Brahmagupta-Fibonacci Identity/Corollary

Theorem

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

Then:

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

Proof

\(\ds \paren {a^2 + b^2} \paren {c^2 + d^2}\)

\(=\)

\(\ds \paren {a c + b d}^2 + \paren {a d - b c}^2\)

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 $b$

\(\ds \leadsto \ \ \)

\(\ds \paren {a^2 + b^2} \paren {c^2 + d^2}\)

\(=\)

\(\ds \paren {a c - b d}^2 + \paren {a d + b c}^2\)

$\blacksquare$

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.