is used. Ordinal numbers must agree in number and definite status like other adjectives; the cardinal number precedes the noun, except for the number one. 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 additive 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 generate numbers from 500 and above. Alternatively, the 22-letter Hebrew numeral set is sometimes extended to 27 by using 5 sofit forms of the Hebrew letters. By convention, the numbers 15 and 16 are represented as ט״ו and ט״ז in order to refrain from using the two-letter combinations י-ה and י-ו, which are alternate written forms for the Name of God in everyday writing.
In the calendar, this manifests every full moon. Combinations which would spell out words with negative connotations are sometimes avoided by switching the order of the letters. For instance, 744 which should be written as תשמ״ד might instead be written as תשד״מ or תמש״ד; the Hebrew numeral system has sometimes been extended to include the five final letter forms—ך, ם, ן, ף and ץ —which are used to indicate the numbers from 500 to 900. The ordinary forms for 500 to 900 are: ת״ק, ת״ר, ת״ש, ת״ת and תת״ק. Gershayim are inserted before the last letter to indicate that the sequence of letters represents a number rather than a word; this is used in the case. A single Geresh is appended after a single letter to indicate that the letter represents a number rather than a word; this is used in the case. Note that Geresh and Gershayim indicate "not a word." Context determines whether they indicate a number or something else. An alternative method found in old manuscripts and still found on modern-day tombstones is to put a dot above each letter of the number.
In print, Arabic numerals are emplo
Hindu–Arabic numeral system
The Hindu–Arabic numeral system is a positional decimal numeral system, is the most common system for the symbolic representation of numbers in the world. It was invented between the 4th centuries by Indian mathematicians; the system was adopted in Arabic mathematics by the 9th century. Influential were the books of Al-Kindi; the system spread to medieval Europe by the High Middle Ages. The system is based upon ten glyphs; the symbols used to represent the system are in principle independent of the system itself. The glyphs in actual use are descended from Brahmi numerals and have split into various typographical variants since the Middle Ages; these symbol sets can be divided into three main families: Western Arabic numerals used in the Greater Maghreb and in Europe, Eastern Arabic numerals used in the Middle East, the Indian numerals used in the Indian subcontinent. The Hindu-Arabic numerals were invented by mathematicians in India. Perso-Arabic mathematicians called them "Hindu numerals", they came to be called "Arabic numerals" in Europe, because they were introduced to the West by Arab merchants.
The Hindu–Arabic system is designed for positional notation in a decimal system. In a more developed form, positional notation uses a decimal marker, a symbol for "these digits recur ad infinitum". In modern usage, this latter symbol is a vinculum. In this more developed form, the numeral system can symbolize any rational number using only 13 symbols. Although found in text written with the Arabic abjad, numbers written with these numerals place the most-significant digit to the left, so they read from left to right; the requisite changes in reading direction are found in text that mixes left-to-right writing systems with right-to-left systems. Various symbol sets are used to represent numbers in the Hindu–Arabic numeral system, most of which developed from the Brahmi numerals; the symbols used to represent the system have split into various typographical variants since the Middle Ages, arranged in three main groups: The widespread Western "Arabic numerals" used with the Latin and Greek alphabets in the table, descended from the "West Arabic numerals" which were developed in al-Andalus and the Maghreb.
The "Arabic–Indic" or "Eastern Arabic numerals" used with Arabic script, developed in what is now Iraq. A variant of the Eastern Arabic numerals is used in Urdu; the Indian numerals in use with scripts of the Brahmic family in India and Southeast Asia. Each of the dozen major scripts of India has its own numeral glyphs; as in many numbering systems, the numerals 1, 2, 3 represent simple tally marks. After three, numerals tend to become more complex symbols. Theorists believe that this is because it becomes difficult to instantaneously count objects past three; the Brahmi numerals at the basis of the system predate the Common Era. They replaced the earlier Kharosthi numerals used since the 4th century BC. Brahmi and Kharosthi numerals were used alongside one another in the Maurya Empire period, both appearing on the 3rd century BC edicts of Ashoka. Buddhist inscriptions from around 300 BC use the symbols that became 1, 4, 6. One century their use of the symbols that became 2, 4, 6, 7, 9 was recorded.
These Brahmi numerals are the ancestors of the Hindu–Arabic glyphs 1 to 9, but they were not used as a positional system with a zero, there were rather separate numerals for each of the tens. The actual numeral system, including positional notation and use of zero, is in principle independent of the glyphs used, younger than the Brahmi numerals; the place-value system is used in the Bakhshali Manuscript. Although date of the composition of the manuscript is uncertain, the language used in the manuscript indicates that it could not have been composed any than 400; the development of the positional decimal system takes its origins in Hindu mathematics during the Gupta period. Around 500, the astronomer Aryabhata uses the word kha to mark "zero" in tabular arrangements of digits; the 7th century Brahmasphuta Siddhanta contains a comparatively advanced understanding of the mathematical role of zero. The Sanskrit translation of the lost 5th century Prakrit Jaina cosmological text Lokavibhaga may preserve an early instance of positional use of zero.
