Instruct: Given a question, retrieve relevant documents that can help answer the question.
Query: When was the  Brāhmasphuṭasiddhānta written?
The Brāhmasphuṭasiddhānta ("Correctly Established Doctrine of Brahma", abbreviated BSS) is the main work of Brahmagupta, written c. 628.[1] Τhis text of mathematical astronomy contains significant mathematical content, including a good understanding of the role of zero, rules for manipulating both negative and positive numbers, a method for computing square roots, methods of solving linear and quadratic equations, and rules for summing series, Brahmagupta's identity, and Brahmagupta’s theorem.
Some common transposing instruments sound a major ninth lower than written.
These include the tenor saxophone, the bass clarinet, the baritone/euphonium when written in treble clef, and the trombone when written in treble clef (British brass band music).
In the 7th century Brahmagupta used different colours to represent the unknowns in algebraic equations in the "Brāhmasphuṭasiddhānta". One section of this book is called "Equations of Several Colours".
In the early ninth century, the caliph Al-Ma'mun ordered translations of many Greek mathematical works and at least one Sanskrit work (the "Sindhind",
which may or may not be Brahmagupta's Brāhmasphuṭasiddhānta).
Diophantus's main work, the "Arithmetica", was translated into Arabic by Qusta ibn Luqa (820–912).
Part of the treatise "al-Fakhri" (by al-Karajī, 953 – ca. 1029) builds on it to some extent. According to Rashed Roshdi, Al-Karajī's contemporary Ibn al-Haytham knew what would later be called Wilson's theorem.
During the 600s, negative numbers were in use in India to represent debts. Diophantus' previous reference was discussed more explicitly by Indian mathematician Brahmagupta, in "Brāhmasphuṭasiddhānta" 628, who used negative numbers to produce the general form quadratic formula that remains in use today. However, in the 12th century in India, Bhaskara gives negative roots for quadratic equations but says the negative value "is in this case not to be taken, for it is inadequate; people do not approve of negative roots."
Usually terms with the highest power (exponent), are written on the left, for example, formula_10 is written to the left of formula_11. When a coefficient is one, it is usually omitted (e.g. formula_12 is written formula_10). Likewise when the exponent (power) is one, (e.g. formula_14 is written formula_15). When the exponent is zero, the result is always 1 (e.g. formula_16 is always rewritten to formula_17). However formula_18, being undefined, should not appear in an expression, and care should be taken in simplifying expressions in which variables may appear in exponents.
The last two of these rules are notable as the earliest attempt to define division by zero, even though they are not compatible with modern number theory (division by zero is undefined for a field).[5]
Brahmagupta () (born , died ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the "Brāhmasphuṭasiddhānta" (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical treatise, and the "Khaṇḍakhādyaka" ("edible bite", dated 665), a more practical text.
By convention, terms with the highest power (exponent), are written on the left, for example, formula_10 is written to the left of formula_11. When a coefficient is one, it is usually omitted (e.g. formula_12 is written formula_10). Likewise when the exponent (power) is one, (e.g. formula_14 is written formula_15), and, when the exponent is zero, the result is always 1 (e.g. formula_16 is written formula_17, since formula_18 is always formula_19).
Mahāvīra (or Mahaviracharya, "Mahavira the Teacher") was a 9th-century Jain mathematician possibly born in or close to the present day city of Mysore, in southern India. He was the author of "Gaṇitasārasan̄graha" (or "Ganita Sara Samgraha", c. 850), which revised the Brāhmasphuṭasiddhānta. He was patronised by the Rashtrakuta king Amoghavarsha. He separated astrology from mathematics. It is the earliest Indian text entirely devoted to mathematics. He expounded on the same subjects on which Aryabhata and Brahmagupta contended, but he expressed them more clearly. His work is a highly syncopated approach to algebra and the emphasis in much of his text is on developing the techniques necessary to solve algebraic problems. He is highly respected among Indian mathematicians, because of his establishment of terminology for concepts such as equilateral, and isosceles triangle; rhombus; circle and semicircle. Mahāvīra's eminence spread in all South India and his books proved inspirational to other mathematicians in Southern India. It was translated into Telugu language by Pavuluri Mallana as "Saar Sangraha Ganitam".
The first known documented use of zero dates to AD 628, and appeared in the "Brāhmasphuṭasiddhānta", the main work of the Indian mathematician Brahmagupta. He treated 0 as a number and discussed operations involving it, including division. By this time (the 7th century) the concept had clearly reached Cambodia as Khmer numerals, and documentation shows the idea later spreading to China and the Islamic world.
Along with Yaʿqūb ibn Ṭāriq and his father he helped translate the Indian astronomical text by Brahmagupta (fl. 7th century), the "Brāhmasphuṭasiddhānta", into Arabic as "Az-Zīj ‛alā Sinī al-‛Arab"., or the "Sindhind". This translation was possibly the vehicle by means of which the Hindu numerals were transmitted from India to Islam.
Another feature of Pitman's shorthand allows most vowels to be omitted in order to speed up the process of writing. As mentioned above, each vowel is written next to the consonant stroke at the beginning, middle or end of the stroke. Pitman's shorthand is designed to be written on lined paper and when a word's first vowel is a "first position" vowel (i.e. it is written at the beginning of the stroke), the whole shorthand outline for the word is written above the paper's ruled line. When it is a second position vowel, the outline is written on the line. And when it is a third position vowel it is written through the line. In this way, the position of the outline indicates that the first vowel can only be one of four possibilities. In most cases, this means that the first and often all the other vowels can be omitted entirely.
