Brahmagupta-Fibonacci Identity
Theorem
Let $a, b, c, d$ be numbers.
Then:
- $\paren {a^2 + b^2} \paren {c^2 + d^2} = \paren {a c + b d}^2 + \paren {a d - b c}^2$
Corollary
- $\paren {a^2 + b^2} \paren {c^2 + d^2} = \paren {a c - b d}^2 + \paren {a d + b c}^2$
General result
Let $a, b, c, d, n$ be numbers.
- $\paren {a^2 + n b^2} \paren {c^2 + n d^2} = \paren {a c + n b d}^2 + n \paren {a d - b c}^2$
Extension
Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$ be integers.
Then:
- $\ds \prod_{j \mathop = 1}^n \paren { {a_j}^2 + {b_j}^2} = c^2 + d^2$
where $c, d \in \Z$.
That is: the product of any number of sums of two squares is also a sum of two squares.
Proof 1
\(\ds \) | \(\) | \(\ds \paren {a c + b d}^2 + \paren {a d - b c}^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\paren {a c}^2 + 2 \paren {a c} \paren {b d} + \paren {b d}^2} + \paren {\paren {a d}^2 - 2 \paren {a d} \paren {b c} + \paren {b c}^2}\) | 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 | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a^2 + b^2} \paren {c^2 + d^2}\) |
$\blacksquare$
Proof 2
From the more general version of Brahmagupta-Fibonacci Identity:
- $\paren {a^2 + n b^2} \paren {c^2 + n d^2} = \paren {a c + n b d}^2 + n \paren {a d - b c}^2$
The result follows by setting $n = 1$.
$\blacksquare$
Proof 3
Lagrange's Identity gives:
- $\ds \paren {\sum_{k \mathop = 1}^n {a_k}^2} \paren {\sum_{k \mathop = 1}^n {b_k}^2} - \paren {\sum_{k \mathop = 1}^n a_k b_k}^2 = \sum_{i \mathop = 1}^{n - 1} \sum_{j \mathop = i + 1}^n \paren {a_i b_j - a_j b_i}^2$
Setting $n = 2$:
\(\ds \paren {\sum_{k \mathop = 1}^2 {a_k}^2} \paren {\sum_{k \mathop = 1}^2 {b_k}^2} - \paren {\sum_{k \mathop = 1}^2 a_k b_k}^2\) | \(=\) | \(\ds \sum_{i \mathop = 1}^1 \sum_{j \mathop = i + 1}^2 \paren {a_i b_j - a_j b_i}^2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren { {a_1}^2 + {a_2}^2} \paren { {b_1}^2 + {b_2}^2} - \paren {a_1 b_1 + a_2 b_2}^2\) | \(=\) | \(\ds \paren {a_1 b_2 - a_2 b_1}^2\) | expanding out summations | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {a^2 + b^2} \paren {c^2 + d^2} - \paren {a c + b d}^2\) | \(=\) | \(\ds \paren {a d - b c}^2\) | renaming $a_1 \to a, a_2 \to b, b_1 \to c, b_2 \to d$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {a^2 + b^2} \paren {c^2 + d^2}\) | \(=\) | \(\ds \paren {a c + b d}^2 + \paren {a d - b c}^2\) |
$\blacksquare$
Proof 4
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 $(1)$ |
$\blacksquare$
Also see
Lagrange's Identity, of which this the case where $n = 2$.
Source of Name
This entry was named for Brahmagupta and Leonardo Fibonacci.
Also known as
This identity is also known as Diophantus's Identity, for Diophantus of Alexandria.
Some sources also give it as as Fibonacci's Identity.
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
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $13$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $13$