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Fibonacci Identity

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Since

|(a+ib)(c+id)|=|a+ib||c+di|
(1)
|(ac-bd)+i(bc+ad)|=sqrt(a^2+b^2)sqrt(c^2+d^2),
(2)

it follows that

(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(bc+ad)^2
(3)
=e^2+f^2.
(4)

This identity implies the two-dimensional Cauchy's inequality.

See also

Cauchy's Inequality, Euler Four-Square Identity, Lebesgue Identity

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References

Petkovšek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A K Peters, p. 9, 1996. http://www.cis.upenn.edu/~wilf/AeqB.html.

Referenced on Wolfram|Alpha

Fibonacci Identity

Cite this as:

Weisstein, Eric W. "Fibonacci Identity." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FibonacciIdentity.html

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