Brahmagupta-Fibonacci Identity/Corollary

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Let $a, b, c, d$ be numbers.


$\paren {a^2 + b^2} \paren {c^2 + d^2} = \paren {a c - b d}^2 + \paren {a d + b c}^2$


\(\ds \paren {a^2 + b^2} \paren {c^2 + d^2}\) \(=\) \(\ds \paren {a c + b d}^2 + \paren {a d - b c}^2\) Brahmagupta-Fibonacci Identity
\(\ds \leadsto \ \ \) \(\ds \paren {a^2 + \paren {-b}^2} \paren {c^2 + d^2}\) \(=\) \(\ds \paren {a c + \paren {-b} d}^2 + \paren {a d - \paren {-b} c}^2\) substituting $-b$ for $b$
\(\ds \leadsto \ \ \) \(\ds \paren {a^2 + b^2} \paren {c^2 + d^2}\) \(=\) \(\ds \paren {a c - b d}^2 + \paren {a d + b c}^2\)


Source of Name

This entry was named for Brahmagupta‎ and Leonardo Fibonacci‎.

Historical Note

Both Brahmagupta‎ and Leonardo Fibonacci‎ described what is now known as the Brahmagupta-Fibonacci Identity in their writings:

  • 628: Brahmagupta: Brahmasphutasiddhanta (The Opening of the Universe)
  • 1225: Fibonacci: Liber quadratorum (The Book of Squares)

However, it appeared earlier than either of those in Diophantus of Alexandria's Arithmetica of the third century C.E.