The ''Brāhma-sphuṭa-siddhānta'' ("Correctly Established

Doctrine Doctrine (from , meaning 'teaching, instruction') is a codification (law), codification of beliefs or a body of teacher, teachings or instructions, taught principles or positions, as the essence of teachings in a given branch of knowledge or in a ...

Brahma Brahma (, ) is a Hindu god, referred to as "the Creator" within the Trimurti, the triple deity, trinity of Para Brahman, supreme divinity that includes Vishnu and Shiva.Jan Gonda (1969)The Hindu Trinity, Anthropos, Bd 63/64, H 1/2, pp. 212– ...

", abbreviated BSS) is a main work of

Brahmagupta Brahmagupta ( – ) was an Indian Indian mathematics, mathematician and Indian astronomy, astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established Siddhanta, do ...

, written c. 628. This text of

mathematical astronomy Theoretical astronomy is the use of analytical and computational models based on principles from physics and chemistry to describe and explain astronomical objects and astronomical phenomena. Theorists in astronomy endeavor to create theoretica ...

contains significant

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

content, including the first good understanding of the role of

zero 0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...

, rules for manipulating both negative and positive numbers, a method for computing

square root In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...

s, methods of solving

linear In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x) ...

and

quadratic equation In mathematics, a quadratic equation () is an equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where the variable (mathematics), variable represents an unknown number, and , , and represent known numbers, where . (If and ...

s, and rules for summing

series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...

Brahmagupta's identity In algebra, Brahmagupta's identity says that, for given n, the product of two numbers of the form a^2+nb^2 is itself a number of that form. In other words, the set of such numbers is closed under multiplication. Specifically: :\begin \left(a^2 + ...

, and Brahmagupta theorem. The book was written completely in verse and does not contain any kind of mathematical notation. Nevertheless, it contained the first clear description of the

quadratic formula In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation. Other ways of solving quadratic equations, such as completing the square, yield the same solutions. Given a general quadr ...

(the solution of the quadratic equation).Bradley, Michael. ''The Birth of Mathematics: Ancient Times to 1300'', p. 86 (Infobase Publishing 2006).

Positive and negative numbers

''Brāhmasphuṭasiddhānta'' is one of the first books to provide concrete ideas on

positive number In mathematics, the sign of a real number is its property of being either positive, negative, or 0. Depending on local conventions, zero may be considered as having its own unique sign, having no sign, or having both positive and negative sign. ...

negative number In mathematics, a negative number is the opposite (mathematics), opposite of a positive real number. Equivalently, a negative number is a real number that is inequality (mathematics), less than 0, zero. Negative numbers are often used to represe ...

s, and zero. For example, it notes that the sum of a positive number and a negative number is their difference or, if they are equal, zero; that subtracting a negative number is equivalent to adding a positive number; that the product of two negative numbers is positive. Some of the notions of fractions differ from the modern

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...

system. For example, Brahmagupta allows

division by zero In mathematics, division by zero, division (mathematics), division where the divisor (denominator) is 0, zero, is a unique and problematic special case. Using fraction notation, the general example can be written as \tfrac a0, where a is the di ...

resulting in a fraction with a in the denominator, and defines . In modern mathematics, division by zero is undefined for any field.

Influence

Ashadhara, the son of Rihluka, wrote ''Graha-jnana'' with tables based on ''Brahma-sphuta-siddhanta'' in 1132. This work is also known by the names ''Graha-ganita'', ''Brahma-tulyanayana'', ''Bhaumadi-panchagraha-nayana'', ''Kshanika-grahanayana'', or simply ''Ashadhara''. Harihara wrote an extended version of the ''Graha-jnana'' around 1575 CE.

References

External links

''Brahmasphutasiddhanta''
at GRETIL (mathematical chapters: 12, 18-20, 21.17-23) * {{DEFAULTSORT:Brahmasphutasiddhanta Indian mathematics 7th-century Sanskrit literature 7th-century manuscripts History of algebra Treatises Indian astronomy texts Brahmagupta