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Cassini And Catalan Identities
__notoc__ Cassini's identity (sometimes called Simson's identity) and Catalan's identity are mathematical identities for the Fibonacci numbers. Cassini's identity, a special case of Catalan's identity, states that for the ''n''th Fibonacci number, : F_F_ - F_n^2 = (-1)^n. Note here F_0 is taken to be 0, and F_1 is taken to be 1. Catalan's identity generalizes this: :F_n^2 - F_F_ = (-1)^F_r^2. Vajda's identity generalizes this: :F_F_ - F_F_ = (-1)^nF_F_. History Cassini's formula was discovered in 1680 by Giovanni Domenico Cassini, then director of the Paris Observatory, and independently proven by Robert Simson (1753). However Johannes Kepler presumably knew the identity already in 1608. Eugène Charles Catalan found the identity named after him in 1879. The British mathematician Steven Vajda (1901–95) published a book on Fibonacci numbers (''Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications'', 1989) which contains the identity carrying his name. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity (mathematics)
In mathematics, an identity is an equality relating one mathematical expression ''A'' to another mathematical expression ''B'', such that ''A'' and ''B'' (which might contain some variables) produce the same value for all values of the variables within a certain range of validity. In other words, ''A'' = ''B'' is an identity if ''A'' and ''B'' define the same functions, and an identity is an equality between functions that are differently defined. For example, (a+b)^2 = a^2 + 2ab + b^2 and \cos^2\theta + \sin^2\theta =1 are identities. Identities are sometimes indicated by the triple bar symbol instead of , the equals sign. Common identities Algebraic identities Certain identities, such as a+0=a and a+(-a)=0, form the basis of algebra, while other identities, such as (a+b)^2 = a^2 + 2ab +b^2 and a^2 - b^2 = (a+b)(a-b), can be useful in simplifying algebraic expressions and expanding them. Trigonometric identities Geometrically, trigonometri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
