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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.

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 (the solution of the quadratic equation).^[2]^[3]

Brāhmasphuṭasiddhānta rules for numbers

Brahmasphutasiddhanta was the very first book to introduce zero as a number, modern way of conducting the four fundamental operations (addition, subtraction, Multiplication, and division), negative numbers, modern way of solving the quadratic equation and much more new mathematical ideas.

Brāhmasphuṭasiddhānta is the first books to provide concrete ideas on positive numbers, negative numbers, and zero. He wrote the following rules:^[4]

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

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]

References

^ "Brahmagupta | Indian astronomer". Encyclopedia Britannica.
^ Bradley, Michael. The Birth of Mathematics: Ancient Times to 1300, p. 86 (Infobase Publishing 2006).
^ Mackenzie, Dana. The Universe in Zero Words: The Story of Mathematics as Told through Equations, p. 61 (Princeton University Press, 2012).
^ Henry Thomas Colebrooke. Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara, London 1817, p. 339 (online)
^ Kaplan, Robert (1999). The nothing that is: A natural history of zero. New York: Oxford University Press. pp. 68–75. ISBN 0-19-514237-3. The book has Twenty arithmetic operations (Parikarma) 1) Samkalitha – Addition 2) Vyavakalitha – Subtraction 3) Ganana – Multiplication 4) Bhagabara – Division 5) Varga – Squaring 6) Varga Mula – Square root 7) Ghana – Cube 8) Ghana mula – Cube root 9 to 13) - Pancha jathi- - Five rules of reduction relating to the five standard form of fractions 14) Trirasika – Rule of three 15) Vyasta Trirasika – Inverse rule of three 16) Pancha rasika – Rule of five 17) Sapta rasika – Rule of seven 18) Navarasika – Rule of nine 19) Eka dasa rasika – Rule of eleven 20) Bhanda – prathi bhanda - Barter and exchange Eight determinations (Vyavahara) 1) Misraka – Mixture 2) Sredhi – Progression or series 3) Ksetra – Plane figure 4) Khata – Excavations 5) Citi – Stock 6) Krakacika – Saw 7) Rasi – Mound 8) Chaya – Shadow

External links

Brahmasphutasiddhanta at GRETIL (mathematical chapters: 12, 18-20, 21.17-23)
O'Connor, John J.; Robertson, Edmund F., "Brahmagupta", MacTutor History of Mathematics Archive, University of St Andrews

[1] "Brahmagupta | Indian astronomer". Encyclopedia Britannica.

[Bradley-2] Bradley, Michael. The Birth of Mathematics: Ancient Times to 1300, p. 86 (Infobase Publishing 2006).

[3] Mackenzie, Dana. The Universe in Zero Words: The Story of Mathematics as Told through Equations, p. 61 (Princeton University Press, 2012).

[4] Henry Thomas Colebrooke. Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara, London 1817, p. 339 (online)

[Kaplan-5] Kaplan, Robert (1999). The nothing that is: A natural history of zero. New York: Oxford University Press. pp. 68–75. ISBN 0-19-514237-3. The book has Twenty arithmetic operations (Parikarma) 1) Samkalitha – Addition 2) Vyavakalitha – Subtraction 3) Ganana – Multiplication 4) Bhagabara – Division 5) Varga – Squaring 6) Varga Mula – Square root 7) Ghana – Cube 8) Ghana mula – Cube root 9 to 13) - Pancha jathi- - Five rules of reduction relating to the five standard form of fractions 14) Trirasika – Rule of three 15) Vyasta Trirasika – Inverse rule of three 16) Pancha rasika – Rule of five 17) Sapta rasika – Rule of seven 18) Navarasika – Rule of nine 19) Eka dasa rasika – Rule of eleven 20) Bhanda – prathi bhanda - Barter and exchange Eight determinations (Vyavahara) 1) Misraka – Mixture 2) Sredhi – Progression or series 3) Ksetra – Plane figure 4) Khata – Excavations 5) Citi – Stock 6) Krakacika – Saw 7) Rasi – Mound 8) Chaya – Shadow

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