# Brāhmasphuṭasiddhānta

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The main work of Brahmagupta, Brāhmasphuṭasiddhānta ("Correctly Established Doctrine of Brahma"), written c. 628, contains ideas including a good understanding of the mathematical 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 the Brahmagupta’s theorem. The book was written completely in verse rather than using mathematical symbols. Nevertheless, it contained the first clear description of the quadratic formula (which is the solution of the quadratic equation).[1][2]

## Brahmasphuta-siddhantas rules for numbers

Brhmasphuta-siddhanta is one of the first mathematical books to provide concrete ideas on positive numbers, negative numbers, and zero. He wrote the following rules:[3]

• 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
• A positive or negative number when divided by zero is a fraction with the zero as denominator
• 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
• Zero divided by zero is zero

The last of these rules is obsolete, as division by zero is undefined for a field; however it is notable that it was the earliest attempt to define division by 0.[4]

## References

1. ^ Bradley, Michael. The Birth of Mathematics: Ancient Times to 1300, p. 86 (Infobase Publishing 2006).
2. ^ Mackenzie, Dana. The Universe in Zero Words: The Story of Mathematics as Told through Equations, p. 61 (Princeton University Press, 2012).
3. ^ Henry Thomas Colebrooke. Algebra with Arithmetic of Brahmagupta and Bhaskara. London 1817.
4. ^ 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.