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
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory