Brahmagupta-Fibonacci Identity/Proof 1

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

\(\ds \)

\(\)

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

\(\ds \)

\(=\)

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

Square of Sum, Square of Difference

\(\ds \)

\(=\)

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

multiplying out

\(\ds \)

\(=\)

\(\ds a^2 c^2 + a^2 d^2 + b^2 c^2 + b^2 d^2\)

simplifying

\(\ds \)

\(=\)

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

$\blacksquare$

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.