These Indian developments were taken up in Islamic mathematics in the 8th century, as recorded in al-Qifti's Chronology of the scholars. The numeral system came to be known to both the Persian mathematician Khwarizmi, who wrote a book, On the Calculation with Hindu Numerals in about 825, the Arab mathematician Al-Kindi, who wrote four volumes, On the Use of the Hindu Numerals around 830; these earlier texts did not use the Hindu numerals. Kushyar ibn L
Suzhou numerals
The Suzhou numerals known as Suzhou mazi, is a numeral system used in China before the introduction of Arabic numerals. The Suzhou numerals are known as huama, jingzima and shangma; the Suzhou numeral system is the only surviving variation of the rod numeral system. The rod numeral system is a positional numeral system used by the Chinese in mathematics. Suzhou numerals are a variation of the Southern Song rod numerals. Suzhou numerals were used as shorthand in number-intensive areas of commerce such as accounting and bookkeeping. At the same time, standard Chinese numerals were used in formal writing, akin to spelling out the numbers in English. Suzhou numerals were once popular in Chinese marketplaces, such as those in Hong Kong along with local transportation before the 1990s, but they have been supplanted by Arabic numerals; this is similar to what had happened in Europe with Roman numerals used in ancient and medieval Europe for mathematics and commerce. Nowadays, the Suzhou numeral system is only used for displaying prices in Chinese markets or on traditional handwritten invoices.
In the Suzhou numeral system, special symbols are used for digits instead of the Chinese characters. The digits of the Suzhou numerals are defined between U +3029 in Unicode. An additional three code points starting from U+3038 were added later; the numbers one and three are all represented by vertical bars. This can cause confusion. Standard Chinese ideographs are used in this situation to avoid ambiguity. For example, "21" is written as "〢一" instead of "〢〡" which can be confused with "3"; the first character of such sequences is represented by the Suzhou numeral, while the second character is represented by the Chinese ideograph. The digits are positional; the full numerical notations are written in two lines to indicate numerical value, order of magnitude, unit of measurement. Following the rod numeral system, the digits of the Suzhou numerals are always written horizontally from left to right when used within vertically written documents; the first line contains the numerical values, in this example, "〤〇〢二" stands for "4022".
The second line consists of Chinese characters that represents the order of magnitude and unit of measurement of the first digit in the numerical representation. In this case "十元" which stands for "ten yuan"; when put together, it is read as "40.22 yuan". Possible characters denoting order of magnitude include: qiān for thousand bǎi for hundred shí for ten blank for oneOther possible characters denoting unit of measurement include: yuán for dollar máo or for 10 cents lǐ for the Chinese mile any other Chinese measurement unitNotice that the decimal point is implicit when the first digit is set at the ten position. Zero is represented by the character for zero. Leading and trailing zeros are unnecessary in this system; this is similar to the modern scientific notation for floating point numbers where the significant digits are represented in the mantissa and the order of magnitude is specified in the exponent. The unit of measurement, with the first digit indicator, is aligned to the middle of the "numbers" row.
In the Unicode standard version 3.0, these characters are incorrectly named Hangzhou style numerals. In the Unicode standard 4.0, an erratum was added which stated: The Suzhou numerals are special numeric forms used by traders to display the prices of goods. The use of "HANGZHOU" in the names is a misnomer. All references to "Hangzhou" in the Unicode standard have been corrected to "Suzhou" except for the character names themselves, which cannot be changed once assigned, according to the Unicode Stability Policy. In the episode "The Blind Banker" of the 2010 BBC television series Sherlock, Sherlock Holmes erroneously refers to the number system as "Hangzhou" instead of the correct "Suzhou."
Decimal
The decimal numeral system is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral system; the way of denoting numbers in the decimal system is referred to as decimal notation. A decimal numeral, or just decimal, or casually decimal number, refers to the notation of a number in the decimal numeral system. Decimals may sometimes be identified for containing a decimal separator. "Decimal" may refer to the digits after the decimal separator, such as in "3.14 is the approximation of π to two decimals". The numbers that may be represented in the decimal system are the decimal fractions, the fractions of the form a/10n, where a is an integer, n is a non-negative integer; the decimal system has been extended to infinite decimals, for representing any real number, by using an infinite sequence of digits after the decimal separator. In this context, the decimal numerals with a finite number of non–zero places after the decimal separator are sometimes called terminating decimals.
