Brahmagupta-Fibonacci Identity/Proof 4

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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

Let $z_1 = a - b i, z_2 = c + d i$ be complex numbers.

Let $\cmod z$ denote the complex modulus of a given complex number $z \in \C$.


By definition of complex multiplication:

$(1): \quad z_1 z_2 = \paren {a c + b d} + \paren {a d - b c} i$


Then:

\(\ds \cmod {z_1 z_2}\) \(=\) \(\ds \cmod {z_1} \cmod{z_2}\) Complex Modulus of Product of Complex Numbers
\(\ds \leadsto \ \ \) \(\ds \cmod {\paren {a - b i} \paren {c + d i} }^2\) \(=\) \(\ds \cmod {a - b i}^2 \cmod {c + d i}^2\) $z_1 = a - b i, z_2 = c + d i$
\(\ds \leadsto \ \ \) \(\ds \paren {a c + b d}^2 + \paren {a d - b c}^2\) \(=\) \(\ds \paren {a^2 + b^2} \paren {c^2 + d^2}\) Definition of Complex Modulus, and from $(1)$

$\blacksquare$


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.


Sources

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