Brahmagupta's identity

In algebra, Brahmagupta's identity says that, for given Template:Mvar, the product of two numbers of the form Template:Math is itself a number of that form. In other words, the set of such numbers is closed under multiplication. Specifically:

$$(a^2+n{b}^2)(c^2+n{d}^2)=(ac\pm nbd{)}^2+n(ad\mp bc{)}^2.$$

This identity holds in both the ring of integers and the ring of rational numbers, and more generally in any commutative ring.

Also, it can be viewed as a special case of the following identity:

$$(h{a}^2+k{b}^2)(h{c}^2+k{d}^2)=(hac\pm kbd{)}^2+hk(ad\mp bc{)}^2.$$

History

The identity is a generalization of the so-called Brahmagupta-Fibonacci identity (where Template:Math) 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.

Application to Pell's equation

In its original context, Brahmagupta applied his discovery to the solution of what was later called Pell's equation, namely Template:Math. Using the identity in the form

$$(x_1^2-N{y}_1^2)(x_2^2-N{y}_2^2)=(x_1{x}_2+N{y}_1{y}_2{)}^2-N(x_1{y}_2+x_2{y}_1{)}^2,$$

he was able to "compose" triples Template:Math and Template:Math that were solutions of Template:Math, to generate the new triple

$$(x_1{x}_2+N{y}_1{y}_2,x_1{y}_2+x_2{y}_1,k_1{k}_2).$$

Not only did this give a way to generate infinitely many solutions to Template:Math starting with one solution, but also, by dividing such a composition by Template:Math, integer or "nearly integer" solutions could often be obtained. The general method for solving the Pell equation given by Bhaskara II in 1150, namely the chakravala (cyclic) method, was also based on this identity.

See also

• Brahmagupta polynomials
• Brahmagupta–Fibonacci identity
• Brahmagupta's interpolation formula
• Gauss composition law
• Indian mathematics
• List of Indian mathematicians

External links

• Brahmagupta's identity at PlanetMath
• Brahmagupta Identity on MathWorld
• A Collection of Algebraic Identities Template:Webarchive