A repeating decimal is an infinite decimal that after some place repeats indefinitely the same sequence of digits. An infinite decimal represents a rational number if and only if it is a repeating decimal or has a finite number of nonzero digits. Many numeral systems of ancient civilisations use ten and its powers for representing numbers because there are ten fingers on two hands and people started counting by using their fingers. Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, Chinese numerals. Large numbers were difficult to represent in these old numeral systems, only the best mathematicians were able to multiply or divide large numbers; these difficulties were solved with the introduction of the Hindu–Arabic numeral system for representing integers. This system has been extended to represent some non-integer numbers, called decimal fractions or decimal numbers for forming the decimal numeral system. For writing numbers, the decimal system uses ten decimal digits, a decimal mark, for negative numbers, a minus sign "−".
The decimal digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. For representing a non-negative number, a decimal consists of either a sequence of digits such as 2017, or in full generality, a m a m − 1 … a 0 or two sequence of digits separated by a decimal mark such as 3.14159, 15.00, or in full generality a m a m − 1 … a 0. B 1 b 2 … b n It is assumed that, if m > 0, the first digit am is not zero, but, in some circumstances, it may be useful to have one or more 0's on the left. This does not change the value represented by the decimal. For example, 3.14 = 03.14 = 003.14. If bn =0, it may be removed, conversely, trailing zeros may be added without changing the represented number: for example, 15 = 15.0 = 15.00 and 5.2 = 5.20 = 5.200. Sometimes the extra zeros are used for indicating the accuracy of a measurement. For example, 15.00 m may indicate that the measurement error is less than one centimeter, while 15 m may mean that the length is fifteen meters, that the error may exceed 10 cm. For representing a negative number, a minus sign is placed before am.
The numeral a m a m − 1 … a 0. B 1 b 2 … b n represents the number a m 10 m + a m − 1 10 m − 1 + ⋯ + a 0 10 0 + b 1 10 1 + b 2 10 2 + ⋯ + b n 10 n Therefore, the contribution of each digit to the value of a number depends on its position in the numeral; that is, the decimal system is a positional numeral system The numbers that are represented by decimal numerals are the decimal fractions, that is, the rational numbers that may be expressed as a fraction, the denominator of, a power of ten. For example, the numerals 0.8, 14.89, 0.00024 represent the fractions 8/10, 1489/100, 24/100000. More a decimal with n digits after the separator represents the fraction with denominator 10n, whose numerator is the integer obtained by removing the separator. Expressed as a reduced fraction, the decimal numbers are those whose denominator is a product of a powe
Counting rods
Counting rods are small bars 3–14 cm long, that were used by mathematicians for calculation in ancient East Asia. They are placed either horizontally or vertically to represent any rational number; the written forms based on them are called rod numerals. 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. Chinese arithmeticians used counting rods well over two thousand years ago. In 1954 forty-odd counting rods of the Warring States period were found in Zuǒjiāgōngshān Chu Grave No.15 in Changsha, Hunan. In 1973 archeologists unearthed a number of wood scripts from a tomb in Hubei dating from the period of the Han dynasty. On one of the wooden scripts was written: "当利二月定算"; this is one of the earliest examples of using counting-rod numerals in writing. In 1976 a bundle of Western Han-era counting rods made of bones was unearthed from Qianyang County in Shaanxi; the use of counting rods must predate it. 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 cross-section, but by the time of the Sui dynasty mathematicians used triangular rods to represent positive numbers and rectangular rods for negative numbers. After the abacus flourished, counting rods were abandoned except in Japan, where rod numerals developed into a symbolic notation for algebra. Counting rods represent digits by the number of rods, the perpendicular rod represents five. To avoid confusion and horizontal forms are alternately used. Vertical rod numbers are used for the position for the units, ten thousands, etc. while horizontal rod numbers are used for the tens, hundred thousands etc. It is written in Sunzi Suanjing that "one is vertical, ten is horizontal". Red rods represent black rods represent negative numbers. Ancient Chinese understood negative numbers and zero, though they had no symbol for the latter; the Nine Chapters on the Mathematical Art, composed in the first century CE, stated " subtract same signed numbers, add different signed numbers, subtract a positive number from zero to make a negative number, subtract a negative number from zero to make a positive number".
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. For instance, in Licheng suanjin, 81 was transcribed as, 108 was transcribed as. In the same manuscript, 405 was transcribed as, with a blank space in between for obvious reasons, could in no way be interpreted as "45". In other words, transcribed rod numerals may not be positional, but on the counting board, they are positional. is an exact image of the counting rod number 405 on a table top or floor. The value of a number depends on its physical position on the counting board. A 9 at the rightmost position on the board stands for 9. Moving the batch of rods representing 9 to the left one position gives 9 or 90. Shifting left again to the third position gives 9 or 900; each time one shifts a number one position to the left, it is multiplied by 10. Each time one shifts a number one position to the right, it is divided by 10.
