Brahmagupta's identity
In algebra, Brahmagupta's identity says that the product of two numbers of the form is itself a number of that form. In other words, the set of such numbers is closed under multiplication. Specifically:
Both (1) and (2) can be verified by expanding each side of the equation. Also, (2) can be obtained from (1), or (1) from (2), by changing b to −b.
This identity holds in both the ring of integers and the ring of rational numbers, and more generally in any commutative ring.
The identity is a generalization of the so-called Fibonacci identity (where n=1) which is actually found in Diophantus' Arithmetica (III, 19). That identity was rediscovered by Brahmagupta (598–668), an Indian mathematician and astronomer, who generalized it and used it in his study of what is now called Pell's equation. His Brahmasphutasiddhanta was translated from Sanskrit into Arabic by Mohammad al-Fazari, and was subsequently translated into Latin in 1126. The identity later appeared in Fibonacci's Book of Squares in 1225.
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