The year and day are always written in Arabic numerals, followed by a full stop. The month can be written by its full name or can be abbreviated, or even denoted by Roman or Arabic numerals. Except for the first case (month written by its full name), the month is followed by a full stop. Usually, when the month is written in letters, there is no leading zero before the day. On the other hand, when the month is written in Arabic numerals, a leading zero is common, but not obligatory. Except at the beginning of a sentence, the name of the month always begins with a lower-case letter.
In 628, the Indian mathematician Brahmagupta wrote "Brāhmasphuṭasiddhānta" which includes, among many other things, a study of equations of the form . In particular he considered what is now called Pell's equation, , and found a method for its solution. In Europe this problem was studied by Brouncker, Euler and Lagrange.
Phonetically, it is evident, for example, the predominance of vowel or similar (written "a"), instead of unstressed (written "e"). In Canzés, instead of Milanese nasalization of vowel, there is a velar nasal (written "n") with abbreviation of the vowel. There aren't geminate consonants in words, excepting half-geminate affricate (written "z"), that never change to . The final consonants are always voiceless. Written "v" have a very weak sound, almost semivocalic. There are also vowel allophones as and , and (both written "a") and more open (written "ü", sometimes "i" when variant of ), in addition to basic Western Lombard vowels: (written "a"), (written "é"), (written "è"), (written "i"), (written "ó"), (written "ò"), (written "ö") with (written "ö", sometimes "ü" when variant of ), (written "u") and normal (written "ü"). Vowels and are inverted (for example: "cóo", head; "cuut", whetstone) as to many others Brianzöö and Milanese varieties. Syllables closed by and based on vowel "a", often change it with (written "ò"), that, like other rounded consonants also in other Western Lombard varieties, change to when unstressed. Letter "s" before consonant is usually aspirated. There are many phaenomena of assimilation or adaptation, caused by meeting of two words, especially in crashes of consonants. In the word "culzùn", trousers, you can see the adesinential plural, the use of , not , the conservation of "z", the mutation "cal-" > "còl-" > "cul-".
Brahmagupta's Brāhmasphuṭasiddhānta is the first book that mentions zero as a number, hence Brahmagupta is considered the first to formulate the concept of zero. The current system of the four fundamental operations (addition, subtraction, multiplication and division) based on the Hindu-Arabic number system also first appeared in Brahmasphutasiddhanta. It was also one of the first texts to provide concrete ideas on positive and negative numbers.
The Hellenistic mathematicians Hero of Alexandria and Diophantus[11] as well as Indian mathematicians such as Brahmagupta continued the traditions of Egypt and Babylon, though Diophantus' Arithmetica and Brahmagupta's Brāhmasphuṭasiddhānta are on a higher level.[12] For example, the first complete arithmetic solution (including zero and negative solutions) to quadratic equations was described by Brahmagupta in his book Brahmasphutasiddhanta. Later, Persian and Arabic mathematicians developed algebraic methods to a much higher degree of sophistication. Although Diophantus and the Babylonians used mostly special ad hoc methods to solve equations, Al-Khwarizmi's contribution was fundamental. He solved linear and quadratic equations without algebraic symbolism, negative numbers or zero, thus he had to distinguish several types of equations.[13]
The sum of two positive quantities is positive The sum of two negative quantities is negative The sum of zero and a negative number is negative The sum of zero and a positive number is positive The sum of zero and zero is zero The sum of a positive and a negative is their difference; or, if they are equal, zero In subtraction, the less is to be taken from the greater, positive from positive In subtraction, the less is to be taken from the greater, negative from negative When the greater however, is subtracted from the less, the difference is reversed When positive is to be subtracted from negative, and negative from positive, they must be added together The product of a negative quantity and a positive quantity is negative The product of two negative quantities is positive The product of two positive quantities is positive Positive divided by positive or negative by negative is positive Positive divided by negative is negative. Negative divided by positive is negative Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator A positive or negative number when divided by zero is a fraction with the zero as denominator Zero divided by zero is zero
When names are written in English, the patronymic is not equivalent to an English middle name. When the name is written in English, the patronymic may be omitted with the given name written out in full or abbreviated (Vladimir Putin or V. Putin), both the first name and the patronymic may be written out in full (Vladimir Vladimirovich Putin), both the first name and the patronymic may be abbreviated (V. V. Putin) or the first name may be written out in full with the patronymic abbreviated (Vladimir V. Putin).
The "Zīj", written around 770, was based on a Sanskrit work, thought to be similar to the "Brāhmasphuṭasiddhānta". This work was brought to the court of al-Mansūr from Sindh, reportedly by a Sindhi astronomer named Kankah.
Brāhmasphuṭasiddhānta is one of the first books to provide concrete ideas on positive numbers, negative numbers, and zero. He wrote the following rules:[4]
There are also spreads in which unequal number of options are simultaneously purchased and written. When more options are written than purchased, it is a ratio spread. When more options are purchased than written, it is a backspread.
The "Brāhmasphuṭasiddhānta" of Brahmagupta (598–668) is the earliest known text to treat zero as a number in its own right and to define operations involving zero. The author could not explain division by zero in his texts: his definition can be easily proven to lead to algebraic absurdities. According to Brahmagupta,
The baritone is pitched in concert B, meaning that when no valves are actuated, the instrument will produce partials of the B harmonic series. Music for the baritone horn can be written in either the bass clef or the treble clef. When written in the bass clef, the baritone horn is a non-transposing instrument. However, when written in the treble clef, it is often used as transposing instrument, transposing downward a major ninth from the music as written, so that written middle C for the baritone is concert B below low C (B in scientific pitch notation), with the fingerings thus matching those of the trumpet but sounding an octave lower. It is often used to play parts written for the similarly pitched tenor trombone or euphonium.