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algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, the Brahmagupta–Fibonacci identity expresses the product of two sums of two squares as a sum of two squares in two different ways. Hence the set of all sums of two squares is closed under multiplication. Specifically, the identity says :\begin \left(a^2 + b^2\right)\left(c^2 + d^2\right) & = \left(ac-bd\right)^2 + \left(ad+bc\right)^2 & & (1) \\ & = \left(ac+bd\right)^2 + \left(ad-bc\right)^2. & & (2) \end For example, :(1^2 + 4^2)(2^2 + 7^2) = 26^2 + 15^2 = 30^2 + 1^2. The identity is also known as the Diophantus identity, Daniel Shanks, Solved and unsolved problems in number theory, p.209, American Mathematical Society, Fourth edition 1993. as it was first proved by Diophantus of Alexandria. It is a special case of Euler's four-square identity, and also of Lagrange's identity.
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 ...
proved and used a more general Brahmagupta identity, stating :\begin \left(a^2 + nb^2\right)\left(c^2 + nd^2\right) & = \left(ac-nbd\right)^2 + n\left(ad+bc\right)^2 & & (3) \\ & = \left(ac+nbd\right)^2 + n\left(ad-bc\right)^2. & & (4) \end This shows that, for any fixed ''A'', the set of all numbers of the form ''x''2 + ''Ay''2 is closed under multiplication. These identities hold for all
integers An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
, as well as all
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 ...
s; more generally, they are true in any
commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
. All four forms of the identity 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'', and likewise with (3) and (4).


History

The identity first appeared in
Diophantus Diophantus of Alexandria () (; ) was a Greek mathematician who was the author of the '' Arithmetica'' in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through algebraic equations. Although Jose ...
' ''
Arithmetica Diophantus of Alexandria () (; ) was a Greek mathematics, Greek mathematician who was the author of the ''Arithmetica'' in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through algebraic equations ...
'' (III, 19), of the third century A.D. It was rediscovered by Brahmagupta (598–668), an
Indian mathematician Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta, ...
and
astronomer An astronomer is a scientist in the field of astronomy who focuses on a specific question or field outside the scope of Earth. Astronomers observe astronomical objects, such as stars, planets, natural satellite, moons, comets and galaxy, galax ...
, who generalized it to
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 used it in his study of what is now called
Pell's equation Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive Square number, nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian ...
. His '' Brahmasphutasiddhanta'' was translated from
Sanskrit Sanskrit (; stem form ; nominal singular , ,) is a classical language belonging to the Indo-Aryan languages, Indo-Aryan branch of the Indo-European languages. It arose in northwest South Asia after its predecessor languages had Trans-cultural ...
into
Arabic Arabic (, , or , ) is a Central Semitic languages, Central Semitic language of the Afroasiatic languages, Afroasiatic language family spoken primarily in the Arab world. The International Organization for Standardization (ISO) assigns lang ...
by Mohammad al-Fazari, and was subsequently translated into
Latin Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
in 1126. The identity was introduced in western Europe in 1225 by
Fibonacci Leonardo Bonacci ( – ), commonly known as Fibonacci, was an Italians, Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages". The name he is commonly called, ''Fibonacci ...
, in '' The Book of Squares'', and, therefore, the identity has been often attributed to him.


Related identities

Analogous identities are Euler's four-square related to
quaternions In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quaternion ...
, and Degen's eight-square derived from the
octonions In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have ...
which has connections to
Bott periodicity In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by , which proved to be of foundational significance for much further research, in particular in K-theory of stable compl ...
. There is also Pfister's sixteen-square identity, though it is no longer bilinear. These identities are strongly related with Hurwitz's classification of
composition algebra In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies :N(xy) = N(x)N(y) for all and in . A composition algebra includes an involution ...
s. The Brahmagupta–Fibonacci identity is a special form of Lagrange's identity, which is itself a special form of Binet–Cauchy identity, in turn a special form of the
Cauchy–Binet formula In mathematics, specifically linear algebra, the Cauchy–Binet formula, named after Augustin-Louis Cauchy and Jacques Philippe Marie Binet, is an identity for the determinant of the product of two rectangular matrices of transpose shapes (so th ...
for matrix determinants.


Multiplication of complex numbers

If ''a'', ''b'', ''c'', and ''d'' are
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s, the Brahmagupta–Fibonacci identity is equivalent to the multiplicative property for absolute values of
complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
: : , a+bi , \cdot , c+di , = , (a+bi)(c+di) , . This can be seen as follows: expanding the right side and squaring both sides, the multiplication property is equivalent to : , a+bi , ^2 \cdot , c+di , ^2 = , (ac-bd)+i(ad+bc) , ^2, and by the definition of absolute value this is in turn equivalent to : (a^2+b^2)\cdot (c^2+d^2)= (ac-bd)^2+(ad+bc)^2. An equivalent calculation in the case that the variables ''a'', ''b'', ''c'', and ''d'' are
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 ...
s shows the identity may be interpreted as the statement that the
norm Norm, the Norm or NORM may refer to: In academic disciplines * Normativity, phenomenon of designating things as good or bad * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy), a standard in normative e ...
in the field Q(''i'') is multiplicative: the norm is given by : N(a+bi) = a^2 + b^2, and the multiplicativity calculation is the same as the preceding one.


Application to Pell's equation

In its original context, Brahmagupta applied his discovery of this identity to the solution of
Pell's equation Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive Square number, nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian ...
''x''2 − ''Ay''2 = 1. Using the identity in the more general form :(x_1^2 - Ay_1^2)(x_2^2 - Ay_2^2) = (x_1x_2 + Ay_1y_2)^2 - A(x_1y_2 + x_2y_1)^2, he was able to "compose" triples (''x''1, ''y''1, ''k''1) and (''x''2, ''y''2, ''k''2) that were solutions of ''x''2 − ''Ay''2 = ''k'', to generate the new triple :(x_1x_2 + Ay_1y_2 \,,\, x_1y_2 + x_2y_1 \,,\, k_1k_2). Not only did this give a way to generate infinitely many solutions to ''x''2 − ''Ay''2 = 1 starting with one solution, but also, by dividing such a composition by ''k''1''k''2, 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.


Writing integers as a sum of two squares

When used in conjunction with one of Fermat's theorems, the Brahmagupta–Fibonacci identity proves that the product of a square and any number of primes of the form 4''n'' + 1 is a sum of two squares.


See also

* Brahmagupta matrix *
Indian mathematics Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta, ...
* Brahmagupta polynomials *
List of Indian mathematicians Indian mathematicians have made a number of contributions to mathematics that have significantly influenced scientists and mathematicians in the modern era. One of such works is Hindu numeral system which is predominantly used today and is likely ...
* List of Italian mathematicians *
Sum of two squares theorem In number theory, the sum of two squares theorem relates the prime decomposition of any integer to whether it can be written as a sum of two Square number, squares, such that for some integers , . An integer greater than one can be written as a ...


Notes


References

* *


External links


Brahmagupta's identity
at
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Brahmagupta Identity
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A Collection of Algebraic Identities
{{DEFAULTSORT:Brahmagupta-Fibonacci identity Brahmagupta Algebraic identities Squares in number theory