This applies to multiple-digit numbers. Song dynasty mathematician Jia Xian used hand-written Chinese decimal orders 步十百千萬 as rod numeral place value, as evident from a facsimile from a page of Yongle Encyclopedia, he arranged 七萬一千八百二十四 as 七一八二四 萬千百十步He treated the Chinese order numbers as place value markers, 七一八二四 became place value decimal number. He wrote the rod numerals according to their place value: In Japan, mathematicians put counting rods on a counting board, a sheet of cloth with grids, used only vertical forms relying on the grids. An 18th-century Japanese mathematics book has a checker counting board diagram, with the order of magnitude symbols "千百十一分厘毛“. Examples: Rod numerals are a positional numeral system made from shapes of counting rods. Positive numbers are written as they are and the negative numbers are written with a slant bar at the last digit; the vertical bar in the horizontal forms 6–9 are drawn shorter to have the same character height. A circle is used for 0. Many historians think it was imported from Indian numerals by Gautama Siddha in 718, but some think it was created from the Chinese text space filler "□", others think that the Indians acquired it from China, because it resembles a Confucian philosophical symbol for nothing.
In the 13th century, Southern Song mathematicians changed digits for 4, 5, 9 to reduce strokes. The new horizontal forms transformed into Suzhou numerals. Japanese continued to use the traditional forms. Examples: In Japan, Seki Takakazu developed the rod num
Binary number
In mathematics and digital electronics, a binary number is a number expressed in the base-2 numeral system or binary numeral system, which uses only two symbols: "0" and "1". The base-2 numeral system is a positional notation with a radix of 2; each digit is referred to as a bit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by all modern computers and computer-based devices; the modern binary number system was studied in Europe in the 16th and 17th centuries by Thomas Harriot, Juan Caramuel y Lobkowitz, Gottfried Leibniz. However, systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt and India. Leibniz was inspired by the Chinese I Ching; the scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions. Horus-Eye fractions are a binary numbering system for fractional quantities of grain, liquids, or other measures, in which a fraction of a hekat is expressed as a sum of the binary fractions 1/2, 1/4, 1/8, 1/16, 1/32, 1/64.
Early forms of this system can be found in documents from the Fifth Dynasty of Egypt 2400 BC, its developed hieroglyphic form dates to the Nineteenth Dynasty of Egypt 1200 BC. The method used for ancient Egyptian multiplication is closely related to binary numbers. In this method, multiplying one number by a second is performed by a sequence of steps in which a value is either doubled or has the first number added back into it; this method can be seen in use, for instance, in the Rhind Mathematical Papyrus, which dates to around 1650 BC. The I Ching dates from the 9th century BC in China; the binary notation in the I Ching is used to interpret its quaternary divination technique. It is based on taoistic duality of yin and yang.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, although he did not intend his arrangement to be used mathematically.
Viewing the least significant bit on top of single hexagrams in Shao Yong's square and reading along rows either from bottom right to top left with solid lines as 0 and broken lines as 1 or from top left to bottom right with solid lines as 1 and broken lines as 0 hexagrams can be interpreted as sequence from 0 to 63. The Indian scholar Pingala developed a binary system for describing prosody, he used binary numbers in the form of long syllables, making it similar to Morse code. Pingala's Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter; the binary representations in Pingala's system increases towards the right, not to the left like in the binary numbers of the modern, Western positional notation. 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 Asia. Sets of binary combinations similar to the I Ching have been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy.
In the late 13th century Ramon Llull had the ambition to account for all wisdom in every branch of human knowledge of the time. For that purpose he developed a general method or ‘Ars generalis’ based on binary combinations of a number of simple basic principles or categories, for which he has been considered a predecessor of computing science and artificial intelligence. In 1605 Francis Bacon discussed a system whereby letters of the alphabet could be reduced to sequences of binary digits, which could be encoded as scarcely visible variations in the font in any random text. For the general theory of binary encoding, he added that this method could be used with any objects at all: "provided those objects be capable of a twofold difference only. John Napier in 1617 described a system he called location arithmetic for doing binary calculations using a non-positional representation by letters. Thomas Harriot investigated several positional numbering systems, including binary, but did not publish his results.
The first publication of the system in Europe was by Juan Caramuel y Lobkowitz, in 1700. Leibniz studied binary numbering in 1679. Leibniz's system uses 1, like the modern binary numeral system. An example of Leibniz's binary numeral system is as follows: 0 0 0 1 numerical value 20 0 0 1 0 numerical value 21 0 1 0 0 numerical value 22 1 0 0 0 numerical value 23Leibniz interpreted the hexagrams of the I Ching as evidence of binary calculus; as a Sinophile, Leibniz was aware of