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.